Grade 4 Mathematics Curriculum Guide

Mathematics Grade Four

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Table of Contents Acknowledgements…………………………………………….…………….… Foreword………………………………………………………..………………. Background ………………………………………………………..…………… Introduction Purpose of the Document………………………………………………. Beliefs About Students and Mathematics Learning……………………. Affective Domain………………………………………………………. Early Childhood………………………………………………………… Goal for Students……………………………………………………….. Conceptual Framework for K–9 Mathematives Mathematical Processes………………………………………………... Nature of Mathematics…………………………………………………. Strands………………………………………………………………….. Outcomes and Achievement Indicators………………………………... Summary……………………………………………………………….. Instructional Focus Planning for Instruction………………………………………………… Resources……………………………………………………………….. General and Specific Outcomes........................................................................... General and Specific Outcomes by Strand Grades 3 - 5……………….…………………………………………… Outcomes with Achievement Indicators Unit: Numeration Unit…………………………………….……………. Unit: Addition and Subtraction………………………………………… Unit: Patterns in Mathematics………..………………………………..... Unit: Data Relationships…………………………………………….….. Unit: 2-D Geometry …………………………………………….……… Unit: Multiplication and Division Facts………………..………………. Unit: Fractions and Decimals…………………………………………... Unit: Measurement…….……………………………………………….. Unit: Multiplying Multi-Digit Numbers………………..………………. Unit: Dividing Multi-digit Numbers………………..…………………... Unit: 3-D Geometry …………………………………………………….

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Appendix A: Outcomes with Achievement Indicators (Strand)…………….

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Appendix B: References………………………….………………………...

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Acknowledgements The Department of Education would like to thank Western and Northern Canadian Protocol (WNCP) for Collaboration in Education. The Common Curriculum Framework for K-9 Mathematics – May 2006 and The Common Curriculum Framework for Grades 10-12 – January 2008. Reproduced (and/or adapted) by permission. All rights reserved. We would also like to thank the provincial Grade 4 Mathematics curriculum committee, the Alberta Department of Education, the New Brunswick Department of Education, and the following people for their contribution: Trudy Porter, Program Development Specialist - Mathematics, Division of Program Development, Department of Education Elizabeth Dubeau, Teacher / Assistant Principal -Woodland Elementary, Dildo/ New Harbour Dina Healey, Teacher – Lakeside Academy, Buchans Gail Keats, Teacher – Cowan Heights Elementary, St. John’s Gillian Normore, Teacher – Paradise Elementary, Paradise John Power, Numeracy Support Teacher – Eastern School District Sharon Power, Numeracy Support Teacher – Eastern School District Tracy Templeman, Numeracy Support Teacher – Nova Central School District Patricia Maxwell – Program Development Specialist Mathematics, Division of Program Development, Department of Education

Every effort has been made to acknowledge all sources of contribution to the development of this document. Any omissions or errors will be amended in final print.

K-9 Mathematics Curriculum NL

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Foreword The Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics released in 2006 by the National Council of Teachers in Mathematics (NCTM) and the WNCP Common Curriculum Frameworks for Mathematics K – 9 (WNCP, 2006), assists provinces in developing a mathematics curriculum framework. Newfoundland and Labrador has used this curriculum framework to direct the development of this curriculum guide. This curriculum guide is intended to provide teachers with the overview of the outcomes framework for mathematics education. It also includes suggestions to assist teachers in designing learning experiences and assessment tasks.


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BACKGROUND The province of Newfoundland and Labrador commissioned a review of mathematics curriculum in the summer of 2007. This review resulted in a number of significant recommendations. In March of 2008 it was announced that this province accepted all recommendations. The first and perhaps most significant of the recommendations were as follows: That the WNCP Common Curriculum Frameworks for Mathematics K – 9 and Mathematics 10 – 12 (WNCP, 2006 and 2008) be adopted as the basis for the K – 12 mathematics curriculum in this province. That implementation commence with Grades K, 1, 4, 7 in September 2008, followed by in Grades 2, 5, 8 in 2009 and Grades 3, 6, 9 in 2010. That textbooks and other resources specifically designed to match the WNCP frameworks be adopted as an integral part of the proposed program change. That implementation be accompanied by an introductory professional development program designed to introduce the curriculum to all mathematics teachers at the appropriate grade levels prior to the first year of implementation. As recommended, implementation at grades K, 1, 4 and 7 begins in September 2008. All teachers assigned to those grades in the spring of 2008 received a two-day professional development opportunity related to the new curriculum and resources. Newly hired teachers will have the same opportunity in September. All teachers will receive follow-up professional development in late fall.


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INTRODUCTION

PURPOSE OF THE DOCUMENT The Mathematics Curriculum Guides for Newfoundland and Labrador have been derived from The Common Curriculum Framework for K-9 Mathematics: Western and Northern Canadian Protocol, May 2006 (the Common Curriculum Framework). These guides incorporate the conceptual framework for Kindergarten to Grade 9 Mathematics and the general outcomes, specific outcomes and achievement indicators established in the common curriculum framework. They also include suggestions for teaching and learning, suggested assessment strategies, and an identification of the associated resource match between the curriculum and authorized as well as recommended resource materials.

BELIEFS ABOUT STUDENTS AND MATHEMATICS LEARNING Mathematical understanding is fostered when students build on their own experiences and prior knowledge.

Students are curious, active learners with individual interests, abilities and needs. They come to classrooms with varying knowledge, life experiences and backgrounds. A key component in successfully developing numeracy is making connections to these backgrounds and experiences. Students learn by attaching meaning to what they do, and they need to construct their own meaning of mathematics. This meaning is best developed when learners encounter mathematical experiences that proceed from the simple to the complex and from the concrete to the abstract. Through the use of manipulatives and a variety of pedagogical approaches, teachers can address the diverse learning styles, cultural backgrounds and developmental stages of students, and enhance within them the formation of sound, transferable mathematical understandings. At all levels, students benefit from working with a variety of materials, tools and contexts when constructing meaning about new mathematical ideas. Meaningful student discussions provide essential links among concrete, pictorial and symbolic representations of mathematical concepts. The learning environment should value and respect the diversity of students’ experiences and ways of thinking, so that students are comfortable taking intellectual risks, asking questions and posing conjectures. Students need to explore problem-solving situations in order to develop personal strategies and become mathematically literate. They must realize that it is acceptable to solve problems in a variety of ways and that a variety of solutions may be acceptable.

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AFFECTIVE DOMAIN

To experience success, students must be taught to set achievable goals and assess themselves as they work toward these goals.

A positive attitude is an important aspect of the affective domain and has a profound impact on learning. Environments that create a sense of belonging, encourage risk taking and provide opportunities for success help develop and maintain positive attitudes and self-confidence within students. Students with positive attitudes toward learning mathematics are likely to be motivated and prepared to learn, participate willingly in classroom activities, persist in challenging situations and engage in reflective practices. Teachers, students and parents need to recognize the relationship between the affective and cognitive domains, and attempt to nurture those aspects of the affective domain that contribute to positive attitudes. To experience success, students must be taught to set achievable goals and assess themselves as they work toward these goals. Striving toward success and becoming autonomous and responsible learners are ongoing, reflective processes that involve revisiting the setting and assessing of personal goals.

EARLY CHILDHOOD

Curiosity about mathematics is fostered when children are actively engaged in their environment.

Young children are naturally curious and develop a variety of mathematical ideas before they enter Kindergarten. Children make sense of their environment through observations and interactions at home, in daycares, in preschools and in the community. Mathematics learning is embedded in everyday activities, such as playing, reading, beading, baking, storytelling and helping around the home. Activities can contribute to the development of number and spatial sense in children. Curiosity about mathematics is fostered when children are engaged in, and talking about, such activities as comparing quantities, searching for patterns, sorting objects, ordering objects, creating designs and building with blocks. Positive early experiences in mathematics are as critical to child development as are early literacy experiences.


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GOALS FOR STUDENTS

Mathematics education must prepare students to use mathematics confidently to solve problems.

The main goals of mathematics education are to prepare students to: • use mathematics confidently to solve problems • communicate and reason mathematically • appreciate and value mathematics • make connections between mathematics and its applications • commit themselves to lifelong learning • become mathematically literate adults, using mathematics to contribute to society. Students who have met these goals will: gain understanding and appreciation of the contributions of mathematics as a science, philosophy and art • exhibit a positive attitude toward mathematics • engage and persevere in mathematical tasks and projects • contribute to mathematical discussions • take risks in performing mathematical tasks • exhibit curiosity. •

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CONCEPTUAL FRAMEWORK FOR K–9 MATHEMATICS The chart below provides an overview of how mathematical processes and the nature of mathematics influence learning outcomes.

MATHEMATICAL PROCESSES There are critical components that students must encounter in a • • •

• • • •

Communicatio mathematics program in order to achieve the goals of mathematics n [C] education and embrace lifelong learning in mathematics. Connections Students are expected to: [CN] • communicate in order to learn and express their understanding Mental • connect mathematical ideas to other concepts in mathematics, to Mathematics everyday experiences and to other disciplines and Estimation [ME] • demonstrate fluency with mental mathematics and estimation Problem • develop and apply new mathematical knowledge through problem Solving [PS] solving Reasoning [R] • develop mathematical reasoning Technology • select and use technologies as tools for learning and for solving [T] problems Visualization • develop visualization skills to assist in processing information, making [V] connections and solving problems. The program of studies incorporates these seven interrelated mathematical processes that are intended to permeate teaching and learning.


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Communication [C]

Students must be able to communicate mathematical ideas in a variety of ways and contexts.

Students need opportunities to read about, represent, view, write about, listen to and discuss mathematical ideas. These opportunities allow students to create links between their own language and ideas, and the formal language and symbols of mathematics. Communication is important in clarifying, reinforcing and modifying ideas, attitudes and beliefs about mathematics. Students should be encouraged to use a variety of forms of communication while learning mathematics. Students also need to communicate their learning using mathematical terminology. Communication helps students make connections among concrete, pictorial, symbolic, oral, written and mental representations of mathematical ideas.

Connections [CN]

Through connections, students begin to view mathematics as useful and relevant.

Contextualization and making connections to the experiences of learners are powerful processes in developing mathematical understanding. This can be particularly true for First Nations, Métis and Inuit learners. When mathematical ideas are connected to each other or to real-world phenomena, students begin to view mathematics as useful, relevant and integrated. Learning mathematics within contexts and making connections relevant to learners can validate past experiences and increase student willingness to participate and be actively engaged. The brain is constantly looking for and making connections. “Because the learner is constantly searching for connections on many levels, educators need to orchestrate the experiences from which learners extract understanding.… Brain research establishes and confirms that multiple complex and concrete experiences are essential for meaningful learning and teaching” (Caine and Caine, 1991, p. 5).

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Mental Mathematics and Estimation [ME] Mental mathematics is a combination of cognitive strategies that enhance flexible thinking and number sense. It is calculating mentally without the use of external memory aids. Mental mathematics enables students to determine answers without paper and pencil. It improves computational fluency by developing efficiency, accuracy and flexibility.

Mental mathematics and estimation are fundamental components of number sense.

“Even more important than performing computational procedures or using calculators is the greater facility that students need—more than ever before—with estimation and mental math” (National Council of Teachers of Mathematics, May 2005). Students proficient with mental mathematics “become liberated from calculator dependence, build confidence in doing mathematics, become more flexible thinkers and are more able to use multiple approaches to problem solving” (Rubenstein, 2001, p. 442). Mental mathematics “provides the cornerstone for all estimation processes, offering a variety of alternative algorithms and nonstandard techniques for finding answers” (Hope, 1988, p. v). Estimation is used for determining approximate values or quantities or for determining the reasonableness of calculated values. It often uses benchmarks or referents. Students need to know when to estimate, how to estimate and what strategy to use. Estimation assists individuals in making mathematical judgements and in developing useful, efficient strategies for dealing with situations in daily life.


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Problem Solving [PS]

Learning through problem solving should be the focus of mathematics at all grade levels.

Learning through problem solving should be the focus of mathematics at all grade levels. When students encounter new situations and respond to questions of the type How would you …? or How could you …?, the problem-solving approach is being modelled. Students develop their own problem-solving strategies by listening to, discussing and trying different strategies. A problem-solving activity must ask students to determine a way to get from what is known to what is sought. If students have already been given ways to solve the problem, it is not a problem, but practice. A true problem requires students to use prior learnings in new ways and contexts. Problem solving requires and builds depth of conceptual understanding and student engagement. Problem solving is a powerful teaching tool that fosters multiple, creative and innovative solutions. Creating an environment where students openly look for, and engage in, finding a variety of strategies for solving problems empowers students to explore alternatives and develops confident, cognitive mathematical risk takers.

Reasoning [R] Mathematical reasoning helps students think logically and make sense of mathematics. Students need to develop confidence in their abilities to reason and justify their mathematical thinking. High-order questions challenge students to think and develop a sense of wonder about mathematics.

Mathematical reasoning helps students think logically and make sense of mathematics.

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Mathematical experiences in and out of the classroom provide opportunities for students to develop their ability to reason. Students can explore and record results, analyze observations, make and test generalizations from patterns, and reach new conclusions by building upon what is already known or assumed to be true. Reasoning skills allow students to use a logical process to analyze a problem, reach a conclusion and justify or defend that conclusion.


Technology [T] Technology contributes to the learning of a wide range of mathematical outcomes and enables students to explore and create patterns, examine relationships, test conjectures and solve problems.

Technology Calculators and computers can be used to: contributes to the • explore and demonstrate mathematical relationships and patterns learning of a wide • organize and display data range of • extrapolate and interpolate mathematical • assist with calculation procedures as part of solving problems outcomes and • decrease the time spent on computations when other mathematical enables students learning is the focus to explore and • reinforce the learning of basic facts create patterns, • develop personal procedures for mathematical operations examine • create geometric patterns relationships, test • simulate situations conjectures and • develop number sense. solve problems. Technology contributes to a learning environment in which the growing curiosity of students can lead to rich mathematical discoveries at all grade levels.

Visualization [V] Visualization “involves thinking in pictures and images, and the ability to perceive, transform and recreate different aspects of the visual-spatial world” (Armstrong, 1993, p. 10). The use of visualization in the study of mathematics provides students with opportunities to understand mathematical concepts and make connections among them.

Visualization is fostered through the use of concrete materials, technology and a variety of visual representations.

Visual images and visual reasoning are important components of number, spatial and measurement sense. Number visualization occurs when students create mental representations of numbers. Being able to create, interpret and describe a visual representation is part of spatial sense and spatial reasoning. Spatial visualization and reasoning enable students to describe the relationships among and between 3-D objects and 2-D shapes. Measurement visualization goes beyond the acquisition of specific measurement skills. Measurement sense includes the ability to determine when to measure, when to estimate and which estimation strategies to use (Shaw and Cliatt, 1989).


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NATURE OF MATHEMATICS • • • • • • •

Change Constancy Number Sense Patterns Relationships Spatial Sense Uncertainty

Mathematics is one way of trying to understand, interpret and describe our world. There are a number of components that define the nature of mathematics and these are woven throughout this program of studies. The components are change, constancy, number sense, patterns, relationships, spatial sense and uncertainty.

Change It is important for students to understand that mathematics is dynamic and not static. As a result, recognizing change is a key component in understanding and developing mathematics.

Change is an integral part of mathematics and the learning of mathematics.

Within mathematics, students encounter conditions of change and are required to search for explanations of that change. To make predictions, students need to describe and quantify their observations, look for patterns, and describe those quantities that remain fixed and those that change. For example, the sequence 4, 6, 8, 10, 12, … can be described as: • the number of a specific colour of beads in each row of a beaded design • skip counting by 2s, starting from 4 • an arithmetic sequence, with first term 4 and a common difference of 2 • a linear function with a discrete domain (Steen, 1990, p. 184).

Constancy

Constancy is described by the terms stability, conservation, equilibrium, steady state and symmetry.

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Different aspects of constancy are described by the terms stability, conservation, equilibrium, steady state and symmetry (AAAS–Benchmarks, 1993, p. 270). Many important properties in mathematics and science relate to properties that do not change when outside conditions change. Examples of constancy include the following: • The ratio of the circumference of a teepee to its diameter is the same regardless of the length of the teepee poles. • The sum of the interior angles of any triangle is 180°. • The theoretical probability of flipping a coin and getting heads is 0.5. Some problems in mathematics require students to focus on properties that remain constant. The recognition of constancy enables students to solve problems involving constant rates of change, lines with constant slope, direct variation situations or the angle sums of polygons.


Number Sense Number sense, which can be thought of as intuition about numbers, is the most important foundation of numeracy (British Columbia Ministry of Education, 2000, p. 146).

An intuition about number is the most important foundation of a numerate child.

A true sense of number goes well beyond the skills of simply counting, memorizing facts and the situational rote use of algorithms. Mastery of number facts is expected to be attained by students as they develop their number sense. This mastery allows for facility with more complex computations but should not be attained at the expense of an understanding of number. Number sense develops when students connect numbers to their own reallife experiences and when students use benchmarks and referents. This results in students who are computationally fluent and flexible with numbers and who have intuition about numbers. The evolving number sense typically comes as a by-product of learning rather than through direct instruction. However, number sense can be developed by providing rich mathematical tasks that allow students to make connections to their own experiences and their previous learning.

Patterns Mathematics is about recognizing, describing and working with numerical and non-numerical patterns. Patterns exist in all strands of this program of studies.

Mathematics is about recognizing, describing and working with numerical and non-numerical patterns.

Working with patterns enables students to make connections within and beyond mathematics. These skills contribute to students’ interaction with, and understanding of, their environment. Patterns may be represented in concrete, visual or symbolic form. Students should develop fluency in moving from one representation to another. Students must learn to recognize, extend, create and use mathematical patterns. Patterns allow students to make predictions and justify their reasoning when solving routine and nonroutine problems. Learning to work with patterns in the early grades helps students develop algebraic thinking, which is foundational for working with more abstract mathematics in higher grades.


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Relationships Mathematics is used to describe and explain relationships.

Mathematics is one way to describe interconnectedness in a holistic worldview. Mathematics is used to describe and explain relationships. As part of the study of mathematics, students look for relationships among numbers, sets, shapes, objects and concepts. The search for possible relationships involves collecting and analyzing data and describing relationships visually, symbolically, orally or in written form.

Spatial Sense Spatial sense involves visualization, mental imagery and spatial reasoning. These skills are central to the understanding of mathematics.

Spatial sense offers a way to interpret and reflect on the physical environment.

Spatial sense is developed through a variety of experiences and interactions within the environment. The development of spatial sense enables students to solve problems involving 3-D objects and 2-D shapes and to interpret and reflect on the physical environment and its 3-D or 2D representations. Some problems involve attaching numerals and appropriate units (measurement) to dimensions of shapes and objects. Spatial sense allows students to make predictions about the results of changing these dimensions; e.g., doubling the length of the side of a square increases the area by a factor of four. Ultimately, spatial sense enables students to communicate about shapes and objects and to create their own representations.

Uncertainty In mathematics, interpretations of data and the predictions made from data may lack certainty. Events and experiments generate statistical data that can be used to make predictions. It is important to recognize that these predictions (interpolations and extrapolations) are based upon patterns that have a degree of uncertainty.

Uncertainty is an inherent part of making predictions.

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The quality of the interpretation is directly related to the quality of the data. An awareness of uncertainty allows students to assess the reliability of data and data interpretation. Chance addresses the predictability of the occurrence of an outcome. As students develop their understanding of probability, the language of mathematics becomes more specific and describes the degree of uncertainty more accurately.


STRANDS • • • •

Number Patterns and Relations Shape and Space Statistics and Probability

The learning outcomes in the program of studies are organized into four strands across the grades K–9. Some strands are subdivided into substrands. There is one general outcome per substrand across the grades K–9. The strands and substrands, including the general outcome for each, follow.

Number •

Develop number sense.

Patterns and Relations Patterns • Use patterns to describe the world and to solve problems. Variables and Equations Represent algebraic expressions in multiple ways.

•

Shape and Space Measurement • Use direct and indirect measurement to solve problems. 3-D Objects and 2-D Shapes Describe the characteristics of 3-D objects and 2-D shapes, and analyze the relationships among them.

•

Transformations Describe and analyze position and motion of objects and shapes.

•

Statistics and Probability Data Analysis • Collect, display and analyze data to solve problems. Chance and Uncertainty • Use experimental or theoretical probabilities to represent and solve problems involving uncertainty.


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OUTCOMES AND ACHIEVEMENT INDICATORS The program of studies is stated in terms of general outcomes, specific outcomes and achievement indicators.

General outcomes

General outcomes are overarching statements about what students are expected to learn in each strand/substrand. The general outcome for each strand/substrand is the same throughout the grades.

Specific outcomes

Specific outcomes are statements that identify the specific skills, understanding and knowledge that students are required to attain by the end of a given grade.

Achievement indicators

Achievement indicators are samples of how students may demonstrate their achievement of the goals of a specific outcome. The range of samples provided is meant to reflect the scope of the specific outcome. Achievement indicators are context-free. In the specific outcomes, the word including indicates that any ensuing items must be addressed to fully meet the learning outcome. The phrase such as indicates that the ensuing items are provided for illustrative purposes or clarification, and are not requirements that must be addressed to fully meet the learning outcome.

SUMMARY The conceptual framework for K–9 mathematics describes the nature of mathematics, mathematical processes and the mathematical concepts to be addressed in Kindergarten to Grade 9 mathematics. The components are not meant to stand alone. Activities that take place in the mathematics classroom should stem from a problem-solving approach, be based on mathematical processes and lead students to an understanding of the nature of mathematics through specific knowledge, skills and attitudes among and between strands.

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Instructional Focus

Planning for Instruction The curriculum is arranged into eleven units. These units are not intended to be discrete units of instruction. The integration of outcomes across strands makes mathematical experiences meaningful. Students should make the connection between concepts both within and across strands. Consider the following when planning for instruction: • Integration of the mathematical processes within each strand is expected. • By decreasing emphasis on rote calculation, drill and practice, and the size of numbers used in paper and pencil calculations, more time is available for concept development. • Problem solving, reasoning and connections are vital to increasing mathematical fluency and must be integrated throughout the program. • There is to be a balance among mental mathematics and estimation, paper and pencil exercises, and the use of technology, including calculators and computers. Concepts should be introduced using manipulatives and be developed concretely, pictorially and symbolically. • Students bring a diversity of learning styles and cultural backgrounds to the classroom. They will be at varying developmental stages. .

Resources The resource selected by Newfoundland and Labrador for students and teachers is Math Focus 4 (Nelson). Schools and teachers have this as their primary resource offered by the Department of Education. Column four of the curriculum guide references Math Focus 4 for this reason. Teachers may use any resource or combination of resources to meet the required specific outcomes listed in column one of the curriculum guide.


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GENERAL AND SPECIFIC OUTCOMES GENERAL AND SPECIFIC OUTCOMES BY STRAND (pages 17 – 30) This section presents the general and specific outcomes for each strand, for Grades 3, 4 and 5 GENERAL AND SPECIFIC OUTCOMES WITH ACHIEVEMENT INDICATORS (pages 95 – 106) This section presents general and specific outcomes with corresponding achievement indicators and is organized by unit. The list of indicators contained in this section is not intended to be exhaustive but rather to provide teachers with examples of evidence of understanding that may be used to determine whether or not students have achieved a given specific outcome. Teachers may use any number of these indicators or choose to use other indicators as evidence that the desired learning has been achieved. Achievement indicators should also help teachers form a clear picture of the intent and scope of each specific outcome. Refer to Appendix A for the general and specific outcomes with corresponding achievement indicators organized by strand.

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GENERAL AND SPECIFIC OUTCOMES BY STRAND (Grades 3, 4 and 5) [C] Communication [CN] Connections [ME] Mental Mathematics and Estimation

[PS] [R] [T] [V]

Problem Solving Reasoning Technology Visualization

Number

Grade 3 General Outcome Develop number sense. Specific Outcomes


1. Say the number sequence 1. Represent and describe whole numbers to 10 000, 0 to 1000 forward and pictorially and symbolically. backward by: • 5s, 10s or 100s, using any [C, CN, V] starting point • 3s, using starting points 2. Compare and order numbers to 10 000. that are multiples of 3 • 4s, using starting points [C, CN, V] that are multiples of 4 • 25s, using starting points 3. Demonstrate an understanding of addition of that are multiples of 25. numbers with answers to [C, CN, ME] 10 000 and their corresponding subtractions 2. Represent and describe (limited to 3- and 4-digit numbers to 1000, numerals) by: concretely, pictorially and • using personal strategies symbolically. for adding and subtracting [C, CN, V] • estimating sums and differences 3. Compare and order • solving problems numbers to 1000. involving addition and [C, CN, R, V] subtraction. [C, CN, ME, PS, R] 4. Estimate quantities less than 1000, using referents. [ME, PS, R, V]


Grade 5 General Outcome Develop number sense. Specific Outcomes 1. Represent and describe whole numbers to 1 000 000. [C, CN, V, T] 2. Use estimation strategies, including: • front-end rounding • compensation • compatible numbers in problem-solving contexts. [C, CN, ME, PS, R, V] 3. Apply mental mathematics strategies and number properties, such as: • skip counting from a known fact • using doubling or halving • using patterns in the 9s facts • using repeated doubling or halving to determine, with fluency, answers for basic multiplication facts to 81 and related division facts. [C, CN, ME, R, V]

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[C] Communication [CN] Connections [ME] Mental Mathematics and Estimation

[PS] [R] [T] [V]


Number (continued) Grade 3 General Outcome Develop number sense. Specific Outcomes 5.


Illustrate, concretely and 4. Explain and apply the properties of 0 and 1 for pictorially, the meaning of multiplication and the place value for numerals to property of 1 for division. 1000. [C, CN, R] [C, CN, R, V]

5. Describe and apply mental mathematics strategies, such as: • skip counting from a known fact • using doubling or halving • using doubling or halving and adding or subtracting one more group • using patterns in the 9s facts • using repeated doubling 7. Describe and apply mental to determine basic mathematics strategies for multiplication facts to subtracting two 2-digit 9 × 9 and related division numerals, such as: facts. • taking the subtrahend to [C, CN, ME, R] the nearest multiple of ten and then compensating • thinking of addition • using doubles. [C, CN, ME, PS, R, V] 6.

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Describe and apply mental mathematics strategies for adding two 2-digit numerals, such as: • adding from left to right • taking one addend to the nearest multiple of ten and then compensating • using doubles. [C, CN, ME, PS, R, V]

Grade 5 General Outcome Develop number sense. Specific Outcomes 4. Apply mental mathematics strategies for multiplication, such as: 1. annexing then adding zero 2. halving and doubling 3. using the distributive property. [C, CN, ME, R, V] 5. Demonstrate, with and without concrete materials, an understanding of multiplication (2-digit by 2-digit) to solve problems. [C, CN, PS, V] 6. Demonstrate, with and without concrete materials, an understanding of division (3-digit by 1-digit), and interpret remainders to solve problems. [C, CN, ME, PS, R, V]



[PS] [R] [T] [V]


Number (continued) Grade 3 General Outcome Develop number sense. Specific Outcomes



6. Demonstrate an 7. understanding of multiplication (2- or 3-digit by 1-digit) to solve problems by: • using personal strategies for multiplication with and without concrete 9. Demonstrate an materials understanding of addition • using arrays to represent and subtraction of numbers multiplication with answers to 1000 • connecting concrete (limited to 1-, 2- and 3representations to digit numerals), symbolic representations concretely, pictorially and 8. • estimating products symbolically, by: • applying the distributive • using personal strategies property. for adding and subtracting [C, CN, ME, PS, R, V] with and without the support of manipulatives 7. Demonstrate an • creating and solving understanding of division problems in context that (1-digit divisor and up to 9. involve addition and 2-digit dividend) to solve subtraction of numbers. problems by: • using personal strategies [C, CN, ME, PS, R, V] for dividing with and without concrete materials • estimating quotients • relating division to 10. multiplication. [C, CN, ME, PS, R, V] 8.

Apply estimation strategies to predict sums and differences of two 2digit numerals in a problem-solving context. [C, ME, PS, R]


Demonstrate an understanding of fractions by using concrete, pictorial and symbolic representations to: • create sets of equivalent fractions • compare fractions with like and unlike denominators. [C, CN, PS, R, V] Describe and represent decimals (tenths, hundredths, thousandths), concretely, pictorially and symbolically. [C, CN, R, V] Relate decimals to fractions and fractions to decimals (to thousandths). [CN, R, V] Compare and order decimals (to thousandths) by using: • benchmarks • place value • equivalent decimals. [C, CN, R, V]

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[PS] [R] [T] [V]





11. Demonstrate an 10. Apply mental mathematics 8. Demonstrate an understanding of fractions understanding of addition strategies and number less than or equal to one by and subtraction of properties, such as: • using doubles using concrete, pictorial and decimals (limited to • making 10 symbolic representations to: thousandths). • using the commutative • name and record fractions [C, CN, PS, R, V] for the parts of a whole or property • using the property of zero a set • thinking addition for • compare and order fractions subtraction • model and explain that for for basic addition facts and different wholes, two related subtraction facts identical fractions may not to 18. represent the same [C, CN, ME, PS, R, V] quantity • provide examples of where fractions are used. [C, CN, PS, R, V] 9. Represent and describe decimals (tenths and hundredths), concretely, pictorially and symbolically. [C, CN, R, V]

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[PS] [R] [T] [V]



Grade 4

Grade 5

General Outcome Develop number sense. Specific Outcomes

11. Demonstrate an 10. Relate decimals to fractions and fractions to decimals understanding of (to hundredths). multiplication to 5 × 5 by: [C, CN, R, V] • representing and explaining multiplication using equal grouping and 11. Demonstrate an understanding of addition arrays and subtraction of decimals • creating and solving (limited to hundredths) by: problems in context that • using personal strategies involve multiplication to determine sums and • modelling multiplication differences using concrete and visual • estimating sums and representations, and differences recording the process • using mental symbolically mathematics strategies • relating multiplication to to solve problems. repeated addition [C, ME, PS, R, V] • relating multiplication to division. [C, CN, PS, R]


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[PS] [R] [T] [V]


Number (continued) Grade 3

Grade 4

Grade 5

General Outcome Develop number sense. Specific Outcomes 12. Demonstrate an understanding of division (limited to division related to multiplication facts up to 5 × 5) by: • representing and explaining division using equal sharing and equal grouping • creating and solving problems in context that involve equal sharing and equal grouping • modelling equal sharing and equal grouping using concrete and visual representations, and recording the process symbolically • relating division to repeated subtraction • relating division to multiplication. [C, CN, PS, R] 13. Demonstrate an understanding of fractions by: • explaining that a fraction represents a part of a whole • describing situations in which fractions are used • comparing fractions of the same whole that have like denominators. [C, CN, ME, R, V]

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[PS] [R] [T] [V]


Patterns and Relations (Patterns) Grade 3 General Outcome Use patterns to describe the world and to solve problems. Specific Outcomes

Grade 4 General Outcome Use patterns to describe the world and to solve problems. Specific Outcomes

Grade 5 General Outcome Use patterns to describe the world and to solve problems. Specific Outcomes

1. Demonstrate an 1. Determine the pattern rule 1. Identify and describe understanding of increasing patterns found in tables and to make predictions about patterns by: charts, including subsequent elements. • describing multiplication chart. [C, CN, PS, R, V] • extending [C, CN, PS, V] • comparing • creating numerical (numbers to 1000) 2. Translate among different representations of a pattern, and non-numerical patterns such as a table, a chart or using manipulatives, concrete materials. diagrams, sounds and actions. [C, CN, V] [C, CN, PS, R, V] 3. Represent, describe and 2. Demonstrate an extend patterns and understanding of decreasing relationships, using charts patterns by: and tables, to solve • describing problems. • extending [C, CN, PS, R, V] • comparing • creating numerical (numbers to 1000) 4. Identify and explain mathematical relationships, and non-numerical patterns using charts and diagrams, to using manipulatives, diagrams, sounds and solve problems. actions. [CN, PS, R, V] [C, CN, PS, R, V] 3. Sort objects or numbers, using one or more than one attribute. [C, CN, R, V]


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[PS] [R] [T] [V]


Patterns and Relations (Variables and Equations) Grade 3 General Outcome Represent algebraic expressions in multiple ways. Specific Outcomes

Grade 4 General Outcome Represent algebraic expressions in multiple ways. Specific Outcomes

4. Solve one-step addition and subtraction equations involving a symbol to represent an unknown number. [C, CN, PS, R, V]

5. Express a given problem as an equation in which a symbol is used to represent an unknown number. [CN, PS, R] 6. Solve one-step equations involving a symbol to represent an unknown number. [C, CN, PS, R, V]

28 /

Grade 5 General Outcome Represent algebraic expressions in multiple ways. Specific Outcomes 2. Express a given problem as an equation in which a letter variable is used to represent an unknown number (limited to whole numbers). [C, CN, PS, R] 3. Solve problems involving single-variable, one-step equations with whole number coefficients and whole number solutions. [C, CN, PS, R]



[PS] [R] [T] [V]


Shape and Space (Measurement) Grade 3 General Outcome

Grade 4 General Outcome

Use direct and indirect Use direct and indirect measurement to solve problems. measurement to solve problems. Specific Outcomes

Specific Outcomes

Grade 5 General Outcome Use direct and indirect measurement to solve problems. Specific Outcomes

1. Relate the passage of time to 1. Read and record time, using 1. Identify 90º angles. digital and analog clocks, common activities, using [ME, V] including 24-hour clocks. nonstandard and standard [C, CN, V] units (minutes, hours, days, 2. Design and construct weeks, months, years). different rectangles, given 2. Read and record calendar [CN, ME, R] either perimeter or area, or dates in a variety of formats. both (whole numbers), and [C, V] 2. Relate the number of make generalizations. seconds to a minute, the [C, CN, PS, R, V] 3. Demonstrate an number of minutes to an understanding of area of hour and the number of days 3. Demonstrate an regular and irregular 2-D to a month in a understanding of measuring shapes by: problem-solving context. length (mm) by: • recognizing that area is [C, CN, PS, R, V] • selecting and justifying measured in square units referents for the unit • selecting and justifying 3. Demonstrate an mm referents for the units cm2 understanding of measuring • modelling and or m2 length (cm, m) by: describing the • selecting and justifying • estimating area, using relationship between referents for cm2 or m2 referents for the units mm and cm units, and • determining and recording cm and m between mm and m • modelling and describing area (cm2 or m2) units. • constructing different the relationship between [C, CN, ME, PS, R, V] rectangles for a given area the units cm and m • estimating length, using (cm2 or m2) in order to demonstrate that many referents • measuring and recording different rectangles may have the same area. length, width and height. [C, CN, ME, PS, R, V] [C, CN, ME, PS, R, V]


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[PS] [R] [T] [V]


Shape and Space (Measurement) (continued) Grade 3 General Outcome Use direct and indirect measurement to solve problems. Specific Outcomes 4.

•

•

• •

5.

Demonstrate an understanding of measuring mass (g, kg) by: selecting and justifying referents for the units g and kg modelling and describing the relationship between the units g and kg estimating mass, using referents measuring and recording mass. [C, CN, ME, PS, R, V]

Demonstrate an understanding of perimeter of regular and irregular shapes by: • estimating perimeter, using referents for cm or m • measuring and recording perimeter (cm, m) • constructing different shapes for a given perimeter (cm, m) to demonstrate that many shapes are possible for a perimeter. [C, ME, PS, R, V]

30 /

Grade 4

Grade 5 General Outcome Use direct and indirect measurement to solve problems. Specific Outcomes 4. Demonstrate an understanding of volume by: • selecting and justifying referents for cm3 or m3 units • stimating volume, using referents for cm3 or m3 • measuring and recording volume (cm3 or m3) • constructing right rectangular prisms for a given volume. [C, CN, ME, PS, R, V] 5. Demonstrate an understanding of capacity by: • describing the relationship between mL and L • selecting and justifying referents for mL or L units • stimating capacity, using referents for mL or L • measuring and recording capacity (mL or L). [C, CN, ME, PS, R, V]



[PS] [R] [T] [V]


Shape and Space (3-D Objects and 2-D Shapes) Grade 3

Grade 4

Grade 5

General Outcome Describe the characteristics of Describe the characteristics of Describe the characteristics of 3-D objects and 2-D shapes, and 3-D objects and 2-D shapes, and 3-D objects and 2-D shapes, and analyze the relationships analyze the relationships among analyze the relationships among among them. them. them. Specific Outcomes Specific Outcomes Specific Outcomes General Outcome

General Outcome

6.

Describe 3-D objects according to the shape of the faces and the number of edges and vertices. [C, CN, PS, R, V]

4. Describe and construct right rectangular and right triangular prisms. [C, CN, R, V]

7.

Sort regular and irregular polygons, including: • triangles • quadrilaterals • pentagons • hexagons • octagons according to the number of sides. [C, CN, R, V]


6. Describe and provide examples of edges and faces of 3-D objects, and sides of 2-D shapes that are: • parallel • intersecting • perpendicular • vertical • horizontal. [C, CN, R, T, V] 7. Identify and sort quadrilaterals, including: • rectangles • squares • trapezoids • parallelograms • rhombuses according to their attributes. [C, R, V]

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[PS] [R] [T] [V]


Shape and Space (Transformations) Grade 3

Describe and analyze position and motion of objects and shapes. Specific Outcomes 5. Demonstrate an understanding of line symmetry by: • identifying symmetrical 2-D shapes • creating symmetrical 2-D shapes • drawing one or more lines of symmetry in a 2-D shape. [C, CN, V] 6.

32 /

Grade 5

Grade 4 General Outcome

Demonstrate an understanding of congruency, concretely and pictorially. [CN, R, V]

General Outcome Describe and analyze position and motion of objects and shapes. Specific Outcomes 8.

Identify and describe a single transformation, including a translation, rotation and reflection of 2-D shapes. [C, T, V]

9.

Perform, concretely, a single transformation (translation, rotation or reflection) of a 2-D shape, and draw the image. [C, CN, T, V]



[PS] [R] [T] [V]


Statistics and Probability (Data Analysis) Grade 3

Grade 4

Grade 5

General Outcome Collect, display and analyze data to solve problems. Specific Outcomes



1. Collect first-hand data and organize it using: • tally marks • line plots • charts • lists to answer questions. [C, CN, PS, V]

1. Demonstrate an understanding of many-to-one correspondence. [C, R, T, V]

1. Differentiate between first-hand and second-hand data. [C, R, T, V]

2. Construct, label and interpret bar graphs to solve problems. [C, PS, R, V]

2. Construct and interpret pictographs and bar graphs involving many-to-one correspondence to draw conclusions. [C, PS, R, V]


2. Construct and interpret double bar graphs to draw conclusions. [C, PS, R, T, V]

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[PS] [R] [T] [V]


Statistics and Probability (Chance and Uncertainty) Grade 3

Grade 4

Grade 5 General Outcome Use experimental or theoretical probabilities to represent and solve problems involving uncertainty. Specific Outcomes 3. Describe the likelihood of a single outcome occurring, using words such as: • impossible • possible • certain. [C, CN, PS, R] 4. Compare the likelihood of two possible outcomes occurring, using words such as: • less likely • equally likely • more likely. [C, CN, PS, R]

34 /


UNIT: NUMERATION

Estimated Completion

September

October

November December January February

March

April

May

June

UNIT: NUMERATION

36

GRADE 4 MATHEMATICS CURRICULUM GUIDE - DRAFT

UNIT: NUMERATION

Unit Overview Background .

Students have already had significant place value experience up to end of Grade 3. While there may be many students who have not mastered the topic completely, most should arrive at grade 4 with a foundation to build upon. The intent in Grade 4 is to extend upon this foundation and develop place value concepts for 4-digit numbers to 10 000. It is important for students to gain an understanding of the relative size (magnitude) of numbers through real life contexts that are personally meaningful. Use numbers from student’s experiences, such as capacity for local arenas, or population of the school/community. Students can use these personal referents to think of other large numbers. Students can also use benchmarks that they may find helpful such as multiples of 100 and 1000, as well as 250, 500, 750, 2500, 5000, and 7500. The focus of instruction should be ensuring students develop flexible thinking with respect to larger whole numbers.

Process Standards Key


GRADE 4 MATHEMATICS CURRICULUM GUIDE – DRAFT

[PS] [R] [T] [V]


37

UNIT: NUMERATION

Strand: Number Specific Outcome

Suggestions for Teaching and Learning While the unit on multiplication and division facts does not begin until around Christmas, it is suggested that the facts with products up to 45 be incorporated early as part of the 5-10 minutes of morning routine.

It is expected that students will: N1 Represent and describe whole numbers to 10 000, concretely, pictorially and symbolically [C, CN, V]

Students should recognize the value represented by each digit in a number, as well as what the number means as a whole. Include situations in which students use money, place value charts, base ten materials and number lines. Money How many $100 bills are there in $8347? Place value charts Thousands

H

T

O

H

T

O

Base ten materials What does 999 look like? What would 1 000 look like? Have students construct it with the base ten blocks. If you were to make a base ten block that could represent 10 000, what would it look like? Why? If you had ten flats, what is the total value? How would you write a given base ten collection, as a numeral? If you could create a new base ten block that would represent 10 000, what would it look like and why? (One suggestion might be to model 10 000 as a long train or tower using 10 of the thousand cubes. It will be a 10 thousand rod. Students should recognize that this also models 10 000 unit cubes.) Mathematical language: Words have special meanings in mathematics. By using a multiplicity of mathematical language, we help children to develop a rich language base conducive to communicating understanding. This will lead to higher order thinking as students move through the grades. For example, when using Base Ten materials, there is a variety of vocabulary that can be used to describe these materials. The thousands block (“block” is a generic term for any of the Base Ten pieces) can be referred to as the large cube, the hundreds block can be referred to as a flat, the tens block can be referred to as a rod or a long and the ones block can be referred to as a unit. Avoid using “thousands cube”, “hundreds flat”, “tens rod” or “ones” as student will need to be flexible in their thinking of models. Later, when the Base Ten materials are used to represent decimals, the individual blocks will take on different leanings. At that point, for example, the rod may represent one.

38


UNIT: NUMERATION

General Outcome: Develop Number Sense Suggested Assessment Strategies Student-Teacher Dialogue N1 Ask questions about the reasonableness of numbers such as, “Would it be reasonable for an elementary school to have 9600 students?” or “Would it be reasonable for an elevator to hold 20 people?” “Would someone be able to drive 26 hundred kilometres in a day?” “Would it be reasonable to pay $5 000 for a boat/book/computer?” Investigate and discuss possible answers. Have students create their own “reasonable” questions about a variety of topics.

Resources/Notes Authorized Resources MathFocus 4 Chapter 2: Numeration Getting Started: Modelling Numbers TR pp. 9 - 12 SB pp. 34-35

Portfolio N1 Exploding the Number: Have the students write any 4-digit number in the center of a large sheet of paper. Ask the students to represent the number in as many ways as possible. This should be repeated any time throughout the unit as children build on their knowledge of number.

Additional Resources:

10 less than 2167

2137+20

two thousand one hundred fifty-seven

2000+100+50+7

2157

149 years from now.

Teaching Student-Centered Mathematics, Grades 3-5, Van de Walle and Lovin, p. 45 -49 Making Math Meaningful to Canadian Students, K-8, Small, p.137 – 148

157 more than 2000


39

UNIT: NUMERATION


Suggestions for Teaching and Learning

It is expected that students will: N1 Represent and describe whole numbers to 10 000, concretely, pictorially and symbolically (Cont’d) [C, CN, V]

Number-line Given a number line with two reference points ask the students to place a given whole number. Include number lines with different starting and ending points. 0

5500 Place 2500 on the number line in relative position


6500

10 000


N1.1 Read a given four-digit numeral without using the word and; e.g., 5321 is five thousand three hundred twenty-one, NOT five thousand three hundred AND twenty-one.

N1.1 The word ‘and’ is reserved for reading decimal numbers. For example, 3.2 will read “three and two tenths”. Set an example by not using “and” when reading numbers. This is simply a convention. Ask students to watch for times when you will use the word “and” in a number, inappropriately. Do this intentionally on occasion to observe who notices. Rewards may be given.

N1.2 Write a given numeral, using proper spacing without commas; e.g.4567 or 4 567, 10 000.

N1.2 Students will see four digit numbers written with or without a space between the thousands and hundreds digits. However, the conventional use of spaces helps children to read larger multi-digit numbers without the use of commas.

N1.4 Represent a given numeral using a place value chart or other models.

1.4 When representing numerals, students should model numbers containing zeros: For example: 1003 means: 1 thousand, 3 ones

40


UNIT: NUMERATION

General Outcome: Develop Number Sense Suggested Assessment Strategies (Cont’d)

Resources/Notes

Student-Teacher Dialogue Tell the student that a snowmobile costs $9130. Ask: If one were to pay for it in $100 bills, how many would be needed? Extend by asking how many $10 dollar bills would be needed. Performance Have students, as a class, create a “ten thousands” chart. Provide each small group of students with hundred grids (or other pictorial representations such as arrays of dots) and have them create a model to represent 1000. Combine these models to create a class representation of 10 000. Performance Have the students find large numbers from newspapers and magazines. Ask them to share and discuss the numbers within their group. Have students read, write, and model these numbers in different ways. Student-Teacher Dialogue Teachers could ask the student to imagine flats placed on top of each other to form a tower. Ask: How many flats would be required to construct a tower representing 10 000? How high would this be? Paper and Pencil Teachers model numbers using base ten materials. For example, show 4 rods, 2 large cubes, 3 units. (Note: It is not always necessary for the blocks to be presented in the typical order). Have the students record the number that is represented.

MathFocus 4 Lesson 1: Modelling Thousands N1 (1.1/ 1.2) TR pp.13 - 15 SB pp.36-37

Student-Teacher Dialogue Ask a student to use base ten materials to model 2046 in three different ways. Have him/her explain the models. Student-Teacher Dialogue Ask: How are 1003 and 103 different? Similar?


41

UNIT: NUMERATION




Achievement Indicators: N1.2 Write a given numeral, using proper spacing without commas; e.g.4567 or 4 567, 10 000.

N1.2 Students will see four digit numbers written with or without a space between the thousands and hundreds digits. However, the conventional use of spaces helps children to read larger multi-digit numbers without the use of commas. While a four digit number can be written with or without the space, five digit numbers should include appropriate spacing.

N1.4 Represent a given numeral using a place value chart or other models.

N1.4 One model of representation may not meet the needs of all students in the class. Acceptable models include place value charts, base ten materials and money. Lesson 2 addresses this, primarily using base ten materials and place value charts. The use of other models appears in subsequent lessons.

N1.7 Explain the meaning of each digit in a given 4-digit numeral, including numerals with all digits the same; e.g., for the numeral 2 222, the first digit represents two thousands, the second digit two hundreds, the third digit two tens and the fourth digit two ones.

N1.7 Students will describe numbers in several ways. Typically, the number 8 347 is read as eight thousand, three hundred forty-seven but might also be expressed as: 8 thousands, 34 tens, 7 ones; 83 hundreds, 4 tens, 7 ones; or 8 thousands, 3 hundreds, 47 ones; etc.

42


UNIT: NUMERATION

General Outcome: Develop Number Sense Suggested Assessment Strategies Paper and Pencil Teachers model numbers using base ten materials. For example, show 4 rods, 2 large cubes, 3 units. (Note: It is not always necessary for the blocks to be presented in the typical order). Have the students record the number that is represented.

Student-Teacher Dialogue Pose a problem such as “Patrick chose 6 base ten blocks. The value of these blocks is more than 4000 and less than 4804. Which blocks might Patrick have chosen?” Have the student model and explain. Extension: Ask if a student can find all possible numbers which fit the criteria. Ask the student to justify his/her answer. Presentation Tell the students that a number has 4 digits. The digit in the thousands place is greater than the digit in the tens place. Ask: What number might this be? Have students share their responses. Ask: What is the greatest number this could be? What is the least number this could be?

Resources/Notes MathFocus 4 Lesson 2: Place Value N1 (1.1/ 1.2/ 1.4/ 1.7) TG. pp. 16-19 SB pp. 38 – 41 To confirm full understanding, give some tasks which include the need for trading to form the numeral such as 4 thousands and 14 hundreds. Also, provide sketches of base ten blocks where the blocks are not presented in the typical order. For example, 4 tens blocks, 2 thousands blocks and three hundreds blocks. Ask the student to write the numeral for this collection of base ten materials.

.

Paper and Pencil Ask students to write a number that has at least 20 tens.


43

UNIT: NUMERATION



It is expected that students will: N1 Represent and describe whole numbers to 10 000, concretely, pictorially and symbolically (Cont’d) [C, CN, V] Achievement Indicators: N1.5 Express a given numeral in expanded notation; e.g., 321 = 300 + 20 + 1.

Indicators N1.5/ N1.6/ N1.7 are addressed together in lesson 3. It is important not to limit teaching examples to: 1 635 = ___ thousands ___hundreds ____tens ____ones

N1.6 Write the numeral represented by a given expanded notation.

N1.7 Explain the meaning of each digit in a given 4-digit numeral, including numerals with all digits the same; e.g., for the numeral 2 222, the first digit represents two thousands, the second digit two hundreds, the third digit two tens and the fourth digit two ones

44

in which the student is likely to put the numbers 1, 6, 3, 5, in the blanks in order, since it is the only information available even if he/she has no idea what this question means. To better assess student understanding of place value, provide numbers such as 1 635. Ask, “Which digit is in the hundreds place?” Student might answer “6”. Ask, “What does the 3 represent?” Student might answer “30” or “3 tens”. This type of questioning tells the teacher more about the student understanding of expanded notation. Students should be given the opportunity to work with numbers involving zeros. (e.g. 4062 - It is important to note that that 4062 does not have a digit in the hundreds place however it still has 40 hundreds.)


UNIT: NUMERATION

General Outcome: Develop Number Sense Suggested Assessment Strategies

Resources/Notes

Pencil-Paper Have students use a reference book to find the populations of 2 towns, where populations are 10 000 or less. Ask students to represent these populations in expanded notation. Journal Say the standard name for a number with four digits (e.g. “four thousand forty six”). Using base ten blocks, students model that number on their desks. Record what they have made, in a journal. Next students enter the number on a calculator (standard form) and record. Finally, students write the number in expanded form. Performance Find Your Partner (card game) – Prepare 2 sets of cards to suit size of group. For example: Set A 2023 2332 223 2230 2032 3202

Set B 2000 + 20 + 3 300 + 2 + 30 + 2000 200 + 20 + 3 2000 + 200 + 30 2 + 30 + 2000 three thousand two hundred two

MathFocus 4 Lesson 3: Expanded Form N1 (1.1/ 1.2/ 1.4/ 1.5/ 1.6/ 1.7)) TG pp. 20 - 23 SB pp. 42 – 44 More practice may be needed. Extra practice in the black line masters can be found on page 11.

In Set A, be sure to include some cards with numbers having “0” as a digit or cards that have the same digit repeated (e.g. 2117). In Set B, cards could include base ten pictures or number lines. Distribute cards, 1 per student, and have students circulate around the room to find the partner with the card that corresponds to their own. As students compare cards, encourage them to discuss why their cards match or do not match. Once students have found their partner, they will read their numbers to the teacher for confirmation. Performance/Journal Have students work in pairs. Provide each pair with 4 number cubes. Roll the cubes and line them up to form a 4 digit numeral. Record the numeral in standard form and in expanded notation. Using the same four numerals, rearrange the cubes to find all possible 4 digit numerals. Record each one in math journal. Extension: Place these numbers on a number line.


45

UNIT: NUMERATION



It is expected that students will: N1 Represent and describe whole numbers to 10 000, concretely, pictorially and symbolically (Cont’d) [C, CN, V] Achievement Indicators: N1.4 Represent a given numeral using a place value chart or other models N1.6 Write the numeral represented by a given expanded notation.

Extending students’ conceptual understanding of numbers beyond 1000 is sometimes difficult to do because physical models for thousands are not commonly available. Encouraging students to extend the patterns in the place value system and to create familiar real-world referents helps students develop a fuller sense of these larger numbers. (Van de Walle & Lovin, Teaching Student Centered Mathematics 3 -5, 2006) Provide interesting tasks using numbers like 10 000 and these will become lasting reference points or benchmarks and will provide meaning to large numbers encountered in everyday life. Big number tasks need not take up a lot of time but can be done as group or school-wide projects. Assuming there about 500 pages in your math book, how many math books would it take to make 10 000 pages? Introduction of the words “ten thousand” should be done when the students have demonstrated that they understand such a number does exist. What does 10 000 look like? Arrange the class into 10 groups and supply each group with hundred grids. Assign each group the task of building a rod to represent 1 000 and tape them end to end. In a large area have students come together to create a 10 000 model using the thousand strips. Ask: How long would it take to count to 10 000? (You may time how long it takes for the students to count 100. Then multiply by 100 using a calculator) Ask: Do you think our school sends home 10 000 newsletters in a year?

46


UNIT: NUMERATION


Resources/Notes

Presentation As a class project, collect some type of object of reaching 10 000. For example, pennies, buttons, bread tags, soup labels, etc. Performance Create a large amount such as 10 000 by asking students to draw 100, 200 or 500 dots on a sheet a paper. Ask students to compile their sheets to form a visual of 10 000 (e.g. book, bulletin board etc.). 10 000 paper chain links can be constructed and hung (with benchmarks indicated) along the hallway. Let the school be aware of the ultimate goal. Performance Use a sheet of 1cm dot paper. Cut the show it shows a 10cm x 20cm array of dots. Ask: • how many dots are on one sheet? • how many sheets are needed to show 1 000 dots? • how many are needed to show 5 000 dots? 10 000 dots? Place the sheets in an array of 50 sheets so that students what an array of 10 000 dots would look like. Journal Students research to find examples of situations where the number 10 000 is used. Make these into posters to display or describe in words and drawings. Performance Show one package of unopened copier paper to the class. Discuss how many sheets are in the package and how many packages would be needed for 10 000 sheets of paper.

MathFocus 4 Lesson 4: Describing 10 000 N1 (1.2/ 1.4) TG pp. 24 - 26 SB p. 45 Mid-Chapter Review N1 (1.4/ 1.5/ 1.6) TG. pp 27-29 SB p46 -47 In the interest of time, these may be done together in one period. Page 28 in TG suggests . . . “Students should be able to complete Questions 1 to 7 in class. If not, assign the rest for homework.” Teachers should use own discretion in assigning class practice and homework. It is not necessary to assign all questions nor is it beneficial for all students to complete unfinished practice at home. Teaching Student-Centered Mathematics, Grades 3-5, Van de Walle and Lovin, p. 50 - 51


47

UNIT: NUMERATION



It is expected that students will: N1 Represent and describe whole numbers to 10 000, concretely, pictorially and symbolically (Cont’d) [C, CN, V] Achievement Indicators: N1.1 Read a given four-digit numeral without using the word and; e.g., 5321 is five thousand three hundred twenty-one, NOT five thousand three hundred AND twenty-one. N1.2 Write a given numeral, using proper spacing without commas; e.g.4567 or 4 567, 10 000.

Lesson 5 in the text deals mainly with cheque writing. Remember that cheque writing is not an outcome. It is simply the context used. In fact, we do not write cheques today as often as in the past because of wide use of debit cards. However, cheque writing is a good example of a practical use of writing numbers in words. More practice will be needed and can be found on page 13 of the black line masters. In addition, when writing numbers from standard form to words and vice versa, it is important to ensure that one or more zeros occur in some of the numbers at a various place value position. Students should be able to represent numbers, which they see or hear, in words. Examples:

N1.3 Write a given numeral 0–10 000 in words

48

•

Say: One thousand nine hundred twenty two people attend a hockey game. Write this in words.

•

Write “2 900” on chart. Have students record in words. (Acceptable answers include: twenty nine hundred or two thousand nine hundred).


UNIT: NUMERATION


Resources/Notes

Performance Have students spin a spinner 3 or 4 times and write the corresponding number in words. Performance Have students cut a 4 digit number from a newspaper or magazine and paste it in their journal. Write the number in words. Performance Write the current year in words. They can also write other years such as their birth year, their expected high school graduation year, the year they will turn forty. Performance What’s My Number? –Provide materials for the students to create a poster with a door or flap in the middle. Ask each student to think of a secret number and write it behind the flap using numerals and/or words. Next instruct students to write clues around the outside of the door that will assist classmates in identifying the secret number.

MathFocus 4 Lesson 5: Writing Number Words N1 (1.1/ 1.2/ 1.3) TG. pp 30 -32 SB pp 48 -49

[C]

Jordan’s Secret Number Greater than 999 Has 4 digits Has repeated digits in the hundreds and thousands place

Less than 2 000

Multiple of 10

Tens place is 3 less than 7

Display posters with a letter assigned to each. Provide students with recording sheets so that they can visit each poster and guess the secret numbers in the display. The intent in this task is that students will record their guesses by using words. Presentation Have each student choose any 4 digit number and create a silly rhyme. These rhymes can be combined in a class booklet. Remind students to use words instead of numerals. For example: “One thousand four hundred eight Too many peas to put on my plate!” “ Twenty four hundred seventy one Happy days spent in the sun.”


49

UNIT: NUMERATION



It is expected that students will: N1 Represent and describe whole numbers to 10 000, concretely, pictorially and symbolically (Cont’d) [C, CN, V] N2 Compare and order whole numbers to 10 000. [C, CN, V]

N2 Comparing and ordering is fundamental to understanding numbers. Students should investigate meaningful contexts to compare and order two or more numbers, both with and without models. Students must realize that when comparing two numbers with the same number of digits, the digit in the greatest place value needs to be addressed first. For example, when asked to explain why one number is greater or less than another, they might say that 2542 < 3653 because 2542 is less than 3 thousands while 3653 is more than 3 thousands. When comparing 6456 and 6546, students will begin comparing the thousands and then move to the right until they notice a difference in place value.

Achievement Indicators: N2.2 Create and order three different 4-digit numerals

N2.2 Assign pairs of students the task of making challenging number cards for their classmates to put in order.

N2.3 Identify the missing numbers in an ordered sequence or on a number line (vertical or horizontal).

Provide the students with opportunities to use number lines with various starting and ending points (0 to10 000). Children will encounter instances of having to read vertical number lines as well as horizontal. Examples include thermometers, measuring cups, distance above/below sea level, growth charts, etc.

N2.4 Identify incorrectly placed numbers in an ordered sequence or on a number line (vertical or horizontal).

0

50

1000 2000 3000 4000 5000 6000

525 450 375 300 225 150 75 0


UNIT: NUMERATION


Resources/Notes

Paper and Pencil Ask the students to each write a number that would fall about half way between 9598 and 10 000.

MathFocus 4 Lesson 6: Locating Numbers on a Number Line N1 (1.1) N2 (2.3/ 2.4) TG. pp 33-36 SB pp 50-52

Performance Provide students with cards that have 4-digit numbers written on them For example:

3000

3300

3003

3303

1300

3033

Ask students to stand in a line in ascending order. Ask a few students to explain why they positioned themselves in that particular spot. Numbers can vary according to student level and small group variations are possible. This task may be repeated using descending order. Ask students to space themselves with respect to number size (in relative position). Performance/Portfolio N2 Provide the following riddle: I am thinking of a number. It is between 8000 and 10 000. All the digits are even and the sum of the digits is 16. What are some possibilities? Have students place their numbers in relative position on an open number line. Challenge pairs of students to write similar riddles for one another and to record answers. Performance Ask two students to hold the ends of a skipping rope representing a number line. Attach 4-digit number cards to the line (using clothespins or fold-over cards). Place several cards out of order. Ask them to identify incorrectly placed numbers and to justify their reasoning. Variation: Repeat having some blank spaces for student to identify the missing 4 digit number in the sequence. For example:

1367

1467

?

1567


Lesson 6 address two outcomes. Work also with number lines that include fewer markings so that students will draw upon their estimation skills over a broader range of numbers. This can be done using a string, a start point card, an end point card and various numbers that students are asked to place in the number line. This can be done by individuals within the context of a whole group activity. Curious Math N1 (1.7) TR pp.37 -38 SB p. 53 (may be omitted)

1667

51

UNIT: NUMERATION




N2 Compare and order whole numbers to 10 000 (Cont’d) [C, CN, V]

Achievement Indicators: N2.2 Create and order three different 4-digit numerals

N2.2 Students must recognize that when comparing the size of a number, the 4 in 4289 has a greater value than the 9 and they should be able to provide an explanation. Tape numbers on students’ backs and have students identify their number by asking classmates questions. E.g. Am I greater than 1000? Am I less than 750? Am I an even or odd number? Am I a multiple of…? Once students have identified their number, have them correctly place it on a class number line. Discuss solutions posed by students.

N2.1 Order a given set of numbers in ascending or descending order, and explain the order by making references to place value

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The focus of lesson 8 is on communication in relation to ordering numbers. Recognize that mathematical communication takes on various forms including oral, written, symbolic, graphical, pictorial and physical. (Small p.62) It is important to encourage students to communicate the process used to compare and order numbers. Instruction should enable students to: • Organize and consolidate their mathematical thinking through communication. • Communicate their mathematical thinking coherently and clearly to peers, teachers and others. • Analyze and evaluate the mathematical thinking and strategies of others. • Use the language of mathematics to express mathematical ideas precisely. (NCTM , 2000)


UNIT: NUMERATION


Resources/Notes

Performance Provide a stack of 4 sets of shuffled cards numbered 0 - 9. Ask the students to select 4 cards and arrange them to make the greatest possible number. Have students record and read the number. Then rearrange the cards to make the least possible number.

MathFocus 4 Lesson 7 : Comparing and Ordering Numbers N1 (1.4/ 1.7) N2 (2.2) TG pp 39 – 42 SB pp 54 - 56

Performance Have students use a reference book to find the populations of 2 communities, where populations are 10 000 or less. Ask them to find another population that is greater than that of one of the communities, but less than that of the other. Paper and Pencil Tell the students that you are thinking of a 4-digit number that has 4 hundreds, a greater number of tens, and an even greater number of ones. Ask them to give three possibilities. Paper and Pencil N2.2 Ask the student to record two numbers: the first has 3 in the thousands place, but is less than the second which has 3 in the hundreds place. Paper and Pencil N2 Ask the students to find 3 ways to fill in the blanks to make the following statement true: __245 > 7__84

Student-Teacher Dialogue Tell the student that Bethany’s number had 9 hundreds, but Fran’s had only 6 hundreds. Fran’s number was greater. Explain how this was this possible? Student-Teacher Dialogue: Ask: Which number below must be greater? Explain why. 4__2

9_ 3

Paper and Pencil Given a set of whole numbers, record them on a number line and provide justification for the ordering of the numbers.

Math Game: Target Game N2 (2.2) TG. pp 43 - 44 SB. p 57 While the outcomes reference comparing and ordering, the practice in the text does little to require a student to order a set of objects. It is recommended that for Student Book (page 56, #4), an extra task can be added, which says: “c) Order the set of world records from least to greatest.” Likewise in Student Book ( page 56, #5) an extra task can be added which says: “c) Create all possible numbers using the four digits given and then arrange them from greatest to least.”

MathFocus 4 Lesson 8: Communicating about Ordering Numbers N1 (1.7) N2 (2.1) TR pp 45 -47 SB pp 58-59

Additional Reading: Making Math Meaningful to Canadian Students, K-8, Small, p.61 – 82


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UNIT: NUMERATION



It is expected that students will: N1 Represent and describe whole numbers to 10 000, concretely, pictorially and symbolically (Cont’d) [C, CN, V] N2 Compare and order whole numbers to 10 000 (Cont’d) [C, CN, V] N2.1 Order a given set of numbers in ascending or descending order, and explain the order by making references to place value. (Cont’d)

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N2.1 Display a 4-digit number on an overhead calculator (or on a card, or on the board). Have students enter on their calculators a number which differs from the displayed number by 1 digit. Have them read their numbers and ask others to determine if they are greater than or less than the number on the overhead and by how much. Collect five of their numbers and ask the students to order them. Ask for an explanation of the order.


UNIT: NUMERATION


Resources/Notes Lesson 8 (Cont’d): Communicating about Ordering Numbers N1 (1.7) N2 (2.1) TR pp 45 -47 SB pp 58-59

N2.1Provide 10-sided dice or prepared cards marked 0 to 9. Shuffle the cards. Have students select 4 cards each (or toss a die 4 times) and make the greatest (least) possible number. Have students lay out the cards (with or without the space after the thousands) and read their numbers to their groups. Have some students write their numbers on the board and read them. Ask: Who thinks they might have the greatest number? How far from the greatest possible number is yours? Would it be possible for someone to have the greatest and the least possible numbers when the cards are rearranged? What digits would you want in order to have the greatest difference between your greatest and your least numbers? Provide the students with opportunities to practice comparing numbers such as 3998 and 3010 and ask them to explain their reasoning. Chapter Review N1 (1.2/ 1.3/ 1.4/1.5/ 1.6) N2 (2.1/ 2.2/ 2.3/ 2.4) TR p. 48 - 50 SB p. 60 – 62 Chapter Task: Creating a Puzzle N1 (1.1/ 1.2) N2 (2.1) TR p.51-52 SB p. 63 Other Chapter Assessments


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UNIT: NUMERATION

56


ADDITION AND SUBTRACTION October


September


March

April

May

June

UNIT: ADDITION AND SUBTRACTION

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Unit Overview Introduction

The focus in this unit is for students to be able to work efficiently and flexibly with both traditional algorithms and personal strategies in the addition and subtraction of larger numbers. In order for this to happen, students will have to apply previous knowledge of addition and subtraction facts and the meanings of those operations, understand the basic principles underlying the place value system (Numeration unit) and know how to add and subtract multiples of 10, 100, 1000, etc. (Small, 2008) Students would have worked significantly with 3-digit addition and subtraction in Grade 3. In Grade 4, they will need some review but the focus will be on addition and subtraction of 4-digit numbers. Students need to recognize that estimation is a useful skill in their lives. To be efficient when estimating sums and differences mentally, students need a variety of strategies from which to choose and must be able to access a strategy quickly. Encourage students to estimate prior to calculating the answer. Use a variety of models such as base-ten blocks and number lines to assist in estimation. Some strategies to consider are using benchmarks, rounding, front-end addition / subtraction (left-to-right calculations), and clustering of compatible numbers. Students should have many opportunities to solve and create word problems for the purpose of answering real-life questions, preferably choosing topics of interest to them. These opportunities provide students with a chance to practice their computational skills and clarify their mathematical thinking. “Computational fluency is a balance between conceptual understanding (thinking about the structure of numbers and the relationship among numbers and the operations) and computational proficiency (includes both efficiency and accuracy)”(NCTM, 2000, p.35)




[PS] [R] [T] [V]


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Suggestions for Teaching and Learning While the unit on multiplication and division facts does not begin until around Christmas, it is suggested that the facts with products up to 45 be incorporated early as part of the 5-10 minutes of morning routine.

It is expected that students will: N3. Demonstrate an understanding of addition of whole numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by: - using personal strategies for adding and subtracting - estimating sums and differences - solving problems involving addition and subtraction [C, CN, ME, PS, R]

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Consider the following guidelines for teaching addition and subtraction: Teach in a problem-solving context. Research shows that by solving problems using addition and subtraction, the students create personal strategies for computing and develop understanding about the relationship between the operations and their properties (NCTM 2000, p. 153). Choose problems that relate to the children's own lives (Van de Walle 2001). Provide a variety of problems representing the different addition and subtraction situations with varying degrees of difficulty to differentiate instruction. Work with the whole group initially and have the students paraphrase the problem to enhance understanding (Willis et al. 2006) and to recognize which numbers in a problem refer to a part or to a whole. Have the students estimate the answer to the problem before calculating so that they are better able to determine the reasonableness of their answers. Have base ten materials available for the students to use as needed. Provide time for the students to create their personal strategies to solve the problem and share these strategies with members of their group or with the entire class. Guide the discussion by asking questions to encourage thinking about number relationships, the connection between addition and subtraction, and their personal strategies. Have the students compare their answers to the estimates that they made before solving the problems. Challenge the students to solve the problem another way, do a similar problem without models or clarify the explanation of their personal strategies. Have the students critique their personal strategies as well as those of their classmates to decide which strategy works best for them and why. Have the students create problems for a variety of number sentences illustrating addition and subtraction.




Resources/Notes

Authorized Resources Math Focus 4 Chapter 3: Addition and Subtraction Chapter Opener TR p.8 SB pp.64-65 Getting Started: Counting Students (optional) TR pp.9-11 SB pp.66-67


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It is expected that students will: N3. Demonstrate an understanding of addition of whole numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by: - using personal strategies for adding and subtracting - estimating sums and differences - solving problems involving addition and subtraction (Cont’d) [C, CN, ME, PS, R] Achievement Indicators: N3.3 Describe a situation in which an estimate rather than an exact answer is sufficient.

N3.3 Students need to know that an exact answer is not always necessary. In a given situation, whether or not estimation is appropriate will depend on the context and the numbers or operations involved. For example: Will a container that holds 2000 mL be large enough to hold 350 mL of water from another container as well as 1015 mL of water from a different container?

N3.4 Estimate sums and differences, using different strategies; (e.g., front-end estimation and compensation.)

N3.5 Refine personal strategies to increase their efficiency

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N3.4 Provide students with a variety of estimation strategies, including: benchmarks: 207 - 126 would give an answer between 75 (200 - 125), and 85 (210 - 125) rounding: 439 + 52 is approximately 440 + 50. front-end addition: 138 + 245 = 370 (100 + 200 is 300, 30 + 40 is 70 for an estimate of 370). Some students may include the ones in their estimate making their answer 380. front-end subtraction: 476 - 348 = 130 (400 - 300 is 100, 70 - 40 is 30, 6 and 8 are about the same so I’ll ignore them; my estimate is 130. clustering: cluster the 29, 35, and 42 together to make 100. N3.5 Multiple opportunities to implement personal strategies should be ongoing throughout the year across math strands and integrated with other subject areas. Proficiency with diverse personal strategies contributes to computational fluency and overall number sense.



General Outcome: Develop Number Sense Suggested Assessment Strategies Performance N3.3 Thumbs up ? Thumbs Down? Give students an example of a real life situation in which an estimate, rather than an exact answer, is sufficient when adding or subtracting. If an estimate is sufficient ask students to put their thumbs up. If an estimate is insufficient and an exact answer is required, then students put their thumbs down. After the class response, choose an individual to justify his/her answer.

Resources/Notes Authorized Resources MathFocus 4

Portfolio N3.4 Give the students two estimates for the same sum or difference and have them discuss the advantages and disadvantages of the different estimates. Performance Present the students with problems and have them decide which problems can be answered with an estimate only and which problems require calculation as well as an estimate. Examples of problems: •You are travelling 1265 km to visit relatives. If you travel 568 km the first day, will you have to travel more or less than 700 km the next day to reach your destination in two days? •A book contains 458 pages and you have read 225 pages the first day and 125 pages the second day. After these two days, how many more pages do you have to read to finish this book? •Your three pet rocks weigh a total of 1625 g. If the first rock weighs 980 g and the second rock weighs 320 g, what is the weight of the third rock? •A toad jumps 135 cm on the first jump and 158 cm on the second jump. About how far does it jump in all? Student-Teacher Interview N3.4 Teachers could ask the student to explain how he or she would estimate the difference between the cost of two items. For example, a $599 item and a $378 item.


Lesson 1: Solving Problems by Estimating N3 (3.3/ 3.4) TB pp. 12-15 SB pp.68-69 Lesson 2: Estimating Sums N3 (3.4/ 3.5) TR pp.16-19 SB pp.70-72 Treat these lessons together for instruction.

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It is expected that students will: N3 Demonstrate an understanding of addition of whole numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by: • using personal strategies for adding and subtracting • estimating sums and differences • solving problems involving addition and subtraction (Cont’d) [C, CN, ME, PS, R] Achievement Indicators: N3.4 Estimate sums and differences, using different strategies; (e.g., front-end estimation and compensation.)

N3.4/3.5 Students should be encouraged to be flexible in their use of computational strategies. Personal strategies may, in cases, be faster and less susceptible to error than use of the traditional algorithm. For example, when computing 4000 – 3997, the strategy of counting on/counting back would be more efficient for many learners whereas the traditional algorithm lends itself to errors with regrouping.

N3.6 Solve problems that involve addition and subtraction of more than 2 numbers.

“Not all students invent their own strategies. Strategies invented by class members are shared, explored, and tried out by others. However, an effort should be made to ensure that students understand a strategy before using it. Students should be encouraged to use estimation as a tool to determine reasonableness of results acquired through computation. Example: 2988 + 1987 is about 5000 Provide the students with an addition number sentence such as the following: 328 + 462 = 330 + 460. Have them decide if the number sentence is true or false and why they think so. Encourage the students to think of the equal sign as "the same as" so that they are deciding if the two sides balance each other. In the example above, the statement is true because 2 is taken away from 462 and added to 328, thereby keeping the sum constant

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Resources/Notes

N3.6 Jill received $100 for her birthday. She bought a book for $12, a toy for $37, and a T-shirt for $18. How much money does she have left?

Lessons 1 and 2 (Cont’d)

Performance N3.1/3.2 Provide the students with concrete or pictorial representations of two numbers and then ask them to find either the sum or difference and to represent concretely or pictorially and then represent their thinking symbolically. Interview N3.5 Teachers could ask the student how he/she might find the total of 497 and 699 mentally. Interview N3.6 Teachers could pose the following problem: Johnny walked 12 km on Monday, 28 km on Tuesday and 32 km on Wednesday. How far did he walk in three days? Paper and Pencil N3.6 A concert ran for three consecutive nights. 3146 attended on Thursday, 3502 attended on Friday and 2478 attended on Friday. How many people attended the concert over three nights? Extension: Did more than 10 000 attend over the three nights? Performance N3.1/3.2 Ask students to use the digits 1, 3, 4, 5, 6 and 7, and the symbols +, = to arrive at 782 [e.g. 367 + 415 = 782] Performance N3.6 Provide reinforcement by giving the students a list of addition and subtraction number sentences similar to the examples above. Some are true and some are false. Have the students decide which number sentences are true or false and explain how they know. Performance Provide students with equations involving missing addends. For example 385 + ___ = 500 or 500 – 385 = ___

MathFocus 4 Lesson 3: Exploring Addition and Subtraction N3 (3.4/ 3.6) TR pp.20-22 SB pp.73 This is a good explore to use as a lead into lesson #5. Lesson 4: Adding from Left to Right (optional) TR pp.23-26 SB pp.74-76 N3 (3.4/ 3.6) Math Game: Race to 1500 N3 TR pp.27 SB pp.77 This is a good game to support mental math.


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It is expected that students will: N3 Demonstrate an understanding of addition of whole numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by: • using personal strategies for adding and subtracting • estimating sums and differences • solving problems involving addition and subtraction (Cont’d) [C, CN, ME, PS, R] Achievement Indicators: N3.1 Determine the sum of two numbers using a personal strategy. (one example for 1326 + 548, is to record 1300 + 500 + 74) N3.6 Solve problems that involve addition and subtraction of more than 2 numbers.

N3.6 Students will encounter many instances in which they will have to solve problems involving the addition and subtraction of more than two numbers. Examples: shopping lists, collections [Total? Quantity needed?], total distance travelled, perimeter of polygons, time remaining [Hours? Months?] Understanding the operations of addition and subtraction and the connections between them is crucial. Van de Walle and Lovin states, "Addition names the whole in terms of the parts, and subtraction names a missing part" (2001, p. 107). Only addition is used in finding the whole when given the parts; however, either addition or subtraction may be used in finding a missing part when given the whole and the other part(s). He goes on to say that addition and subtraction problems include three main types: o problems involving change—changing a number by adding to it or taking from it o part–part–whole problems—considering two static quantities either separately or combined o comparison problems—determining how much two numbers differ in size. Recognizing which numbers in a problem refer to a part or to a whole helps the students see the inverse relationship between addition and subtraction and understand their properties (Willis et al. 2006). By using a variety of problems, the students will construct their own meaning for the inverse relationship between addition and subtraction and for the following properties: commutative property of addition—numbers can be added in any order identity element for addition—any number added to zero remains unchanged

General Outcome: Develop Number Sense 66



Suggested Assessment Strategies

Resources/Notes

Performance Have the students analyze a word problem to determine which numbers show the whole and the parts. Then have them decide whether it is the whole or the part that is not known in the problem. Problem example:

Authorized Resources

•You are travelling 1 288 km to go see your brother play hockey. If you travel 654 km the first day, how much farther do you have to travel? Will you have to travel more or less than 700 km the next day to reach your destination in two days? Pencil - Paper Create a problem that can be represented by the number sentence: 350 + ___ = 425. Explain how you know your problem matches the number sentence. Pencil - Paper If there are 195 days in a school year and you had perfect attendance since Kindergarten, how many days will you have been in school by the end of this year? MathFocus 4 Lesson 5: Adding from Right to Left N3 (3.6) TR pp.28-31 SB pp.78-80 Choose some of the practice from lesson 4 to do in addition to the practice for this lesson. #2, 5 and 6 page 76 are good for additional support. Also, choose additional practice from the mid-chapter review as needed. Mid-chapter Review (as needed) N3 (3.3/ 3.4/ 3.6) TR pp.32-35 SB pp.81-82

Strand: Number


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Specific Outcome


It is expected that students will: N3 Demonstrate an understanding of addition of whole numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by: • using personal strategies for adding and subtracting • estimating sums and differences • solving problems involving addition and subtraction (Cont’d) [C, CN, ME, PS, R] Achievement Indicators: N3.4 Estimate sums and differences, using different strategies; (e.g., front-end estimation and compensation.) N3.2 Determine the difference of two numbers using a personal strategy (e.g. for 4127 – 238, record 238 + 2 + 60 + 700 + 3000 + 127 or 4127 – 27 – 100 – 100 – 11).

At this point in the unit, the focus shifts to estimating differences. There is no one right strategy when estimating. When student adjust their numbers to make them easier to estimate differences, it is important to allow them enough time to share their strategies. Students who do not discover personal strategies on their own, benefit form conversing with peers. It is important to reinforce proper mathematics vocabulary. “The terms ‘regroup’, ‘trade’ and ‘exchange’ are used rather than ‘carry’ or ‘borrow’. This is because carrying and borrowing have no real meaning with respect to the operation being performed, but the term ‘regroup’ suitably describes the action the student must take.” (Small, 2008, page 170) Provide the students with a subtraction number sentence such as the following: 482 – 348 = 484 – 350 In this subtraction example, the statement is true because 2 is added to each number on the left of the number sentence, thereby keeping the difference constant.

N3.5 Refine personal strategies to increase their efficiency.

Observing a student solving problems will provide valuable data to guide further instruction and assist students in refining their personal strategies. Success in problem solving depends on a positive climate in which the students are confident in taking risks. By building on the understanding that each student already has and accommodating the individual learning styles, success will follow.

General Outcome: Develop Number Sense

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Resources/Notes

Paper and Pencil N3.1/3.2 Ask the students to find two numbers with a difference of 150 and a sum 500. Paper and Pencil N3.4 Ask the students to find two numbers with a difference of about 80 and a sum of about 200. Interview N3.5 Teachers could ask the student to select from the following questions the one he/she thinks is the easiest to solve and to explain why: 600 - 53 143 - 87 264 – 99 Student-Teacher Dialogue Provide reinforcement by giving the students a list of subtraction number sentences. Some are true and some are false. Have the students decide which number sentences are true or false and explain how they know. Choosing Number Sentences - Present the students with a problem and have them choose which of the number sentences provided could be used to solve the problem. Ask why the number sentences chosen can be used to solve the problem. Example: Rose exercised for 398 minutes during one week while Selina runs for 176 minutes that week. How much longer did Rose exercise than Selina during that week? 176 + 398 =  398 – 176 = 

 = 398 – 176 176 +  = 398

Pencil and Paper Create a problem that can be represented by the number sentence: 200 - ___ = 79. Explain how you know your problem matches the number sentence.

MathFocus4 Lesson 6: Estimating Differences N3 (3.4) TR pp.36-38 SB pp.83 Use as a lead in to subtraction. Lesson 7: Subtracting Numbers Close to Hundreds or Thousands N3 (3.6) TR pp.39-42 SB pp.84-86 Curious Math: Subtracting Another Way (optional) N3 TR pp.43 SB pp.87 This may be confusing to parents if it is sent home as it is quite different from the traditional method. Some children from outside North America may have learned this method. It is a valid method and might be the method of choice by certain ESL students. If the student is comfortable and efficient with this method then there is no compelling reason to require them to convert to the traditional North American method.

Strand: Number


69


Specific Outcome


It is expected that students will: N3 Demonstrate an understanding of addition of whole numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by: • using personal strategies for adding and subtracting • estimating sums and differences • solving problems involving addition and subtraction (Cont’d) [C, CN, ME, PS, R]

General Outcome: Develop Number Sense 70




Resources/Notes

Presentation N3.1/3.2 Have the students prepare a report (video, written, oral…) to describe what they know about addition and subtraction including their strategies.

Lesson 8: Regrouping before Subtracting N3 TR pp.44-47 SB pp.88-90

Paper and Pencil N3.1/3.2 Determine the sum/difference of 3185 and 628 using a personal strategy. Explain your strategy to a classmate. Paper and Pencil Tai has 348 hockey cards. He gives 196 cards to Phu and 82 cards to Fung. How many hockey cards does he have left? Show your work and write your answer in a complete sentence. Pencil and Paper Complete the chart below by drawing a diagram, writing a number sentence and using a personal strategy to solve the problem given. Write the answer to the problem in a complete sentence. Story Problem

Models/Diagrams

You have 278 pennies in your coin collection. If you 350 coins in all, how many coins are not pennies? Number Sentence

Personal Strategy

Math Game: Target 3500 N3 (3.4) TR pp.91 Some students will need more practice using the pencil and paper algorithm than is provided here. Lesson 9: Subtracting by Renaming N3 (3.5) TR pp. 49-51 SB pp.92-93 For subtracting from a number such as 3000, the text uses the method of renaming 3000 to 2999 to avoid trading and then adding the 1 to the answer once it is acquired. Another method to consider is: For subtracting 3000 – 1427, simply drop both numbers by 1 to produce 2999 – 1426. This way the answer requires no adjustment. Sometimes children forget to add on the extra 1 using the first method. However, students might be interested in exploring why both methods work.

Performance One of your pet rocks weighs 1418 g. This rock and another pet rock weigh a total of 2196 g. Estimate the weight of the other pet rock. Show the numbers you used in your estimate. What does the other rock weigh? Show your work and write your answer in a complete sentence. Performance You jog for a certain distance and then walk for 1350 m to cover a total distance of 2126 m. How far did you jog? Show your work and write your answer in a complete sentence.

Strand: Number


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Specific Outcome:


It is expected that students will: N3 Demonstrate an understanding of addition of whole numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by: • using personal strategies for adding and subtracting • estimating sums and differences • solving problems involving addition and subtraction (Cont’d) [C, CN, ME, PS, R]

General Outcome: Develop Number Sense

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Suggested Assessment Strategies Performance Classifying Problems: Open Sort Present the students with a variety of addition and subtraction problems that are written on separate pieces of paper. Have the students work in groups to classify the problems into groups, label the groups and explain why the problems fit where they have been placed. Explain that some problems may fit in more than one group. Challenge the students to: • find another way to classify the problems • create other problems and place them into the groups. Some categories used by the students may include the following: • addition, subtraction, both addition and subtraction • part–part–whole, change, compare • only estimation is needed, both estimation and a calculated answer are needed • find the part, find the whole. Classifying Problems: Closed Sort Present the students with a variety of addition and subtraction problems that are written on separate pieces of paper and also provide them with the categories into which they are to group the problems. See the examples of categories given above. The students sort the problems into the categories provided and justify their choices.


Resources/Notes Lesson 10: Communicating about Number Concepts and Procedures N3 (3.6) TR pp. 52-54 SB pp.94-95 Chapter Review N3 (3.3/ 3.4/ 3.6) TR pp. 55-58 SB pp. 96-98 Chapter Task: Counting Calories N3 (3.4/ 3.6) TR pp. 59-60 SB p. 99 Other Chapter Assessments Cumulative Review TR pp. 61-62 SB pp. 100-101 Not all questions are applicable at this time. Be selective.

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PATTERNS IN MATHEMATICS October



September

March

April

May

June

UNIT 3: PATTERNS IN MATHEMATICS


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Mathematics is often referred to as the science of patterns. Patterns, which exist both in the natural world and in manmade creations, are encountered by students in everyday contexts. In the learning of mathematics, students need to be able to recognize, describe and extend patterns and to use them to solve problems. The ability to discern and utilize patterns effectively fosters the development of algebraic thinking. Students at the grade 4 level begin to represent algebraic thinking with algebraic expressions. The various representations of patterns, including the use of symbols and variables, provide valuable tools in making generalizations of mathematical relationships. Patterns permeate every mathematical concept. Patterns abound in our world. The mathematics curriculum should help sensitize students to the patterns they meet every day and to the mathematical descriptions or models of these patterns and relationships (NCMT, Curriculum and Evaluation Standards for School Mathematics, 1989, pp. 100–101 fix reference format). Grade 4 students will extend upon previous knowledge of patterns and relations as they explore the different types of patterns, discover pattern rules, translate between concrete and pictorial representations of patterns and charts or tables; determine equalities; and investigate how patterns using symbols and variables are used mathematically to describe change and to model quantitative relationships.


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[C] Communication [PS] Problem Solving [CN] Connections [R] Reasoning [ME] Mental Mathematics [T] Technology [V] Visualization and Estimation



Strand: Patterns and Relations Specific Outcome:


It is expected that students will:

While the unit on multiplication and division facts does not begin until around Christmas, it is suggested that the facts with products up to 45 be incorporated early as part of the 5-10 minutes of morning routine.

PR1 Identify and describe patterns found in tables and charts, *including a multiplication chart. [C, CN, PS, V]

Grade 4 students will continue to work with, and expand on, the many patterns in different tables and chart. In Getting Started and Lesson One the focus is on patterns in a hundred chart and in an addition table. This material is addressed in detail in Grades 2 and 3. Therefore it is recommended that teachers be very selective with regard to this material as it does not reflect patterns at the grade 4 level. Select only 1 or 2 practice exercises.

* addressed in Multiplication and Division Facts unit Achievement Indicators: PR1.4 Describe the pattern found in a given table or chart

Explore the many patterns in the hundred chart. For example: Select four numbers that form a square. Add the two numbers on the diagonal, such as, 59 + 68 and 58 + 69. The sums are equal. 127 58

59

68

69 127

Extend several hundred charts so they can explore from 1 to 100, 101 to 200, up to 999. On these charts, use colored counters to cover numbers forming a pattern and encourage the students to explore the place value representation of the covered numbers; for example, the Explore patterns found on an addition chart, such as: • only even numbers are located on the main diagonal (upper left to lower right), so the sum of a number with itself is always even • the numbers increase by ones across a row, since one more is added for each step right • all of the 8s are on one diagonal line, since each time an addend is one greater, the other must be one less • there are three 2s, four 3s, five 4s, etc. • the diagonals of any four numbers that form a square will have the same sum Students have been reviewing basic multiplication facts with products to 45 but have not worked extensively with a multiplication chart. A more in-depth treatment of patterns in a multiplication charts will follow in Chapter 6.


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General Outcome: Use Patterns to Describe the World and Solve Problems Suggested Assessment Strategies

Resources/Notes

Ask Paper and Pencil PR1 Using a hundred chart, ask students to find all the multiples of 2 and color them in. Have students describe the pattern. Repeat this for the multiples of 3, 4, 5, 6, 7, 8, and 9. Ask students to describe what changes they notice as the numbers increase. When reviewing student work, notice to what extent students: • identify all • (some or none) of the multiples of the given number • are able to predict and extend the pattern of multiples • describe the pattern (clearly, partially, with difficulty) by relating it to similar designs in the real world.

Authorized Resources Math Focus4 Authorized Resources Math Focus4 Chapter 1: Patterns in Mathematics

Paper and Pencil PR1.1 Ask the student to fill in the missing numbers, explaining the reason for each choice: 4, 8, ___ , 16, 20, __

Getting Started: Comparing Patterns TR pp. 9 - 10 SB pp. 2-3 Lesson 1: Patterns in an Addition Table PR1 (1.2/ 1.4) TR pp.11-14 SB pp.4-6

5, ___ , 15, ___ , 25 3, ___ , ___ , 12, 15 Portfolio PR 1.1 Teachers could have students describe in writing the patterns they can find in a given chart. Encourage students to reflect on how their work demonstrates that they were good mathematicians.(e.g., by looking for patterns, using mathematical vocabulary to describe my thinking, persevering even though the task was difficult, accepting a challenge, and asking good questions.) Paper and Pencil PR1.2 Provide students with a chart with missing numbers and ask the students to identify the missing numbers and explain their reasoning.

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It is expected that students will: PR1 Identify and describe patterns found in tables and charts, *including a multiplication chart. Cont’d) [C, CN, PS, V] * addressed in Multiplication and Division Facts unit Achievement Indicators: PR1.2 Determine the missing element(s) in a given table or chart.

PR2 The hundred chart is a useful model for students to find and describe a variety of patterns. Encourage students to discuss pattern rules as a means of helping them to determine missing elements.

PR1.4 Describe the pattern found in a given table or chart

Ask students to describe a pattern to confirm understanding. This description is referred to as a pattern rule. Students should use vocabulary, such as vertical, horizontal, diagonal, row, column, pattern rule, starting point, increasing, decreasing and repeating patterns to help describe patterns.

PR3 Represent, describe and extend patterns and relationships, using charts and tables, to solve problems [C, CN, PS, R, V] Achievement Indicators: PR3.1 Translate the information in a given problem into a table or chart. PR3.2 Identify and extend the patterns in a table or chart to solve a given problem

PR3 Growing patterns also have a numeric component, for example, the number of objects in each step. A table or “T-chart” can be constructed to represent the pattern. Once a table is used for the growing pattern, the materials may become unnecessary. Students should then realize that they can extend a pattern without building a model each time. This also leads to the next step which would be to predict what will happen at a particular step (Van de Walle and Lovin 2006, p. 293-294). Have students not only practice extending patterns with materials and drawings but more importantly, to translate patterns from one medium to another. For example, if a pattern is created using red and blue pattern blocks students should know that these blocks can also be represented using letters. Discuss with students how these patterns are mathematically alike.


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Resources/Notes

Performance Using a hundred chart, place a colored chip on numbers 21, 28, 36 and 45. Use addition and subtraction to extend the pattern in both directions. Continue using the chips to complete the hundred chart. Explain the pattern using pictures, numbers and words. 1

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MathFocus 4 Lesson 1: Patterns in an Addition Table (Cont’d) TR pp.11-14 SB pp.4-6

Math Game: Patterns in Charts PR1 (1.4) PR2 (2.1) TR pp.15-16 SB pp.7

Performance PR1.2/1.4 Given this numerical pattern, have students extend the pattern and explain: • how they determined the pattern and its missing elements • what real world situation could be described by this pattern? A 1 2 3 4

B 2 4

MathFocus 4 Lesson 2: 8 Extending Patterns in Tables PR1 (1.2/ 1.4) PR3 (3.1/ 3.2) TR pp.17-20 Have students use manipulatives to illustrate this pattern. Ask them to SB pp.8-11 describe how the concrete representation illustrates this pattern. This lesson is a strong match with outcome PR3.

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Strand: Patterns and Relations Specific Outcome


It is expected that students will: PR3 Represent, describe and extend patterns and relationships, using charts and tables, to solve problems. (Cont’d) [C, CN, PS, R, V] .

When beginning their study of the concept of relations, students need hands-on experience with manipulatives, charts and diagrams used within contexts that are engaging and meaningful to them. Furthermore, they need ample opportunities to connect patterns to number ideas. Once a table or chart is developed, students have two representations of a pattern: the one created with the drawing or materials and the numeric version that is in the table/ grid. When looking for relationships, some students focus on the table while others will focus on the physical pattern. It is important for students to see that relationships discovered exist in a variety of forms. When a relationship is found in a table, challenge students to see how that pattern is represented using concrete materials. (Van de Walle and Lovin 2006, p. 295).

It is expected that students will: PR2.Translate among different representations of a pattern, such as a table, a chart or concrete materials. [C, CN, V] Achievement Indicators: PR2.1 Create a concrete representation of a given pattern displayed in a table or chart

Step

1

2

3

4

5

…

10

# of squares

2

6

12

20

?

…

?

PR2.1 When given a pattern displayed in a table or chart, students should reproduce it using concrete materials. Conversely, when given a pattern made with concrete materials, students should create a table or chart. Students should be provided with ample opportunities to construct growing patterns using concrete materials (toothpicks, pattern blocks, multi-link cubes, etc.) and create a table /chart to represent the pattern. Students should be asked to describe what is happening as the pattern grows and how the new step is related to the previous one.

PR1.3 Identify the error(s) in a given table or chart.

It is helpful for students to think of a pattern rule and apply it when analyzing tables or charts for errors.


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Resources/Notes

PR2.2 Performance Present students with a geometric design series and have them extend the pattern and develop a “T- chart” to go with it. Ask students what the 10th step would be, 12th, 20th, etc. For example:

Lesson 2: Extending Patterns in Tables (Cont’d)

Squares are the basic building blocks for this series #1

#2

#3

Design # 1 2 3 4

#4

# of squares 1 2 3 4

Performance PR2.1 Provide a table involving one arithmetic operation in the pattern, such as the one below. Describe what the data could be about, complete the table and create a concrete representation using linking cubes. 1 3

2 6

3 9

4 12

5 ?

6 ?

7 ?

8 ?

9 ?

Performance PR2 Provide students with a problem that involves a pattern and ask them to represent the pattern in at last two different ways and explain their reasoning. Journal PR1.3 Give students a chart such as the following: 1 4 2 8 3 12 4 18 5 20 6 22 7 28 8 32 9 36

Lesson 3: PR1 (1.3) PR2 (2.2) PR3 (3.2) TR pp.21-24 SB pp.12-14 While some students can answer the questions posed without actually representing their solution, it would be advisable to insist that they represent at least one of the practice items since the outcome itself emphasizes representing as one of its important components. Curious Math: Number Chains PR1 (1.2/ 1.4) TG p.14 SB p.15 Good match with outcomes but should be considered as optional.

Mid-Chapter Review PR1 (1.2/ 1.4) PR2 (2.1) PR3 (3.1/ 3.2) TG pp.26-28 SB pp.16-17 Use as needed but remember to Ask students to identify (in their math journal) where the pattern has focus on material relevant to errors. Have students explain in writing how they know they are right. lesson #2 and 3

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It is expected that students will: PR1.Identify and describe patterns found in tables and charts, including a multiplication chart. [C, CN, PS, V] (Cont’d) Achievement Indicators: PR1.4 Describe the pattern found in a given table or chart PR3 Represent, describe and extend patterns and relationships, using charts and tables, to solve problems [C, CN, PS, R, V]

PR3 As students’ abilities to recognize and create patterns become more refined, they are better prepared to use this knowledge. Their ability to solve problems is further developed as they systematically investigate a variety of patterns. Students move from a basic recognition of patterns to a more sophisticated use of patterns as a problem solving strategy.

Achievement Indicators: PR3.2 Identify and extend the patterns in a table or chart to solve a given problem.

PR3.2 Students should have opportunities, through problem solving, to make connections between physical patterns and information displayed in a chart or table. Meaningful, real-life situations should be regularly provided to ensure that students have sufficient practice to extend patterns found in a table in order to solve a given problem. For example: Chad was trying out for the swimming team. He had to swim 30 laps by the end of the second week. He was not able to swim on weekends. On the first day he swam 1 lap; on the second day, 5 laps; on the third day, 9 laps; and so on. Was Chad able to swim enough laps at the end of the 2 weeks to make the team? Day Monday Tuesday Wednesday Thursday


Number of Laps 1 5 9 13

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Resources/Notes

Performance PR3.1 Ask students to solve the following problem. John was making trains using linking cubes:

MathFocus 4 Lesson 4 : Solving Problems using Patterns PR1 (1.4) PR3 (3.2) TR pp.29-31 SB pp.18-19

First Train

Second Train

Third Train

More practice may be needed. If he continues to build trains this way, how many blocks will he use in the 7th train? Ask the students to look for a pattern and create a table to display the information and solve the problem. Train Number of Blocks 1 1 2 5 3 9 4 5 6 7 Pencil and Paper PR3.2 Emma agrees to walk dogs for three weeks while Katie is away on holiday. This is what Katie is offering to pay her…. Day 1 2 3 4 … 21

Dollars 2 4 6 8 … ?

Emma says,” I’ll make you a different deal…” Day Cents

1 2

2 4

3 8

4 16

… …

21 ?

Ask: If you were Katie, which deal would you choose? Explain your reasoning.

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In Grade 3 students learned about equations with symbols to represent unknown numbers. Students in Grade 4 will be expected to create and solve equations, with one unknown solve equations using any one of the four operations. At this time, students will use addition and subtraction operations and in Chapter 6, multiplication or division equations with one unknown will be covered in depth. Examples: 28 – 9 =   = 17 – 8 4+5= +2 50 = 20 + Δ

PR5 Express a given problem as an equation in which a symbol is used to represent an unknown number. (Cont’d) [CN, PS, R] Achievement Indicators: PR5.1 Explain the purpose of the symbol in a given addition, subtraction, multiplication or division equation with one unknown; e.g., 36 ÷ = 6

An equation is a mathematical sentence with an equal sign and is used to express relationships between two quantities. For some students, the equal sign poses a difficulty. Although they are comfortable with, for example, the sentence 4 + 5 = □, they interpret the equality sign to mean “find the answer”. Therefore, when students see the sentence □ - 4 = 5, they may not be sure what to do as they think the answer is already there. Similarly, students might solve 4 + □ =5 by adding 4 and 5 to “get the answer”. The notion of an equation as an expression of balance is not apparent to them. It is important for students to recognize that the equal sign should be viewed as a way to say that the same number has two different names, one on either side of the equals sign. (Marian Small, 2008, p.586). The equal sign is "a symbol of equivalence and balance" (NCTM 2000, p. 39). Students should be comfortable using various symbols to represent the unknown, for example, a square, circle, or triangle. 29 + Δ = 56 or 29 + Δ = 56 Display a number of samples of balance scales, such as those below. Remind the students that since the scale is balanced, an equation can be written to represent the situation illustrated. Have students write an equation for each balance scale and then solve it. For example, 8+

20

8 + = 20, so = 12.

-8

4

- 8 = 4, so = 20

Provide examples using symbols for the unknown with varying levels of difficulty. Varying the format will strengthen the students understanding of equality.


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Resources/Notes

Student- Teacher Dialogue PR5.1 Solve the following equation and explain your thinking. Δ – 13 = 20

MathFocus 4 Lesson 5: Solving Equations PR5 (5.2/ 5.2) PR6 (6.1/ 6.3) TR pp.32-35 SB pp.20-23

Student- Teacher Dialogue PR5.1 Lori said that the box in the following equation stands for more than one number. Is Lori correct? Why or why not? 6+8=+4 Student- Teacher Dialogue Explain how to find the missing number in: 25 + ∆ = 100

Encourage students to write the equation for each problem. If this is not done, it is possible to solve each problem but without reinforcing the notion of balance.

Student- Teacher Dialogue Explain the purpose of the box in the following equation: 15 -  = 8

Two children have a collection of hockey cards. Alex has 5 more cards than Josie. If they have 25 hockey cards altogether, how many cards do they each have? (Proulx 2006) Alex

Josie

+5

total number of hockey cards

25

Performance Have students write equations with unknowns in problem situations involving measurement. For example: The perimeter of a triangle is 12 cm. One side is 3 cm and another side is 4 cm. What is the length of the third side? Have students write equations with unknowns in problem situations involving data such as: The librarian wanted to know what kind of books to buy for the library. Twenty-three students chose science books and some chose picture books. Forty-eight students chose science books or picture books. How many chose picture books? Have students write equations with unknowns in problem situations involving geometry such as: Gina was making pentagons with toothpicks. She has 30 toothpicks. How many pentagons can she make?

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It is expected that students will: PR5 Express a given problem as an equation in which a symbol is used to represent an unknown number (Cont’d) [CN, PS, R] Achievement Indicators: PR5.2 Express a given pictorial or concrete representation of an equation in symbolic form

PR5.2 Provide students with various representations, such as diagrams, number lines and concrete materials, that can be written as equations. Examples:

36 -  = 32 29

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PR6 Solve one-step equations involving a symbol to represent an unknown number. [C, CN, PS, R, V] Achievement Indicators: PR6.1 Solve a given one-step equation using manipulatives.

36

+

This outcome is a continuation from PR5 with solving the equation as the next step. The multiplication and division aspect of this outcome is addressed later in Chapter 6, Multiplication and Division Facts. Examples of equations involving addition and subtraction: If Yolanda has 18 cards and Karl has 5. Write an equation that tells how many more cards Yolanda has. Solve the equation: 18 - 5 =  5 + = 18 You have 24 marbles and your friend gives you some more marbles. Now you have 32 marbles in all. How many marbles did your friend give you? a. Write an equation to show what is happening in this problem. b. Solve the problem. Explain your thinking.

PR6.3 Describe, orally, the meaning of a given one-step equation with one unknown.

Students develop communication in skills in mathematics as they are given opportunities to share their solutions and respond to the solutions of others.


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Resources/Notes

Performance PR5.2 Model using manipulatives and write the equation in symbolic form • Model 4 sets of 5 counters and have students write the corresponding equation. Possible answers include: 5 + 5 + 5 + 5 = ∆ or 4 x 5 = 20

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MathFocus 4 Lesson 5 (Cont’d): Solving Equations TR pp.32-35 SB pp.20-23 Encourage students to write the equation for each problem. If this is not done, it is possible to solve each problem but without reinforcing the notion of balance.





It is expected that students will: PR5 Express a given problem as an equation in which a symbol is used to represent an unknown number (Cont’d) [CN, PS, R] Achievement Indicators: PR5.3 Identify the unknown in a problem; represent the problem with an equation; and solve the problem concretely, pictorially or symbolically.

PR5.3 Connect the concrete, pictorial and symbolic representations consistently as the students develop and demonstrate understanding of equations. Provide students with a variety of story problems and have them write appropriate equations to represent the situations. For example: • You have 3 boxes of pencils with the same number of pencils in each box. There are 36 pencils in all. •

PR5.4 Create a problem for a given equation with one unknown.

A red ribbon is 36 cm long and a blue ribbon is 63 cm long. The blue ribbon is how much longer than the red ribbon?

PR5.4 Use everyday contexts for problems that the student can relate to so that they can translate the meaning of the problem into an appropriate equation using the symbol to represent the unknown number. Encourage students to create problems using a variety of operations : + - x ÷


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Resources/Notes

Performance PR5.3 Given a story problem, have students write an equation to match the problem (e.g., There are 4 sandwiches on a tray; there were 13 at the start; some are missing). Students create an equation to match (e.g., 4+=13) Performance PR5.3 Provide the students with linking cubes – white, red and blue (or any two other colors). Pose the following problem: “Gregory has 13 red marbles and 22 blue marbles. How many more blue marbles than red marbles does Gregory have?” Have the students model this situation by building two columns with the cubes, one representing the red marbles and the other representing the blue marbles. To find the difference between the two columns, white cubes are added to the red column to represent the difference between the two quantities. Have the students draw a diagram to represent the situation.

MathFocus 4 Lesson 6: Solving Problems with Equations PR5 (5.3/ 5.4) PR6 (6.1/ 6.2/ 6.3/ 6.4) TR pp.36-39 SB pp.24-26

Portfolio PR5.3 After solving an equation, have students write in their math journals, using the following prompts: • I know I am right because I _______. • Some strategies I used to solve problems were_______. • Something I learned was_______. • Something challenging was _______. Performance Given an equation such as 14 +  = 21 or 5 x  =15, students create and solve a story problem. Remember to include equations with +, -, x, ÷. Observe to what extent students were able to : • Create a story to match the equation • Explain the meaning of the unknown variable • Solve the problem in one or more ways Performance Draw a diagram to represent this equation. ∆ + 23 = 48 Solve the equation. Write another equation that is equivalent to ∆ + 23 = 48.

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It is expected that students will: PR6 Solve one-step equations involving a symbol to represent an unknown number. (Cont’d) [ C, CN, PS, R, V] Achievement Indicators: PR6.2 Solve a given one-step equation using “guess and test”.

Model the use of “guess and test” as one strategy in finding the value for the unknown that will balance both sides of the equation. For this strategy, a student guesses an answer and then tests it to see if the guess works. If it doesn’t, the student revises the guess based on what was learned and tries again. This repetitive process continues until the answer is found. Some students are able to think through several guesses at once; others need to go one step at a time. Although we often talk about guessing as bad, this strategy reinforces the value of taking risks and learning from the information that is garnered. (Small 2008, p. 44). Example: There are 8 more girls than boys in a room. Altogether, there are 24 people. How many are boys? Possible Solution: A child might start by thinking the following: 5 boys and 13 girls is 18 people (not enough people) 10 boys and 18 girls is 28 people ( too many people) 8 boys and 16 girls is 24 people (There are 8 boys).

PR6.4 Solve a given equation when the unknown is on the left or right side of the equation.

Students should be given opportunities to write equations where the missing number is in different places. For example: 15 + = 275 260 + 15 = 


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Resources/Notes

Journal N6 Pose this problem for students to solve: You know that + 24 = 35. Could  represent 10? Use words, pictures or symbols to show how you know. This is open ended enough that some students could elect to draw a balance scale . . . .others might draw base ten models. Some might work out 10 + 24 = 34, not 35 . . . and so on. Pencil and Paper Encourage the students to write as many different equations as they can, using symbols to represent this situation. Ask the students to trade their equations and ensure that a wide variety of equations are included. For example: ◊ + 15 = 24 15 +  = 24 24 = 15 + ∆ 24 =  + 15

24 – ⌂ = 15 15 = 24 – ∆

24 – 15 =   = 24 – 15

Lesson 6 (Cont’d): Solving Problems with Equations TR pp.36-39 SB pp.24-26

MathFocus 4 Lesson 7: Equations in a Story PR5 (5..3/ 5.4) PR6 (6.1/ 6.2/ 6.4) TR pp.40-42 SB p.27 If you choose to do Lesson 7 then it would be important to be selective in the practice on Lesson 6 since lessons 5, 6, and 7 are all targeted at the same outcome.

Performance Through discussion, have the students verbalize various ways that equations can be written to represent the situation. Encourage the students to use a variety of symbols. Ensure that the students include equations in which the symbol to represent the unknown quantity is on the left side and other equations in which the unknown quantity is on the right side. 48 +  = 100 ∆ + 48 = 100 100 = ◊ + 48 100 – 48 =  100 – ◊ = 48

100 = 48 + ⌂ ∆ = 100 – 48 48 = 100 – 

Performance What does the ∆ equal in the number sentences shown? ∆-7=6 9 + ∆ = 17 Place numbers in the symbols to make the number sentence true.

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Provide the students with part–part–whole and comparison word problems and have them explore the idea of a symbol representing a specific, unknown quantity as they translate the problems into written equations. Review the meaning of the equal sign as a symbol of equivalence or balance of the two quantities on either side of the equation. Examples of problems:

PR6 Solve one-step equations involving a symbol to represent an unknown number. (Cont’d) [ C, CN, PS, R, V] Achievement Indicators: PR6.5 Represent and solve a given addition or subtraction problem involving a “part-part-whole” or comparison context, using a symbol to represent the unknown.

Part–Part–Whole Whole Unknown Connie has 15 red marbles and 28 blue marbles. How many marbles does she have? Part Unknown Connie has 43 marbles. 15 are red and the rest are blue. How many blue marbles does Connie have? Comparison Difference Unknown Connie has 15 red marbles and 28 blue marbles. How many more blue marbles than red marbles does Connie have? (Compare) Unknown Big Quantity Connie has 15 red marbles and some blue marbles. She has 13 more blue marbles than red ones. How many blue marbles does Connie have? Unknown Small Quantity Connie has 28 blue marbles. She has 13 more blue marbles than red ones. How many red marbles does Connie have? Pair the students and then present the following problem: Stephen is 15 years old. He has a younger brother. The sum of their ages is 25. Write a number sentence that will help you solve this problem. Model writing a number sentence for the problem situation. Use a “think-aloud” strategy to help students understand how to approach the task. Ask students: · What information do we know in this problem? (Stephen is 15, when you add his age to his brother's age you will get 25 as the answer) · What information is unknown? (Age of Sam’s brother) · What operations can be used to solve this problem? Explain to students that they would use a symbol to take the place of the unknown number. Write the equation “15 +  = 25” on the board or chart paper. Ask students what  means in the problem. Have students determine the value of . Provide counters for students as a strategy to model and solve the problem..Allow partners to work together to solve for . Select students to share their answer and explain their strategy. Model the problem using the counters and the equation. Place 15 counters on the overhead projector. Add counters until there are 25. Ask students how many counters were added.


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Resources/Notes

Performance Have students represent and solve these word problems: Jackie is Sheena’s older sister. The difference between their ages is 21 years. Sheena is 37. How old is Jackie? Ms. Jackson allowed 7 students to go to the washroom. There were 15 students left in the room. How many students are in the class? Write as many number sentences as you can to make this true:

∆=8 (Answers might include

20 - ∆ = 14

or ∆ + 30 = 38)

MathFocus 4 Chapter Review : PR1 (1.2/ 1.3/ 1.4) PR2 (2.1) PR3 (3.1/ 3.2) PR5 (5.1/ 5.2/ 5.3/ 5.4) PR6 TR pp.43-46 SB pp.28-30 Chapter Task PR1 (1.1/ 1.4) PR2 (2.1) PR3 (3.1/ 3.2) TR pp.47-49 SB pp.31 Other Unit Assessment

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DATA RELATIONSHIPS Suggested Time: approx. 3 weeks October

November December January February Estimated Completion

September

March

April

May

June

UNIT: DATA RELATIONSHIPS


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Graphing is a way to present data, concisely and visually. Using graphs or charts can make it easier to see relationships in data. A great deal of information is communicated using graphs and charts in newspapers, television, books and other places. The ability to read graphs and charts is necessary in making sense of the world we live in. As adults, the ability to understand and evaluate arguments presented through a graph on various issues is important to our individual and collective well being. The ability to undertake investigations, prepare reports and present cases is an essential part in many occupations, in business and in our personal lives. In preparation for learning to interpret graphs and use them as a problem-solving tool, children need to learn how to make graphs themselves first. Making graphs requires collecting, then sorting and interpreting data. Children have had some experiences with this in previous grades. Although the focus during this unit is data relationships, it is important that students are given opportunities to practise what they have learned on an ongoing basis throughout the year depending on special occasions and events that naturally occur (i.e. Halloween, Sports events, Autumn ) to practice what they have learned. It is important that we avoid gathering data for the sole purpose of making a graph. The gathering and analysis of meaningful data includes asking and answering questions to help children make sense of their world. It is important that children apply their knowledge of data relationships to an organizational tool for sorting. Charts, diagrams and graphs are useful as tools to understand mathematical relationships and solve mathematical problems. Examples of such charts covered in this unit include Venn or Carroll diagrams. These tools should be used within meaningful contexts throughout the year.


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[PS] Problem Solving [R] Reasoning [T] Technology [V] Visualization



Strand: Statistics and Probability (Data Analysis) Specific Outcome



While the unit on multiplication and division facts does not begin until around Christmas, it is suggested that the facts with products up to 25 be incorporated early as part of the 5-10 minutes of daily/morning routine. Strategies for Multiplication Facts has been included in Appendix C for ongoing reference throughout the year.

SP1 Demonstrate an understanding of many-toone correspondence. [C, R, T, V]

Prior to Grade Four, students have had opportunities to collect and display data in pictographs and bar graphs. As they investigate a wider range of topics, they may discover that the data they collect is too large to display in a graph using a one-to-one correspondence (i.e. having each symbol or number on the bar graph represent one piece of data).

Achievement Indicators:

Students need to be introduced to the concept of using a many-to-one correspondence [or scale] when they are creating graphs to display large amounts of data. Students should begin to make decisions about what symbol to use and what that symbol should represent. These decisions are based on the data being used.

SP1.1 Compare graphs in which the same data has been displayed using one-to-one and many-to-one correspondences, and explain how they are the same and different. SP1.2 Explain why many-to-one correspondence is sometimes used rather than one-to-one correspondence.

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Students need to be given many opportunities to explore what scale is most appropriate for their set of data. For example, if they want to display a graph to show their marble collection and they have 36 blue, 24 red, and 42 clear, students may decide to draw a pictograph where each symbol represents 2 marbles or one where each symbol represents 6 marbles. In cases where the numbers are all less than 20, it is usually more appropriate to use a one-to-one correspondence. For larger numbers; however, students may find it better to use intervals [increments] of 10, 25, 100, or 1000 based on the data being graphed. Students should discuss their data displays and be able to explain why they chose the scale they did. Students would not be expected to use the term, ‘interval’ in their explanations, but may justify their choice by telling how they ‘skip counted’. It is important for students to ensure that the interval in their data display is consistent. For example, if they are creating a bar graph that has a scale with an interval of 2, all of the numbers need to increase by 2 (2, 4, 6, 8, 10, 12 … and not 2, 4, 6, 7, 8, 9, 10, 12…). Depending on the data and the scale that is selected, it may become necessary to create partial symbols and bars that fall between numbers.



General Outcome: Collect, Display, and Analyze Data to Solve Problems Suggested Assessment Strategies

Resources/Notes

Presentation Provide cubes in the following sizes and colors: small red cubes, medium blue cubes and large yellow cubes (you may substitute paper squares). Ask each student to pick a cube in their favorite color. After all students have chosen their favorite color cube have them place them into three towers and decide which color is the most popular in the class. Some children might answer based on the size of the towers without realizing that the size of the cubes may account for the difference in the size of the towers. Ask which answer is correct: • Most students like yellow because the bar for yellow is tallest • More students like blue because there are more blue squares than red squares. Discuss how bar graphs can be misleading if the scale for the bars does not remain constant.

Presentation: Present two graphs representing the same data such as in the examples below.

MathFocus 4 Chapter 4 – Data Relationships Chapter Opener: TR pp.8 SB p.102-103 Getting Started: Sorting Creatures TR pp.9-11 SB pp. 104-105 The ‘Chapter Opener’ and ‘Getting Started’ of this unit represent familiar material for students, as pictographs and bar graphs were taught in previous grades. It should be addressed briefly – approx. ½ class. Be selective.

Lesson 1: Interpreting and Comparing Pictographs SP1 (1.1/ 1.2) TR pp.12-15 SB pp. 106-109 Be selective

Ask: Do the pictographs show the same data? Why do they look different? Are they both accurate? The scale used on a graph can give you the wrong idea about the data! What wrong idea might a person get by looking at these graphs?

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It is expected that students will: SP1 Demonstrate an understanding of many-toone correspondence. (Cont’d) [C, R, T, V] Achievement Indicators: SP1.1 Compare graphs in which the same data has been displayed using one-to-one and many-to-one correspondences, and explain how they are the same and different.

SP1.2 Explain why many-to-one correspondence is sometimes used rather than one-to-one correspondence.

As students compare given graphs from various sources, they should examine how the graphs are similar and different. Students should discuss why they think the particular correspondence was chosen and what other correspondence may have also been used. Deciding on what scale to use allows students to apply their knowledge of multiplication and therefore, it is very helpful for students to have a good knowledge of basic facts. As students begin to work with greater amounts of data, it becomes inconvenient to draw a symbol to represent every piece of data. Using a scale allows a single symbol to represent a number of items, a situation referred to as many-to-one correspondence. Once students are introduced to the concept of scale, they need to learn how to choose one that is appropriate for a given situation. (Small 2008, p 478).


101



Resources/Notes

Performance Pose a question such as the following: How much television do grade four students watch?

Lesson 1 (Cont’d): Interpreting and Comparing Pictographs SP1 (1.1/ 1.2) TR pp.12-15 SB pp. 106-109 Be selective

Have students estimate about how many hours of television (or video games/computer time) they have watched in a week. Have students construct two pictographs for the same data. The intervals in one can be constructed using one-to-one correspondence and the other using many-to-one correspondence [e.g., О (circle) = 5 hours]. Have students explain which of the two graphs they prefer. Students should give reasons for their choices.

Presentation To assess students’ understanding of the importance of scale, use an overhead projector to present the pictograph and key below. Girls and Boys Viewing Soccer Game:

Ask: Do you think there are more girls or boys watching the soccer game? After listening to the responses, show two possible responses and ask which one they think is correct: • There are more girls because the row of dots for the girls is longer. • There are more boys because 5 + 5 + 5 = 15 boys whereas 2 + 2 + 2 + 2 + 2 + 2 = 12 girls. 15 is more than 12. Discuss the answers.

102

Navigating through Data Analysis and Probability, NCTM, 2004





It is expected that students will: SP2 Construct and interpret pictographs and bar graphs involving many-to-one correspondence to draw conclusions. [C, PS, R, V]


SP2.1 Identify an interval and correspondence for displaying a given set of data in a graph, and justify the choice. SP2.2 Create and label (with categories, title and legend) a pictograph to display a given set of data, using many-to-one correspondence, and justify the choice of correspondence used.

SP2.4 Answer a given question using a given graph in which data is displayed using many-to-one correspondence.

“The emphasis or goal [of this instruction] should be to help students see that graphs and charts tell about information, that different types of representations tell different things about the same data. The value of having students actually construct their own graphs is not so much that they learn the techniques but that they are personally invested in the data and that they learn how a graph conveys information. Once a graph is constructed, the most important activity is discussing what it tells the people who see it especially those who were not involved in making the graph. Discussions about graphs of real data that students have themselves been involved in gathering will help them interpret other graphs and charts that they see in newspapers and on TV.” (Van de Walle and Lovin 2006, p. 329) When students are creating bar graphs and pictographs, it is important to allow opportunities for them to decide on which scales to use for their graphs. By choosing a scale, an interval and a correspondence will be identified. Suggest that students create a graph that shows the most popular authors, movies, types of food, etc. of class members. Have some students create a bar graph that shows the collected data using a scale of 2 and other groups use a scale of 3, 4, and 5. Have students explain which scale was the most appropriate to display the data. If many-to-one correspondence or scale is used, the scale must be clearly stated in a scale statement, or legend. If a scale is used, the symbol chosen should allow for partial symbols that are easy to interpret. A circle or square is the ideal symbol, as it can easily be divided into quarter and half symbols that are easy to interpret. When creating pictographs and bar graphs, it is important for students displays to include a title, labels, and a legend or key (when applicable) on their displays. Questioning should be ongoing throughout tasks to encourage students to interpret the data presented and to draw inferences. It is important to ask questions that go beyond simplistic reading of a graph. Both literal questions and inferential questions should be posed. For example: • How many ….? • How many more/less than….? • Order from least to greatest/ greatest to least… • Based on the information presented in the graph, what other conclusions can you make? • Why do you think . . . ? Have students discuss what kinds of information they can get from reading given bar graphs and pictographs that display the use of many-to-one correspondence.


103



Resources/Notes

Performance Have the students collect a set of data from another subject area or related to a personal interest. Some suggestions to collect data are listed in the chart below. Using the data collected, ask the students to create a graph using a many-to-one correspondence. Students should also provide an explanation as to why the particular correspondence they used was appropriate. Then have students draw one conclusion based upon their graphs.

Lesson 2: Constructing Pictographs SP1 (1.2) SP2 (2.1/ 2.2/ 2.4) TR pp.16-20 SB pp. 110-113

Throughout the year you may wish to use some of the following sample questions to gather data and/or interpret created graphs: To describe or summarize what you have learned from a set of data: How many glasses of water/milk do you drink in a day/week? How many pencils are in your desk? How often does the average fourth grader wash his or her hands in a day? How many pieces of paper were on the floor in each classroom? How much did students in the class weigh when they were born? Which class read the most books last month? To determine preferences and opinions from a set of data: What is your favorite …. (music group)? Which is the tastiest . . . (brand of chocolate)? What traits do you value in a friend? To compare and contrast two or more sets of data: What are the similarities and differences between third and fifth grade students favorite rides at the fair? Does the number of chores expected by parents differ between third and fourth graders? What is the relationship between the temperature of water and the amount of time it takes to dissolve a cube of sugar? To generalize and make predictions from a set of data: What is the typical type of book read by students in the class? Can you predict you neck measurement from your wrist measurement? How would you react to going to school year-round? Is there a trend between the months of the year and the number of student absences? (Navigations through Data Analysis and Probability in Grades 3- 5)

Creating graphs is time consuming. Be selective with practice but ensure questions which ask to create a pictograph be given priority in this lesson.

Navigating through Data Analysis and Probability in Grades 3 – 5. NCTM, 2002

Performance Create and label (with categories, title and legend) a pictograph using the tally chart below about Favourite Category of Movies using many-to-one correspondence, and justify the choice of correspondence used. In order to gather large numbers which are conducive to ‘many-to-one correspondence’ data may be gathered by surveying other classes. Question asked: What kinds of movies do you like best?

104





It is expected that students will: SP2 Construct and interpret pictographs and bar graphs involving many-to-one correspondence to draw conclusions (Cont’d) [C, PS, R, V] Achievement Indicators: SP2.3 Create and label (with axes and title) a bar graph to display a given set of data, using many-to-one correspondence, and justify the choice of interval used.

Show a graph like the one below. Explain that the spacing between each horizontal line represents 2 people and ask questions, such as: • How many people like apple juice? • How many more like apple juice than tomato juice? • How many students answered the questions about their favourite juice? • Why was the interval of 2 chosen? • Why wouldn’t you use an interval of 10? • How would you order the juice by preference? Favourite Juice 10

Number Of Students

8

6

4

2

0 apple

orange

tomato

grape

Types of Juice

Reminders: • If a many-to-one correspondence is used, the scale must be clearly shown along a numbered vertical or horizontal axis. • Both axes should be labeled and include headings. Students extend their understanding of constructing graphs and interpreting data from previous grades by exploring vertical and horizontal displays that require a many-to-one correspondence.


105



Resources/Notes

Pencil and Paper Below is a table which represents the genres of books elementary students check out at the library over the period of one week:

Lessons 3 and 4 deal mainly with bar graphs. Bar graphs are displays that use lengths of bars to represent quantities.

G enre of B ooks Non-F iction F olk T ales F antas y His torical F iction

B ooks C hec ked O ut During O ne Week

Lesson 3: Interpreting and Comparing Bar Graphs SP1 (1.1/ 1.2) SP2 (2.4) TR pp.21-24 SB pp. 114-117

70 100 65 45

Use the given information, or create your own, to engage students in this task. Discuss which correspondence (one-to-one or many-to-one) would be most appropriate for this data set. Someone may suggest or you may suggest using a one-to-one correspondence. Proceed to draw the bar graph and it will become obvious that one-to-one was not the best choice because of the large numbers. Provide students with copies of grid paper (Teacher’s Resource Master’s Booklet – page 24). Have students work in pairs or individually to decide on a more efficient correspondence and construct their own bar graphs. Explain why the correspondence chosen is more efficient. Journal / Portfolio Throughout the unit provide opportunities for students to self assess their graphs. Here are some suggestions for students to complete: • I know I constructed a good graph because… • Some things that are similar between my graph and my classmate’s graph are… • Some things that are different about my graph and my classmate’s graph are… • When I make a graph I choose intervals of 2 or 5 or 10 when… • When I make a graph, I choose to use an interval of 1 when…

Lesson 4: Constructing Bar Graphs SP2 (2.1/ 2.3/ 2.4) TR pp.25-28 SB pp. 118-120

Math Game: Matching Data SP1 (1.1) SP2 TR pp.29 SB p.121

Mid-chapter Review (be selective) SP1 (1.1/ 1.2) SP2 (2.1/ 2.2/ 2.3/ 2.4) TR pp.30-34 SB 122-125

Performance/Presentation Go on an autumn walk and have students collect a variety of fallen leaves. Ask them to classify the leaves according to color, and display class results on a tally chart. Divide students into groups and instruct each group to create and label a bar graph using ‘many-to-one correspondence’. Observe discussions between students as they make choices as to which scale is best to use, how to label the data, choice of title, which symbol to use, etc. • Have students interpret graph results by posing questions such as: Are there less green leaves than the other colors? If so, how many less? • Have students create their own questions about data in the graph. For a cross-curricular connection ask the following question: • Why do you think leaves change color in autumn? (discuss photosynthesis from Science unit, Healthy Habitats) Display completed graphs and ask students to compare how the different groups displayed the same information (direct focus to the choice of scale).

106





It is expected that students will: SP1 Demonstrate an understanding of many-toone correspondence. (Cont’d) [C, R, T, V] Achievement Indicators: SP1.3 Find examples of graphs in which many-to-one correspondence is used in print and electronic media, such as newspapers, magazines and the internet, and describe the correspondence used.

Students may be asked to check at home for graphs in newspapers, magazines, internet, pamphlets, posters or books. Students might enjoy discussing these graphs, as they appeal to their personal interests. For example:

It is expected that students will: SP2 Construct and interpret pictographs and bar graphs involving many-to-one correspondence to draw conclusions [C, PS, R, V] It is expected that students will: PR4 Identify and explain mathematical relationships, using charts and diagrams, to solve problems. [CN, PS, R, V]

In everyday life we sort things by comparison relationships (for example colour and size). Such relationships also apply to number, as numbers have certain characteristics which make them similar or different from other numbers. Students need to explore this particular concept of number by being involved in experiences where they are expected to recognize, describe and identify relationships and number characteristics. Sorting is the action of grouping (or organizing) objects (or data). Classification (or categorization) is the naming of the groups of objects (or data). Create a list of possible attributes (or characteristics) of numbers such as even, odd/ greater than 500, less than 500 /skip counting by 5’s and 10’s and so on. Display the list, encourage students add more attributes and to refer to the list, regularly, as they examine Venn or Carroll diagrams involving numbers. By grade 4, students are expected to use more sophisticated sorting tools such as a Carroll or Venn diagram. These organizational tools are particularly useful as a form of data display when the categories for the sorting situation overlap. Prior to grade four, students would have worked with Venn diagrams but not with Carroll diagrams.


107



Resources/Notes

Presentation Ask students to find an example of a graph from either newspapers, magazines, or the Internet to present to classmates. Teachers can pose questions to elicit information about the type of correspondence used in the graph.

Lesson 5: Graphs in the Media SP1 (1.1/ 1.2/ 1.3) SP2 (2.1/ 2.4) TR pp.35-37

Curious Math: Logic Puzzles PR4 (4.7) TR pp.38-39 SB p.127 (optional but may be used with Lesson 8)

Performance Students work with a partner. Provide students with cards displaying various numerals and pieces of different colored string. Have students create a Venn Diagram using the string. One student selects six cards, chooses two mystery attributes and sorts them according to the attributes. The other student then attempts to guess the sorting rule. Performance Have students create a set of ten 3-digit or 4-digit numbers and sort them using two attributes. Request that they write the sorting rule.

108

Lesson 6: Using Venn Diagrams PR4 (4.4/ 4.5/ 4.6) TR pp.40-43 SB pp. 128-131 Lessons #6-8 represent content which is totally new for grade 4 students and will therefore require a greater focus for instruction.



Strand: Patterns and Relations (Patterns) Specific Outcome


It is expected that students will: PR4 Identify and explain mathematical relationships, using charts and diagrams, to solve problems [CN, PS, R, V]

Students need practice in reading graphs and diagrams. By recognizing and explaining the relationship between attributes of a given set of data, students are strengthening their reasoning skills.

PR4.4 Identify a sorting rule for a given Venn diagram.

There are three types of Venn diagrams: • Two separate circles (when the items being sorted do not share common attributes) • Overlapping circles (when the items being sorted share common attributes) • Circle contained within a circle (if the inner circle is a subset of the outer)

PR4.5 Describe the relationship shown in a given Venn diagram when the circles intersect, when one circle is contained in the other and when the circles are separate.

Introduce the notion of how cross-classification is shown in Venn diagrams by using loops of string and a set of number cards in a context such as the one below. Ensure the sorting rules and numerals, to be sorted, lend themselves to cross-classification. For example: Kim sorted the number of stickers the students in her class collected: 7, 10, 37, 42, 78, 91, 107, 301, 532, 1233


PR4.6 Determine where new elements belong in a given Venn diagram.

Overlapping Circles

Two Separate Circles

Circle contained within a Circle

Reinforce the use of proper mathematical language during sorting activities. The word “and” indicates that each item in the group would have all attributes of both categories where “or” makes the distinction between the two categories under consideration. Ensure that students include all of the data being considered from their sorting situation in their Venn diagram. Whenever students create a Venn diagram, ensure that they draw a rectangle around the circle(s).


109



Resources/Notes

Paper and Pencil Provide the students with data and have them complete the problem. Jennifer listed the numbers for her raffle tickets in the Spring Fair: 723, 694, 496,501, 360, 999, 222 Sort these numbers using all three types of Venn Diagrams. Include labels.

Lesson 6 (Cont’d): Using Venn Diagrams PR4 (4.4/ 4.5/ 4.6) TR pp.40-43 SB pp. 128-131

Ask students to explain the relationships between the 3 types of diagrams. Performance Materials needed: attribute blocks, labeled cards and loops of string. Students work in pairs. Player A sets up a Venn diagram with one, two or three loops and secretly selects cards with sorting rules written on them. Cards are then placed face down in the loops. Player B selects an object and asks player A in which set it belongs until all objects are correctly placed or Player B chooses to guess the rule. If he guesses correctly, a point is awarded and they reverse roles.

110





It is expected that students will: PR4 Identify and explain mathematical relationships, using charts and diagrams, to solve problems. (Cont’d) [CN, PS, R, V] Achievement Indicators: PR4.1 Complete a Carroll diagram by entering given data into correct squares to solve a given problem.

Carroll diagrams are tables that work much like Venn diagrams and are used for the purpose of cross-classification. For Carroll diagrams, two attributes are being used for sorting, with one attribute of each characteristic being the focus (Small 2008, p. 521). The categories of a Carroll diagram should be mutually exclusive unless the diagram is established to fit a specific context. E.g. even and not even. In a Carroll Diagram, numbers or objects are either categorized as having an attribute or not having an attribute. A table is created with four cells to show the four possible combinations of these two attributes.

PR4.2 Determine where new elements belong in a given Carroll diagram.

Either the items themselves, or the count of how many items of each type, are put in the cells. Sort the following numbers in the Carroll Diagram provided: 953, 888, 1501, 8000, 201, 2542, 450, 9349 odd

even 888

< 1000

953 201

1501

450

8000

> 1000 9349


2542

111



Resources/Notes

Pencil/paper:

Lesson 7: Using Carroll Diagrams PR4 (4.1/ 4.2) TR pp.44-47 SB pp.132-133

Performance/Presentation Provide students with various numeral cards containing numbers up to 4-digits and have students create and complete a Carroll diagram. Using the same numeral cards, create another Carroll diagram using a different sorting rule. Have students share the different Carroll diagrams they created. Have them explain their sorting rule for placing each number in the diagram.

112





It is expected that students will: PR4 Identify and explain mathematical relationships, using charts and diagrams, to solve problems. (Cont’d) [CN, PS, R, V] Achievement Indicators: PR4.3 Solve a given problem using a Carroll diagram PR4.7 Solve a given problem by using a chart or diagram to identify mathematical relationships

Once students are familiar with various classification methods using a Venn or Carroll Diagram, they should be given opportunities to apply these tools to problem solving situations. This will develop their logical reasoning skills. Discuss with students how to choose which diagram to use in given situations. The situation, at times may make one diagram more favorable. There may be times when either could be used.


113



Resources/Notes

Paper and Pencil Students can use Venn Diagrams to solve problems. In a class of 22 students, 10 play hockey and 15 play basketball. a) Is it possible that there are some students who play neither sport? What is the greatest possible number of students who do not play either sport? Explain your answer. Show your work. b) Is it possible that all 22 students are involved in one sport or the other or both? Explain your answer. Show your work.

MathFocus 4 Lesson 8: Solving Problems Using Diagrams PR4 (4.1/ 4.2/ 4.3/ 4.6/ 4.7) TR pp.48-51 SB pp.

Curious Math: Logic Puzzles PR4 (4.7) TR pp.38-39 SB p.127 (Optional)

Chapter Review: SP1 (1.1/ 1.2) SP2 (2.1/ 2.2/ 2.3/ 2.4) PR4 (4.1/ 4.5/ 4.7) TR pp.52-55 SB pp.139 Chapter Task: Promoting Fire Safety SP1 (1.1) SP2 (2.1/ 2.2/ 2.3/ 2.4) TR pp.56-57 SB pp.99 Other Unit Assessment Be selective.

114




115

2-D GEOMETRY Suggested Time: approx. 1 and1/2 weeks October



September

March

April

May

June

UNIT: 2-D GEOMETRY


117

UNIT: 2-D GEOMETRY


This unit introduces students to symmetry and provides them opportunity to explore symmetry and congruency in 2-D shapes. Both of these properties are important and can be linked to the study of fractions and to the area of regular polygons. When children are learning about symmetry, they need to spend a lot of time manipulating the shapes rather than simply looking at them. Taking the time to allow students to fold, draw and work with models to find properties of 2-D shapes is important, as it promotes visualization and is helpful in problem solving. Teachers should promote precise vocabulary usage and encourage students to use mathematical language regularly in class. Teachers can model this by using the correct terms, in context, repeatedly. Symmetry is quite common in the world. It is relevant for students to realize what asymmetry [not having symmetry] looks like. Review previous learning of 2-D shapes so that the terminology is understood and can be applied as needed. Use everyday contexts to introduce congruence and symmetry, drawing upon the students' prior experiences in the real world.


[C] Communication [PS] Problem Solving [CN] Connections [R] Reasoning [ME] Mental Mathematics [T] Technology and Estimation [V] Visualization

118


UNIT: 2-D GEOMETRY

Strand: Shape and Space (Transformations) Specific Outcome



While the unit on multiplication and division facts does not begin until around Christmas, it is suggested that the facts with products up to 25 be incorporated early as part of the 5-10 minutes of daily/morning routine.

SS5 Demonstrate an understanding of line symmetry by: • identifying symmetrical 2-D shapes • creating symmetrical 2-D shapes • drawing one or more lines of symmetry in a 2-D shape. [C, CN, V]

SS6 Demonstrate an understanding of congruency, concretely and pictorially. [C, CN, V] Achievement Indicators: 6.1 Determine if two given 2-D shapes are congruent and explain the strategy used. 6.2 Create a shape that is congruent to a given 2-D shape

6.3 Identify congruent 2-D shapes from a given set of shapes shown in different orientations. 6.4 Identify corresponding vertices and sides of two given congruent shapes.

Strategies for Multiplication Facts has been included in Appendix C for ongoing reference throughout the year.

Symmetry is a geometric property which is closely related to Outcome SS6 (congruency). Congruency is not directly addressed in the text although it is embedded in the lessons. Knowledge of congruency underpins the understanding of symmetry. Teachers should model the word ‘congruent’ however, students may describe the concept of congruency without always using the explicit term ‘congruent’ (equal parts, same size and shape). Congruency and symmetry can be used to determine what makes some shapes alike and different. Any symmetrical shape can be divided into two congruent parts along the line of symmetry; however, not every composite shape made up of two congruent figures is symmetrical. This regular hexagon is symmetrical. The line of symmetry shown in the diagram divides the hexagon into two congruent shapes, each shape is a pentagon This composite shape is made up of two congruent pentagons. It is not symmetrical. This composite shape is made up of two congruent pentagons. It is symmetrical. Congruence of 2-D Shapes: Two 2-D shapes are congruent if they are identical in shape and size – that is, if one is an exact duplicate of the other. Students sometimes do not understand the difference between the math term congruent and the everyday term the same. It is important to recognize that the term congruent applies only to size and shape. Thus, figures can be different colors, or oriented in different ways, and they will still be congruent as long as they are the same shape and the same size. (Small 2008 p.316)

UNIT: 2-D GEOMETRY

General Outcome: Describe and Analyze Position and Motion of Objects and Shapes Suggested Assessment Strategies

Resources/Notes

Performance Provide diagrams of 2-D shapes some of which are congruent, such as the following:

MathFocus 4 Chapter 5: 2-D Geometry

Chapter Opener TR pp.7 Optional: Given the time frame suggested for this unit, this may be an opportune time to integrate mathematics and art curricula.

Ask the students to:

put a checkmark on shapes that are congruent to

put an X on shapes that are congruent to

shade in the shapes that are congruent to

Have the students explain the strategy they used to determine if the shapes were congruent. Suggest that they trace and cut out the three shapes and then superimpose them on the given shapes to prove congruency. Performance Have the students create two congruent shapes of different orientations on an 11 x 11pin geoboard and draw the designs on square dot paper or geopaper (ensure dot paper is same scale as geoboard). It is important to test for congruency because shapes in different orientations may not appear to be congruent even when they are. Students might choose to cut out one design from the dot paper and superimpose it on the other design for congruency.

120

Getting Started: Exploring Polygons TR pp.8 SB pp.142-143 The outcome SS6 (Congruency) is not directly addressed in the text; however, it can be introduced while doing the Getting Started activities and integrated throughout remaining lessons. Suggestions are also provided in this curriculum guide.


UNIT: 2-D GEOMETRY



It is expected that students will: SS6 Demonstrate an understanding of congruency, concretely and pictorially. (Cont’d) [C, CN, V] Achievement Indicators: 6.1 Determine if two given 2-D shapes are congruent and explain the strategy used. 6.2 Create a shape that is congruent to a given 2-D shape

6.3 Identify congruent 2-D shapes from a given set of shapes shown in different orientations.

Ask students to create a square using geoboards or multilink cubes. Say: Some of these squares are congruent and some are not. Give clues such as Greg’s square is not congruent to Susan’s square but it is congruent to Jane’s. Continue giving clues until students discover what congruence means. Include many hands-on activities to establish the concept of congruence prior to introducing symmetry. Distribute play dough, variety of cookie cutters (some of which are symmetrical and some which are not), a plastic knife and a dowel. Challenge students to press out play dough shapes and explore to see if they are symmetrical, and therefore congruent if one half is folded or flipped and placed exactly on one another.

6.4 Identify corresponding vertices and sides of two given congruent shapes.


121

UNIT: 2-D GEOMETRY


Resources/Notes

Performance Use a blank Frayer Model as shown in the sample below to assess student understanding of congruence.

Chapter Opener TR pp.7 Optional: Given the time frame suggested for this unit, this may be an opportune time to integrate mathematics and art curricula.

Getting Started: Exploring Polygons TR pp.8 SB pp.142-143 The outcome SS6 (Congruency) is not directly addressed in the text; however, it can be introduced while doing the Getting Started activities and integrated throughout remaining lessons. Suggestions are also provided in this curriculum guide.

Performance

Have the students label corresponding vertices and colour-code corresponding sides of congruent pairs of 2-D shapes that they created or are presented to them. Instead of colour-coding the corresponding sides, the students may wish to use markings on the sides as shown below. Include examples that have the congruent shapes in different orientations as shown in the diagram.

Have the students justify that they have identified the corresponding sides and vertices correctly by tracing one shape complete with the markings and superimposing it on the other congruent shape. The labelled vertices and colour-coded or marked sides should match.

122


UNIT: 2-D GEOMETRY



It is expected that students will: SS5 Demonstrate an understanding of line symmetry by: • identifying symmetrical 2-D shapes • creating symmetrical 2-D shapes • drawing one or more lines of symmetry in a 2-D shape. (Cont’d) [C, CN, V] Achievement Indicators: SS5.4 Identify lines of symmetry of a given set of 2-D shapes, and explain why each shape is symmetrical

Students should become familiar with the terms “symmetry”, and “lines of symmetry.” A 2-D figure has line symmetry when it can be divided or folded so that the two parts match exactly. We refer to a fold line as a line of symmetry. Any given line of symmetry divides a figure into equal halves [Relates to Outcome N8 for fractions.] It may also be said that each of the halves are mirror images of each other. Some texts may refer to line symmetry as reflective symmetry or mirror symmetry.

SS5.5 Determine whether or not a given 2-D shape is symmetrical by using an image reflector or by folding and superimposing.

(Small, 2009) Students can use lines of symmetry to define fractions of a 2-D shape, such as halves and quarters.


123

UNIT: 2-D GEOMETRY


Resources/Notes

Performance Have students choose 4 different color crayons or markers and color code corresponding sides. Repeat using various shapes and to match corresponding vertices.

Lesson 1: Lines of Symmetry SS5 (5.4/ 5.5) TR pp.11 SB pp.144-146

Performance Ask students to find and group examples of triangles with symmetry and triangles without symmetry. Various organizers might be used such as T-chart or Yes/No chart. Performance Have students complete a Frayer Model to consolidate their understanding of symmetry. This model can be completed together as a class, in groups, or independently depending on the needs of the students and familiarity with the model. A sample of a completed model is shown below.

Additional Reading: Big Ideas from Dr. Small, Small 2009

124


UNIT: 2-D GEOMETRY




Students have learned about symmetry and now should be provided with opportunities to create their own symmetrical 2-D drawings. Miras (transparent mirrors) are very helpful to students when investigating symmetry. They are useful because they are both transparent and reflective. If a shape is symmetrical along the line where the Mira has been placed, the image on one side of the shape will fall right on top of the other side of the shape.

SS5 Demonstrate an understanding of line symmetry by: • identifying symmetrical 2-D shapes • creating symmetrical 2-D shapes • drawing one or more lines of symmetry in a 2 D shape. (Cont’d) [C, CN, V] Achievement Indicators: SS5.3 Complete a symmetrical 2-D shape, given half the shape and its line of symmetry.

SS5.4 Identify lines of symmetry of a given set of 2-D shapes, and explain why each shape is symmetrical.

SS5.5 Determine whether or not a given 2-D shape is symmetrical by using an image reflector or by folding and superimposing.

Display a simple shape and tell students that it is half of a symmetrical picture. Ask what the whole shape looks like. Allow some time for students to offer suggestions. Then introduce the Mira as one tool to assist them in completing the symmetrical shape. Is there more than one possibility depending on where the Mira is positioned? Give students a Mira and provide them with drawings of half shapes which have a dotted line representing the line of symmetry. Next, have them place the Mira on the dotted line and trace the reflection to complete a symmetrical design. This is an opportunity for students to realize that there is more than one way to determine whether or not a shape is symmetrical. Provide students with Power Polygons or other manipulatives, as well as Miras, paper and scissors. Using a graphic organizer as shown below, have students decide and record whether each given shape is symmetrical or not. (Ask students how they might use folding as a strategy when using hard plastic shapes. They will quickly conclude that to solve this problem, they might trace and cut the shape from paper so that it is foldable). Example: Shape

Symmetrical

Strategy used (Mira? Folding?)

Square (A)

Yes

Folding

Etc. Ask various students to choose one shape to present to the class and explain how they know this shape is symmetrical or not.


125

UNIT: 2-D GEOMETRY


Resources/Notes

Performance Create a class “Symme – Tree”. Distribute several cut outs (some of which are symmetrical and some of which are not) in a baggie to each student. Review the characteristics of symmetry and have each student test (by folding or using a mira) each shape for symmetry. Students then place only the symmetrical shapes on the class “Symme-tree”.

Lesson 2: Using a Symmetry Tool SS5 (5.3/ 5.4/ 5.5) TR pp.15 SB pp.147-150

Performance Students use tiles, fraction pieces or pattern blocks to create a symmetrical design and explain to their partner how their design is symmetrical.

This concept lends itself quite well to center work as children need to explore using a “hands-on” approach. Also, in using this approach you do not require full class sets of manipulatives.

Performance Given one half of a design, students create the other half and identify/explain the line[s] of symmetry. Performance Cut a simple shape out of a picture in a magazine. Fold your shape in half, and then cut along the fold line. Glue your picture to a piece of paper and draw the missing half. (Small 2008, p. 300) Performance Have students draw on squared dot paper examples of the different quadrilaterals. Cut them out and fold them to find the lines of symmetry. Performance Use magazine pictures and check for symmetry using Miras. Share and discuss the lines of symmetry. Performance Give the students a Mira, and a sample of any 2-D manipulative [fraction pieces, tangram pieces, power polygons, pentominoes, fraction circles, or pattern pieces]. Have students work in pairs to explore how a Mira can be used to confirm that a shape is symmetrical by placing it in different positions on top of the shape. If the reflected image falls exactly on top of the other side of the shape, then a line of symmetry has been found. Also ask students to find shapes in the classroom that they can show are symmetrical or not symmetrical using the Mira.

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UNIT: 2-D GEOMETRY



It is expected that students will: SS5 Demonstrate an understanding of line symmetry by: • identifying symmetrical 2-D shapes • creating symmetrical 2-D shapes • drawing one or more lines of symmetry in a 2 D shape. (Cont’d) [C, CN, V] Achievement Indicators: SS5.2 Sort a given set of 2-D shapes as symmetrical and nonsymmetrical.

This indicator is similar to SS5.5. In this lesson, however, the focus is more on students observing symmetry in our environment. Students are encouraged to visualize the matching halves of things in our environment that are not conducive to folding, or testing with a Mira. Observing objects in the environment may require that students look at the 2-D faces of 3-D objects. When looking around the environment, consider: - Where can you find examples of symmetry in your environment? In texts? In visual media? - Why do different shapes have different numbers of lines of symmetry, including none? - Why can a line of symmetry not divide a 2-D shape into thirds?

SS5.7 Provide examples of symmetrical shapes found in the environment, and identify the line(s) of symmetry.

Curriculum Connections: Opportunities may arise in other subject areas to discuss symmetry. Asymmetrical and symmetrical balance is a powerful concept in visual arts. Scientific observations will also reveal many examples of symmetry.


127

UNIT: 2-D GEOMETRY


Resources/Notes

Performance Provide a variety of shapes and ask the students to sort them, grouping those with reflective symmetry and those without reflective symmetry. [Note that it is important to have a sufficient variety of available shapes to sort. Some commercially available products, for example, might be limited to regular 2-D shapes. The inclusion of irregular shapes [polygons] help students develop an understanding of geometric properties. An assortment of shapes for sorting may resemble this set.

Lesson 3: Identifying Symmetrical Shapes (optional) SS5 (5.2/ 5.7) TR pp.19 SB pp.151

Mid-chapter Review SS5 (5.3/ 5.4/ 5.5/ 5.7) TR pp.22 SB pp.152-153 (optional) Review may not be necessary given that students have only covered two lessons at this time. Performance Students can work in groups to determine and record their own sorting rules from a given set of manipulatives. Encourage students to discuss their rationale for the ways that they categorized the manipulatives. Groups will then share their results with the class. This activity provides teachers with opportunities to observe and listen for students’ level of thinking about geometric relationships.

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UNIT: 2-D GEOMETRY




Figures may have multiple lines of symmetry and the lines of symmetry can be vertical, horizontal, or diagonal.

SS5 Demonstrate an understanding of line symmetry by: • identifying symmetrical 2-D shapes • creating symmetrical 2-D shapes • drawing one or more lines of symmetry in a 2 D shape. (Cont’d) [C, CN, V] Achievement Indicators: SS5.8 Sort a given set of 2-D shapes as those that have no lines of symmetry, one line of symmetry or more than one line

Teacher Note: A circle has an infinite number of lines of symmetry. The more sides that a regular polygon has, the greater the number of lines of symmetry there are, as the polygon begins to more closely resemble a circle. Give students some shapes to trace and cut. Have them experiment to find how many lines of symmetry there are and show the class their findings. This provides an opportunity to attend to students’ reasoning so as to identify any misconceptions. Students can use lines of symmetry to define fractions of a 2-D shape, such as halves and quarters. Students may also recognize that there are other fractions of an area that do not identify lines of symmetry (for example, thirds). Students should be given opportunities to predict and then investigate the number of lines of symmetry a given figure has. Folding cut outs of paper polygons, using Miras and geoboards, or creating tile models are all possible ways to explore symmetry. Provide examples of 2-D shapes with one line of symmetry, two lines of symmetry and no lines of symmetry. Have the students draw the lines of symmetry and sort the shapes.


129

UNIT: 2-D GEOMETRY


Resources/Notes

Performance Ask students: Do all triangles have the same number of lines of symmetry? Ask them to record their predictions on paper. Provide sets of triangle cut outs for each pair of students (ensure that each set includes examples of equilateral, scalene and isosceles triangles, although students need not know these terms). Allow ample opportunity for students to prove or disprove their predictions by folding, cutting and superimposing.

Lesson 4: Counting Lines of Symmetry SS5 (5.8) TR pp.25 SB pp.154-156

Math Game: Tangram Shapes SS5 (5.6) Suggestion: may be more suitable to do with lesson 6

Performance: Provide students with the following labeled 2-D shapes.

Have students circle all the symmetrical shapes. Instruct them to draw all the lines of symmetry on the symmetrical shapes. Finally, have the students sort the shapes by the number of lines of symmetry in each shape: no lines of symmetry; one line of symmetry; more than 1 line of symmetry and record answers in table:

Lines of Symmetry

Letter Names for the Shapes

No lines of symmetry One line of symmetry More than 1 line of symmetry . Performance Have the students share their ideas about sorting various sets of 2-D shapes and provide follow-up activities to address any misconceptions that may arise. Performance Encourage flexible thinking by having the students sort sets in more than one way or create symmetrical 2-D shapes in more than one way.

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UNIT: 2-D GEOMETRY




Communication in mathematics, whether silent self-talk or communication with peers or a teacher, is essential to students as they learn; it is also a critical assessment tool for teachers. There are many forms of mathematical communication, including oral, written, physical (through active involvement with manipulatives), and symbolic / pictorial communication. A variety of types of communication should be encouraged. Knowing the right word to use allows students to communicate more effectively and efficiently and, as a result, consolidate their learning. Often, the mathematics cannot be understood or communicated without reference to appropriate vocabulary. (Small 2008, p. 61 - 71)

SS5 Demonstrate an understanding of line symmetry by: • identifying symmetrical 2-D shapes • creating symmetrical 2-D shapes • drawing one or more lines of symmetry in a 2 D shape. (Cont’d) [C, CN, V] Achievement Indicators: SS5.1 Identify the characteristics of given symmetrical and nonsymmetrical 2-D shapes and explain the process.

Consider creating a word wall where words are posted as a reference for students and teachers. The words associated with communicating symmetry might be grouped together and referred to frequently. The words may also be posted in a personal dictionary or personal glossary. A Concept map such as the Frayer Model is also useful to help students consolidate their understanding of symmetry. This model can be completed together as a class, in groups, or independently depending on the needs of the students and familiarity with the model.

Shapes that have one or more lines of symmetry will match exactly when folded along each fold line, though the resulting halves may reveal different figures.

Not all figures when divided in halves will be symmetrical… though they may be congruent.


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UNIT: 2-D GEOMETRY


Resources/Notes

Journals Give students a sentence prompt to communicate what they know about symmetry:

Lesson 5: Communicating about Symmetry SS5 (5.1) TR pp.31 SB pp.158-160

I know that symmetrical shapes are ___________ . . . To check if a shape is symmetrical or not, I could _________ . . . A _____________ is symmetrical because ____________ . . . A _____________ is not symmetrical because _____________. . . Performance Ask students to independently complete a Frayer Model to assess student’s conceptual understanding of a concept of your choice: Definition

Characteristics

Examples

Non-examples

Interview

Ask student, “If you were explaining symmetry to a grade three student, what would you say?”

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UNIT: 2-D GEOMETRY



It is expected that students will: SS5 Demonstrate an understanding of line symmetry by: • identifying symmetrical 2-D shapes • creating symmetrical 2-D shapes • drawing one or more lines of symmetry in a 2 D shape. (Cont’d) [C, CN, V] Achievement Indicators: SS5.6 Create a symmetrical shape with and without manipulatives.

During this unit, prepare an area where children have ongoing opportunities to explore and extend upon their experiences with symmetrical design. Provide geoboards, Miras, pattern blocks, 2-D shapes, geometric dot paper, Pattern Block grid paper (Masters booklet pp.44-49), fabric, wallpaper, etc. Display symmetrical designs versus non-symmetrical designs as an exhibit in the classrooms or compile digital photographs to make a book or a bulletin board display.


133

UNIT: 2-D GEOMETRY


Resources/Notes

Paper and Pencil Have students work in pairs. Provide each group with grid paper. Each student shades a small design of their own choosing in the centre of the page. Students then trade papers. The partner’s new task is to use the original figure to create a new symmetrical design. The partner can decide on the line of symmetry [top or bottom/diagonal/left or right].

Lesson 6: Creating Symmetry SS5 (5.6) TR pp.35 SB pp.161

Interview/ Portfolio Have student choose a symmetrical work sample from their portfolio and ask them to identify and explain the lines of symmetry.

Math Game: Tangram Shapes (optional) SS5 TR pp.29 SB pp.157

Performance Paper Folding Build on the students' knowledge of identifying symmetrical 2-D shapes by tracing the shapes, cutting them out, then folding them to show that the halves are congruent. Ask the students to fold a piece of paper and create a design along the fold line and cut it out. Review that the unfolded shape is symmetrical because the two halves are congruent. Have the students share their symmetrical 2-D shapes. Using paint makes the designs eye appealing for a bulletin board display. Performance Have the students work in pairs. Provide them with pattern blocks and isometric dot paper. Instruct one student, in each pair, to create a design using two pattern blocks. The other student in the pair is then to copy the design (reflect the design) to make a composite symmetrical 2-D design. The students may create the symmetrical design using a vertical, horizontal or oblique axis of symmetry. (Note: The vertical line of symmetry is easiest for students to use in creating symmetrical designs.) Examples: Original Design Composite Symmetric Shape

Curious Math: Folding Paper Shapes SS5 TR pp.38 SB pp.162

Chapter Review: (be selective) SS5 (5.1/ 5.3/ 5.4/ 5.5/ 5.6/ 5.8)) TR pp.39 SB pp.163-164

Chapter Task: Counting Calories SS5 (5.3/ 5.4/ 5.6) TR pp.42 SB pp.165 Other Unit Assessment

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UNIT: 2-D GEOMETRY


135

MULTIPLICATION AND DIVISION FACTS Suggested Time: approx. 3 weeks October

November

December January February


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April

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UNIT: MULTIPLICATION AND DIVISION FACTS


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By the end of Grade 4, demonstrating knowledge of a multiplication fact means giving a quick response (less than three seconds for most students) to a fact question without resorting to non-efficient techniques, such as counting. Flash cards and timed tests are valuable but should only be used after adequate development has taken place. Strategies for Multiplication Facts has been included in Appendix C for ongoing reference throughout the year. The recommended approach for strategy practice and retrieval is to first introduce a strategy, with the use of concrete materials, practice the strategy, and then add and practice new strategies. When students have two or more strategies, it is important to focus on strategy selection. Strategy selection involves choosing the strategy that will be most useful to determine a particular fact. Teachers should allow time for both strategy development and the practice of these strategies to ensure that students can demonstrate they know their facts. Often, fact strategies rely on previously developed strategies or concepts. For example, teachers could remind students of the order (commutative) property (e.g., 6 x 8 = 8 x 6). Stress that 6 x 8 refers to 6 eights, while 8 x 6 refers to 8 sixes; the products, however, are the same. Developing basic multiplication facts to 9 × 9 and related division facts requires that the students have a strong foundation in patterns, number relationships, place value, and the meaning, relationships and properties of operations as described in the key indicator. The meaning of multiplication and division and the connection between the operations is crucial as the students develop understanding of multiplication and division facts. Students who have learned their multiplication facts have automatically learned their division facts. This is ongoing throughout the school year as this knowledge is critical in many mathematical concepts.


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N4 Discovery is the best way for students to understand the properties of 0 and 1 for multiplication, since arbitrary rules may become confused with the addition properties of 0 and 1.

N4 Explain and apply the properties of 0 and 1 for multiplication and the property of 1 for division. [C, CN, R] (Cont’d) Achievement Indicators:

N4.1 Determine the answer to a given question involving the multiplication of a number by one, and explain the answer

N4.2 Determine the answer to a given question involving the multiplication of a number by zero, and explain the answer.

Give each student a calculator to explore products involving factors of 0 or 1 (i.e. 654 x 0, 0 x 54, 3418 x 1, 1 x 26, and 7854 x 1, etc.) and look for patterns. Discussion of their findings will suggest rules for factors of 0 and 1, but not reasons for the rule. Using a number line (0 – 10), ask: • What would 7 hops of 1 (7 x 1) look like? (land on 7)

•

What would 1 hop of 7 (1 x 7) look like? (land on 7)

•

What would 5 hops of 0 (5 x 0) look like? (land on 0 five times).

•

What would 0 hops of 5 (0 x 5) look like? (stay on 0 – no hops)

Use various concrete materials and pictorial representations to demonstrate that: • Any number multiplied by 1 remains unchanged. 1 x  simply means one group of  For example, Elizabeth likes to make pancakes for her family on Saturday mornings. There are 6 people in her family. She places one pancake on each plate. Ask: How many pancakes did she make? • Zero property: any number multiplied by 0 is zero (since many zeros still equal 0) For example, use paper plates for the concept of multiplying by zero. Show six empty plates. Ask: ″How many plates are there?” [Answer: six] Ask: “How many cookies are there on each plate?” [Answer: zero]. Ask: “Six groups of zero are how many?” [Answer: 0] Record as a number sentence: 6 x 0 = 0. Address the misconception that multiplication always makes the product greater: Any number multiplied or divided by 1 remains unchanged.


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General Outcome: Develop number sense Suggested Assessment Strategies

Resources/Notes Math Focus4 Chapter Opener and Getting Started is review and should not require more than ½ period

Interview/Journal Pose the question: • What is special about multiplying by 1? • What is special about multiplying by 0? Use words pictures or numbers to explain. Presentation Show students 2 to 4 number sentences, only one of which matches a given story problem. Ask students to select the number sentence that represents the problem and to draw a picture to explain their choice. Sample: Grandma went Christmas shopping for her 6 grandchildren. She wanted to buy them all a ________________ (insert name of popular DVD/ book title/ game/ CD, etc), only to find out that stores were completely sold out! Which number sentence represents grandma’s problem? • 1x6=6 • 6x1=6 • 0x6=0 • 6 x 0 = 0 (correct answer)

Lesson 1: Multiplying by Skip Counting N4 (4.1/ 4.2) N5 (5.1) PR6 (6.2/ 6.4) TG pp. 12-15 SB pp. 170-173

Lessons 1 and 2 may be combined

Paper and Pencil Using a hundred grid, have students find all the multiples of 2 and colour them in. Have students describe the pattern (sample answer: It looks like a checkerboard.) Repeat this task for the multiples of 3, 4, 5, 6, 7, 8, and 9. Ask students to describe what changes they notice as the numbers increase. When reviewing student work, notice to what extent students: • identify all multiples • identify (some or none) of the multiples of the given number • are able to predict and extend the pattern of multiples • describe pattern (clearly, partially, with difficulty) by relating it to similar designs in the real world. Paper and Pencil Ask the student to fill in the missing numbers, explaining the reason for each choice: 4, 8, ___ , 16, 20, __ 5, ___ , 15, ___ , 25 3, ___ , ___ , 12, 15

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It is expected that students will: N5 Describe and apply mental mathematics strategies, such as: - skip counting from a known fact - using doubling or halving - using doubling or halving and adding or subtracting one more group - using patterns in the 9s facts - using repeated doubling to determine basic multiplication facts to 9 x 9 and related division facts [C, CN, ME, R] Achievement Indicators: N5.1Provide examples for applying mental mathematics strategies: • doubling • skip counting from a known fact

PR6 Solve one-step equations involving a symbol to represent an unknown number. [C, CN, PS, R, V] Achievement Indicators:

PR6.2 Solve a given one-step equation, using guess and test.

PR6.4 Solve a given equation when the unknown is on the left or right side of the equation.

Our number system is full of patterns. Pattern exploration helps students develop multiplication strategies such as skip counting and repeated addition. Possible models for skip counting can include a hundreds chart, a number line, an array or a pictograph. Skip counting is practical for some numbers; however, skip counting by other numbers can be difficult for some students (e.g. skip counting by 8). Students may find it easy to skip count by 5 or may use the clock strategy to help them skip count by 5 (See appendix C)

PR6.2 For this strategy, a student guesses an answer and then tests it to see if the guess works. If it doesn’t, the student revises the guess based on what was learned and tries again. This repetitive process continues until the answer is found. Some students are able to think through several guesses at once; others need to go one step at a time. Although we often talk about guessing as bad, this strategy reinforces the value of taking risks and learning from the information that is garnered. (Small 2008, p. 44).

PR6.4 Students should be given opportunities to write equations where the missing number is in different places. For example: 8 x  = 40  x 7 = 28 4 x 9 = = 6 x 8


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Resources/Notes

Performance Whole class game “Buzz” - Students skip count by a chosen number (for example ‘4’). Students may stand in a row or sit in a circle. The first student begins the count by saying ‘1’, the next student says ‘2’ and this continues until a multiple of 4 is reached. Instead of saying the multiple, the student would say “BUZZ”. Student 1: “1” Student 2: “2” Student 3: “3” Student 4: “BUZZ” Student 5: “5” Student 6: “6” Student 7: “7” Student 8: “BUZZ” Etc.

Math Focus4 Lesson 1 (Cont’d): Multiplying by Skip Counting N4 (4.1/ 4.2) N5 (5.1) PR6 (6.2/ 6.4) TG pp. 12-15 SB pp. 170-173

Encourage students to continue to see how high they can skip count. Display a multiplication poster or hundred charts, as some students may need a visual. Pencil and paper Give students a hundred chart to solve this problem: Sharon invited her whole class of 24 to her movie birthday party. Sharon’s mom helped arrange rides for the children so they could go to a movie. Four passengers could fit in each car. How many cars did they need? Performance Give students calculators and a hundreds chart. Ask them to use the calculator to find out the following: • Will you see 83? Count by 2’s. Press 2 + = = = • Will you see 95? Count by 5’s. Press 5 + = = = • Will you see 36? Count by 3’s. Press 3 + = = = • Will you see 91? Count by 10’s. Press 10 + = = = • How can you make calculator count by 4’s? 7’s?

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It is expected that students will: N5 Describe and apply mental mathematics strategies, such as: - skip counting from a known fact - using doubling or halving - using doubling or halving and adding or subtracting one more group - using patterns in the 9s facts - using repeated doubling to determine basic multiplication facts to 9 x 9 and related division facts [C, CN, ME, R] Achievement Indicators:

N5.1Provide examples for applying mental mathematics strategies: • skip counting from a known fact

Students have had significant experience with skip counting and calculating facts to 25. Review skip counting and model how a known fact can be used along with skip counting to find a product. For example: 7 x 4 =  [Think Aloud: Since I already know that 5 groups of 4 is 20, I can start at 20 and then count by 4’s two more times. I can say 20, 24, 28.]


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Resources/Notes

Performance Make available a visual tool such as a hundred chart to help students practice the strategy of skip counting from a known fact.

Math Focus4 Lesson 2: Building on Multiplication Facts N5 TG pp. 16-18 SB pp. 174-175

Students will mark squares to show their thinking. Solve: 5 x 7 =  “Since I already know that 3 groups of 7 are 21 (circle the numeral 21), I can start at 21 and then count by 7 two more times. I can use check marks to cross out numerals as I skip count. I can say 21, 28, 35.” Performance Have students cover numbers on a hundreds chart to make a pattern (e.g. Cover every second number, every sixth number, etc.) Take one of the patterns you created on the hundred chart and record it on a horizontal line.

6

12

18

24

30

36

42

48

54

60

Cover all the numbers with post it notes and give it to a friend. How many numbers does the friend have to uncover before discovering the pattern?

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It is expected that students will: N5 Describe and apply mental mathematics strategies, such as: - skip counting from a known fact - using doubling or halving - using doubling or halving and adding or subtracting one more group - using patterns in the 9s facts - using repeated doubling

When teaching the strategies, try to plan your lessons so that students invent them and build on strategies they already have mastered and use. Resist the temptation to simply tell them your strategy and have them practice it. It will be more useful if students can connect the strategy to concepts they already understand.

to determine basic multiplication facts to 9 x 9 and related division facts [C, CN, ME, R] Achievement Indicators:

N5.1 Provide examples for applying mental mathematics strategies: • doubling • repeated doubling

Actively engage students actively by posing situations that relate to real life investigations. The purpose is to help students establish a conceptual understanding of what multiplication means. The meaning of multiplication is crucial as the students develop an understanding of the multiplication facts. Word problems and the use of manipulatives (such as counters) are key to developing this understanding. Prior to Grade 4, students will have learned the double facts in the context of addition and can now quite easily relate them to the 2x facts. Other facts can be related to these 2x facts. Every 4x fact can be calculated by doubling a 2x fact; every 8x fact can be calculated by doubling a 4x fact (repeated doubling). Using doubling to multiply by 4. Find 4 x 7: First find 2 x 7, then double: Think: 2 x 7=14 and double 14 is 28. Use repeated doubling to multiply by 8. To find 8 x 6: First think of 2 x 6 = 12, then double 12 is 24 and double 24 is 48.


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Resources/Notes

Performance

Math Focus4

Teachers could place students in pairs to practice the “double” strategy for facts such as 4 x 9. (For example, 4 x 9 is double 2 x 9. Since 2 groups of nine are 18, then 4 groups of 9 are 36.) Students should take turns asking facts and providing answers by using the strategy of repeated doubling.

Lesson 3: Doubling Multiplication Facts N5 (5.1) PR1 (1.1/ 1.2/ 1.4) TG pp. 19-22 SB pp. 176-177

Interview Ask students: Use counters to show how you can use 2 x 3 = 6 to help find the answer to 4 x 3. Performance Present the following problem to the students: Keri puts 6 pencils in each of 4 boxes. Sue puts 6 pencils in each of 8 boxes. How many pencils does each girl have? Guiding questions: • Explain how you could use the answer for the number of pencils Keri has, to find the number of pencils Sue has. (doubling) • Explain how the following number sentences could be used in solving this problem: 4 x 6 =  and 2x4x6=

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It is expected that students will: PR1 Identify and describe patterns found in tables and charts, including a multiplication chart [C, CN, PS, V]

Students should be encouraged to find and explain patterns that occur in a multiplication chart. This will help them remember the multiplication facts. It is important that students understand they can use these patterns to determine unknown products or quotients.

Explore doubling patterns found on a multiplication chart.

Achievement Indicators: PR1.1 Identify and describe a variety of patterns in a multiplication chart.

PR1.2 Determine the missing element(s) in a given table or chart.

PR1.4 Describe the pattern found in a given table or chart

Lead students to discover: How the 2x facts can be used to calculate the 4x facts. How the 4x facts can be used to calculate the 8x facts. How the 3x facts can be used to calculate the 6x facts. How the 10x facts can be used to calculate the 5x facts. PR1.2 Teachers should provide students with many opportunities to find missing elements in a chart. For example:

Students should search for patterns formed by groups of things in their environment. They should record the patterns on a T-table and also as multiplication equations. For example: One foot has how many toes?


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Resources/Notes

Paper and Pencil Provide students with a chart with missing numbers and ask the students to identify the missing numbers and explain their reasoning.

Lesson 3: Doubling Multiplication Facts (Cont’d) N5 (5.1) PR1 (1.1/ 1.2/ 1.4) TG pp. 19-22 SB pp. 176-177

Journal Give students a chart such as the following: 1 2 3 4 5 6 7 8 9

4 8 12 16 20 22 28 32 36

Ask students to identify (in their math journal) where the pattern has errors. Have students explain in writing how they know they are correct. Student-Teacher Dialogue Provide the student with a multiplication chart: • Ask him/her to describe some of the patterns he/she observes. • Ask the student to show how one could use the multiplication chart to practice skip counting. • Ask the student to explain why some columns/rows have both even and odd numbers. (For example: Point to row 7. The student might say “because 7 is an odd number and sometimes it is multiplied by another odd number then the product will be odd. Other times it is multiplied by an even number and then the product will be even). • Ask the student to use the multiplication chart to explain why 4 x 5 plus 2 x 5 is the same as 6 x 5. (a possible answer might be “because 4 groups of 5 plus 2 groups of 5 are the same as 6 groups of five”).

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Seeing the relationships between the easy and hard facts will benefit students. Help students realize that if they don’t know a product, they can figure it out by using something they already know. Students should be challenged to find as many interesting ways as possible to answer a harder multiplication fact. Teachers should validate all efficient strategies that students discover. For more challenging facts, provide opportunities that will lead students to discover new strategies depending on what they already know. Leading the children to “invent” and name their strategy strengthens retention and understanding. Students need opportunities to use concrete materials and mental strategies to relate the facts to one another. They must not be rushed into memorizing the facts before they are ready

N5 Describe and apply mental mathematics strategies, such as: - skip counting from a known fact - using doubling or halving - using doubling or halving and adding or subtracting one more group - using patterns in the 9s facts - using repeated doubling to determine basic multiplication facts to 9 x 9 and related division facts [C, CN, ME, R]

Use a clear 11pin x 11 pin geoboard and an overhead projector to demonstrate solving 7 x 6. Discuss with students how they can use arrays of smaller facts to determine an unknown product:

Achievement Indicators: N5.1Provide examples for applying mental mathematics strategies: • doubling • doubling and adding one more group • doubling and subtracting one group • halving

2 groups of 3 x 6 (doubling) and adding 1 more group of 6 Students will learn to divide an equation into easier parts to multiply. Sometimes this might mean using more than one strategy to find the product. Some examples include: • Use halving (the factor) and then doubling (the product) when one factor is an even number: “I can’t recall the product of 8 x 7, but I know I can halve the 8 to find 4 groups of 7. So 4 groups of 7 is 28 and 28 doubled is 56 (or will give the product of 56”

•

Use halving and doubling then add one group when one factor is an odd number. “I can’t recall the product of 9 x 8, but I know that 4 groups of 8 are 32. To get 8 x 8, I can double 32 to get 64. To get 9 x 8, I can add one more group of 8 to get the product of 72”.


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Resources/Notes

Interview: Ask the student to explain how knowing 6 x 5 helps one figure out 12 x 5.

Math Focus4

Performance Ask the student to explain how knowing 8 x 10 helps one figure out 8 x 9. Show how you know using an array. Performance Model and solve a given multiplication problem using an array and record the process.

Lesson 4: Halving and Doubling Multiplication Facts N5 (5.1) PR6 (6.4/ 6.6)) TG pp. 23-26 SB pp. 178-180

Performance By using an 11pin x 11 pin geoboard and different colored elastics, have students place the elastics on the geoboard to show how they can find a product. For example: 8x7

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Strand: Patterns and Relations (Variables and Equations) Specific Outcome


It is expected that students will: PR6.Solve one-step equations involving a symbol to represent an unknown number. [C, CN, PS, R, V] PR6.4 Solve a given equation when the unknown is on the left or right side of the equation.

PR6.6 Represent and solve a given multiplication or division problem involving equal grouping or partitioning (equal sharing), using a symbol to represent the unknown. N5 Describe and apply mental mathematics strategies, such as: - skip counting from a known fact - using doubling or halving - using doubling or halving and adding or subtracting one more group - using patterns in the 9s facts - using repeated doubling to determine basic multiplication facts to 9 x 9 and related division facts (Cont’d) [C, CN, ME, R] N5.1Provide examples for applying mental mathematics strategies: • use ten facts when multiplying by 9 N5.2 Choose the most efficient mental math strategy for a given situation.

Students should be given opportunities to write equations where the missing number is in different places. Examples of equations involving multiplication: • Children are jumping rope at recess. There are 6 jump ropes and 3 children are playing with each rope. How many children can jump rope?

6x3= ⌂=6x3

All students have been introduced to skip counting by 10 and this knowledge needs to be connected to multiplying by 10. It is important to focus on the meaning of multiplication. Teachers should refrain from telling students that a 0 is added when multiplying by 10 as this shows little connection to the meaning of multiplication. After discussing the 10 facts, ask students how they can use a 10 fact to find the product of a 9 fact. When multiply a factor by 9, the product is always the factor times 10 minus the factor. Examples: Say, if I am trying to figure out 9 x 7 (9 groups of 7), I can first think that 10 x 7 is 70 therefore if I subtract one group of 7, I will have 63 (70 – 7= 63). Likewise when multiplying a factor by 8, the product is always the factor times 10 minus the factor times 2. Examples: Say, if I am trying to figure out 8 x 7 (8 groups of 7), I can first think that 10 x 7 is 70 therefore if I subtract two groups of 7, I will have 56 (70 – 7 – 7 = 56) After students have been focused on some strategies, they should be encouraged to think about strategies and selecting one that is most efficient depending on the situation. Being ‘efficient’ means they can give a quick response without resorting to non-efficient means. This practice in strategy selection should be ongoing so that students do not revert back to counting and ignore more efficient strategies they have learned. The development of multiplication and division fact proficiency in grade 4 allows students to work more efficiently with larger digit equations & problem solving later. If they can recall the facts efficiently, they are more likely to be able to think logically about problem solving without losing their train of thought.


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General Outcome: Represent algebraic expressions in multiple ways Suggested Assessment Strategies

Resources/Notes

Performance SWAT – scatter the possible products of 8 and 9 to a set of multiplication questions on large sheets of paper or on the board. Divide the class into two teams. Each team is given a fly swatter. One member from each team is asked a question and the first team member to “swat” the correct answer on the board gets a point. Team members alternate turns.

Lesson 4 (Cont’d): Halving and Doubling Multiplication Facts N5 (5.1) PR6 (6.4/ 6.6)) TG pp. 23-26 SB pp. 178-180

Performance Ask student to explain how knowing 8 x 10 helps one to figure out 8 x 9. Portfolio Teachers could have students describe in writing the patterns they can find for the products of 8’s and 9’s in a hundreds. Encourage students to reflect on how their work demonstrates that they were good mathematicians.(e.g., by looking for patterns, using mathematical vocabulary to describe my thinking, persevering even though the task was difficult, accepting a challenge, and asking good questions.) Presentation Prepare a list of facts from two or three strategies and ask students to name a strategy that would work for that fact. They should then explain its strategy and demonstrate its use. Presentation When you are comfortable that students are able to use a strategy, mentally, it is time to begin practicing it. To help practice multiplication facts, prepare sets of 24 cards with various multiplication questions on one side (Side ‘A’). On the reverse side (Side ‘B’), write the answer to the previous question. Distribute sets of cards to groups of students – small groups or the whole class. Deal all cards with the answer facing up (Side ‘B’). Decide who begins with the question “Who has…” for example, “…5 x 9?” The player who has the card with 45 on it would respond “I have 45”, turn that card over to ask the question posed on the other side. The cycle continues until the all cards have been turned over and all questions answered.

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Lesson 5 (optional): Using 10s to Multiply Lesson 6: Multiplying by 8 and 9 N5 (5.1) PR1 (1.1/ 1.2/ 1.4) TG pp. 30-33 SB pp. 182-184





It is expected that students will: N5 Describe and apply mental mathematics strategies, such as: - skip counting from a known fact - using doubling or halving - using doubling or halving and adding or subtracting one more group - using patterns in the 9s facts - using repeated doubling to determine basic multiplication facts to 9 9 and related division facts (Cont’d) [C, CN, ME, R] Achievement Indicators:

N5.1Provide examples for applying mental mathematics strategies: • use ten facts when multiplying by 9 N5.2 Choose the most efficient mental math strategy for a given situation

Students may benefit from modeling multiplication using counters on ten frames. Ten frames will be familiar to students from previous grades. They are useful visual models for students when multiplying things that occur in sets of nine. Example: 6x9

Facts with a factor of 9 include the largest products but can be easy to learn. The table of nine facts includes some nice patterns that are fun to discover. Use the following task to help students discover patterns involving 9 as a factor: Show: 9x1=9 9 x 2 = 18 9 x 3 = 27 9 x 4 = 36 9 x 5 = 45 9 x 6 = 56 9 x 7 = 63 9 x 8 = 72 9 x 9 = 81 Ask the students to find as many patterns as possible in the table. As you listen to the students be sure that these two patterns are found: • The tens digit of the product is always one less than the second factor e.g. 9 x 4 = 36 (3 is one less than 4) • The sum of the two digits in the product is always 9. These two ideas to get any 9 fact quickly. For 7 x 9, 1 less than 7 is 6, 6 and 3 make 9 so the answer is 63. “Because two separate rules are involved and a conceptual basis is not apparent, children may confuse the two rules or attempt to apply the idea to other facts. It is not, however, a “rule without reason”. It is an idea based on interesting patterns that exist in the base-ten numeration system.” (Elementary and Middle School Mathematics- Teaching Developmentally 2001, p. 141 -142, Van de Walle)


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Resources/Notes

Performance This activity is to be used when you have observed evidence that students understand and are able to use a strategy, mentally. Remember that as students are building on their repertoire of strategies, it is a good idea to provide opportunities for them to practice prior strategies as well. Provide pairs of students with counters (different color for each player), two paper clips and a 4 by 6 square grid showing the products of various multiplication facts. Below the grid, list 7 factors which correspond with the products on the grid. Player A places a paper clip on one of the factor numbers and Player B then places a paper clip on another factor. Player A multiplies both factors and places his colored counter over the product on the grid. The games continues as Player B chooses a factor, Player A chooses another factor; Player B multiplies both factors, finds the product, covers it, and so on. The winner is the first person to connect four of their counters in a row, horizontally, vertically or diagonally. As you observe the students playing, ask “What strategies helped you with that one?”

Math Focus4

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Performance Provide the students with 2 decks of cards with numerals 0 – 9 on them. Students play in pairs. Player A draws to cards from the deck without letting her opponent see them. Player A secretly multiplies the two numbers and tells player B the product only. Player A then places her hands behind her back (with one card in each hand) and asks Player B to chose a hand. Player B can then be told that number. Player B now knows the product and one factor and then figures out the hidden number. If Player B is correct, she keeps the pair of cards. If she is incorrect, then Player A gets to keep the pair of cards. The person with the most cards at the end of the game is the winner.

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Curious Math Finger Multiplication N5 (5.1) TG pp. 34-35 SB p. 185

56

54

6

Lesson 6 (cont’d): Multiplying by 8 and 9 N5 (5.1) PR1 (1.1/ 1.2/ 1.4) TG pp. 30-33 SB pp. 182-184

Additional Reading: Elementary and Middle School Mathematics- Teaching Developmentally 2001, Van de Walle





It is expected that students will: N7 Demonstrate an understanding of division (1-digit divisor and up to 2-digit dividend) to solve problems by: • using personal strategies for dividing with and without concrete materials • estimating quotients • relating division to multiplication. [C, CN, ME, PS, R, V]

Division is a process that occurs naturally in everyday life such as sharing cookies. It can be effectively taught in conjunction with multiplication and in a problem solving context. Consider reading The Doorbell Rang by Pat Hutchins to introduce division. Students should have many opportunities to solve and create word problems for the purpose of answering real-life questions of personal interest. These opportunities provide students with a chance to practise their computational skills and clarify their mathematical thinking. Two situations, as represented in the picture below, call for the operation of division and students need to know these two meanings (although the end product is the same, the pencils were divided differently) :

Achievement Indicators: N7.1 Solve a given division problem without a remainder, using arrays or base ten materials, and connect this process to the symbolic representation. N7.3 Solve a given division problem, using a personal strategy, and record the process.

•

Equal Sharing: identifying how many in each group E.g. Ms. Brown has set up math centers. She has 32 pencils to be shared equally among 4 centers. How many pencils will each center receive? (To solve this problem, model sharing the 32 pencils by placing one at a time on each plate until all the pencils have been shared) Answer: 32 ÷ 8 = 4

•

Equal Grouping: identifying how many groups: E.g. Ms. Brown is setting up math centers. She has 32 pencils. Ms. Brown knows she needs to put them into equal groups of 8. How many centers can she set up? (To solve the problem this way, model taking groups of 8 pencils and putting them into the centers. She found out that she has enough to make 4 centers.) Answer: 32 ÷ 8 = 4

N7.5 Create and solve a division problem involving a 1- or 2-digit dividend, and record the process.

Both equations look the same but the procedure used to divide the pencils was different. Problems may be modeled with sets of counters, number lines, arrays or counters. In the above example, students may role play using real objects or base ten blocks to represent pencils. Students should record responses using pictures, numbers or words.


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Resources/Notes

Pencil/Paper: Have students create different story problems that demonstrate the two meanings of division (equal sharing and equal grouping). Students could trade problems with a classmate and solve the classmate’s problems. Performance: Draw an array to demonstrate that 18 ÷6 is 3. Performance: Gregory has 35 hockey cards. He shares the cards equally among seven friends. Each friend needs 6 hockey cards to complete their collection. Does Gregory have enough cards? Explain your answer using pictures, numbers and words.

Although, the text does not address the connection between multiplication and division until lesson 8, teachers may deem it appropriate to make the link in lesson 7

Be selective with activities on page 190. Questions 1, 2, 3 and 7 are suggested.

Interview: Ask students to model 36 ÷ 6 using base ten blocks. Performance Ask students to use a model to explain to a classmate how to share 45 marbles among five people. Discuss the different strategies used. Journal Present the following division equation: 64 ÷ 8 = ____ Ask students to solve the equation using two personal strategies. Explain which strategy is most efficient and why.

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Lesson 7: Sharing and Grouping N7 (7.1/ 7.3/ 7.5) TG pp. 40-43 SB pp. 188-190

Math Game (Optional): Comparing Products N5(5.1) TG pp. 44-45 SB pp. 191





It is expected that students will: N4 Apply the properties of 0 and 1 for multiplication and the property of 1 for division. [C, CN, R] 4.1 Determine the answer to a given question involving the multiplication of a number by 1, and explain the answer.

When you divide a number by 1 the answer is the number you started with. For example:

4.2 Determine the answer to a given question involving the multiplication of a number by 0, and explain the answer 4.3 Determine the answer to a given question involving the division of a number by 1, and explain the answer. N5 Describe and apply mental mathematics strategies, such as: - skip counting from a known fact - using doubling or halving - using doubling or halving and adding or subtracting one more group - using patterns in the 9s facts - using repeated doubling to determine basic multiplication facts to 9 x 9 and related division facts (Cont’d) [C, CN, ME, R]

Provide students with opportunities to explore the relationship between multiplication and division. Any multiplication situation can also be viewed as a division situation, and vice versa. To adults this may seem obvious however grade 4 students, when beginning to work with these two concepts may see them as totally different operations. The differences between multiplication and division are subtle and it is not necessary to learn to distinguish between them or label them. Provide opportunities for students to work with problems that are related which will eventually lead them to discover the relationship. One way to approach this concept is by introducing a fact family - a set of four number sentences, or equations, that can be used to described the same situation. Use the following model, with square tiles, to show a fact family:

(Small, 2008 p.123)

N5.1 Provide examples for applying mental mathematics strategies: - relating division to multiplication


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Resources/Notes

Performance N7.1/N7.2 Provide a set of base-ten blocks. Teachers could ask the student to model 3 different division questions of his/her choice and to write the division sentence for each. Performance Write the multiplication and division number sentences shown by the following array.

******* ******* ******* ******* Interview How would you find the answer to the following division fact by relating it to multiplication? 30 ÷ 5 = . Explain your thinking.

Lesson 8: Division and Multiplication N4 (4.3) N5 (5.1) N7 (7.1/ 7.3/ 7.5) PR6 (6.1/ 6.4/ 6.6) TG pp. 46-49 SB pp. 192-195) Lesson 9: Patterns in a Multiplication Table N4 (4.1/ 4.2/ 4.3) N5 (5.1) PR1 (1.1/ 1.2) TG pp. 50-52 SB pp. 196

Performance Using a set of numbers such as 9, 6, 54 write four related facts. Performance Provide students with a set of 24 counters or multilink cubes. Ask them to make an array and record the multiplication and related division equation. (Answer: 3 x 8 = 24, 8 x 3=24, 24 ÷ 3 = 8, 24 ÷ 8 = 3). Rearrange the array to show a different equation. Continue until all possibilities have been explored (1x24, 24x1, 2x12, 12x1, 3x8, 8x3, 4x6, 6x4).

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It might be useful to work with lesson 8 and 9 together. Outcome N4 appears in both lessons but is addressed more overtly in lesson 9.





It is expected that students will: N7 Demonstrate an understanding of division (1-digit divisor and up to 2-digit dividend) to solve problems by: • using personal strategies for dividing with and without concrete materials • estimating quotients • relating division to multiplication. [C, CN, ME, PS, R, V]

There are many types of multiplication and division situations. Each situation can be represented by either a multiplication or division number sentence with one number missing. Each can be solved by either multiplying or dividing the two known quantities. The relationship between multiplication and division facts can be represented by using arrays. Any multiplication situation can also be viewed as a division situation, for example: Model the following array, pretending the stars in the array are chairs that you are setting up in the gym:

******** ******** ******** ********

Achievement Indicators: N7.1 Solve a given division problem without a remainder, using arrays. N7.3 Solve a given division problem, using a personal strategy, and record the process. N7.5 Create and solve a division problem involving a 1- or 2-digit dividend, and record the process.

PR6.Solve one-step equations involving a symbol to represent an unknown number. [C, CN, PS, R, V] Achievement Indicators:

PR6.1 Solve a given one-step equation using manipulatives.

Probe for understanding by asking the following questions: • How many rows of chairs are there? (4) • How many chairs are in each row? (8) • Write a multiplication sentence to show that there are a certain number of rows with a given number of chairs in each row. (4 x 8 = 32) Next, ask students to look at the up-and-down columns in the array. (Point to the columns.) • How many columns are in the array? (8) • How many chairs are in each column? (4) • Write a multiplication sentence to show that there are a certain number of columns with a given number of chairs in each column. (8 x 4 = 32) • Did you get the same answer when you changed the order in which you multiplied the two numbers? Explain. To make a related division sentence, ask the following questions: • How many chairs are there? (32) • How many chairs are in each row? (8) • Write a division sentence to show how many rows there are (32 ÷ 8 = 4) • Write another division fact using the array to show how many chairs are in each row. (32 ÷ 4 = 8)


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Resources/Notes

Performance Divide students into groups and distribute, to each group, 24 pictures/ drawings of people on 4 by 4 cm squares. (Students may enjoy creating their own people by drawing different faces on blank squares!) Ask students to divide their drawings so that there are 12 ‘on a team’. Record the division equation, discussing the meaning of each number in the equation. (The student may explain: “There are 24 people altogether. If I place 12 on each team, there will be 2 teams.) Repeat the activity with 8 on a team, 6 on a team, and 4 on a team. Write a division equation each time that answers “How many teams are there?”

Lesson 8 (Cont’d): Division and Multiplication N4 (4.3) N5 (5.1) N7 (7.1/ 7.3/ 7.5) PR6 (6.1/ 6.4/ 6.6) TG pp. 46-49 SB pp. 192-195) Lesson 9 (Cont’d): Patterns in a Multiplication Table N4 (4.1/ 4.2/ 4.3) N5 (5.1) PR1 (1.1/ 1.2) TG pp. 50-52 SB pp. 196 It might be useful to work with lesson 8 and 9 together. Outcome N4 appears in both lessons but is addressed more overtly in lesson 9.

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Students are familiar with patterns found in a multiplication chart as it has been used earlier in this chapter. Revisit the multiplication chart to help students discover more patterns that can help them recall multiplication facts:

PR1 Identify and describe patterns found in tables and charts, including a multiplication chart [C, CN, PS, V]

Multiplication chart

Achievement Indicators: PR1.1 Identify and describe a variety of patterns in a multiplication chart. PR1.2 Determine the missing element(s) in a given table or chart.

• • •

• • •

•

•

the first row and column are all 0s the ones row and column are the same as the factors the rows and columns match (e.g. the numbers in the 7th row match those in the 7th column factors increase by 1 in each row and column the row and column for the same factor have the same products the numbers on the left-right diagonal increase by 1, 3, 5, 7... row 4 is double row 2, row 6 is double row 3 the grid is symmetrical (i.e., numbers are the same both above and under the left-right diagonal) when you add the corresponding products of rows 2 and 3, you get the product in row 5; for example, 2 × 4 (8) plus 3 × 4 (12) is the same as 5 × 4 (20) when you "cross multiply" any 4 numbers that form a square on the grid, the product is always the same; for example, 2 × 6 = 3 × 4 - also, when you "cross add" these numbers and subtract the sums, you get 1

Using the multiplication chart helps students see the relationship between multiplication and division. For example: How does this chart help you to calculate 24 ÷6? Possible student response: “I looked in the 6 row until I got to 24. Then I looked up the column and found that the quotient is 4.


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General Outcome: Use patterns to describe the world and to solve problems Suggested Assessment Strategies

Resources/Notes

Performance Provide pairs of students with two number cubes with the numbers 4, 5, 6, 7, 8, and 9 on each cube, and one game board as shown below:

35 49 54 30 25

25 30 54 49 35

64 72 81 28 24

24 28 81 72 64

36 56 45 32 20

20 32 45 56 36

42 48 63 40 16

16 40 63 48 42

Player A rolls the number cubes, multiplies the numbers, and covers the product on the game board (there are two separate playing areas on each game board, one for each player). Players alternate turns rolling number cubes, multiplying, and covering products. When a player does not roll an uncovered product, that player loses a turn. When both players have rolled unsuccessfully, two times in a row, the round ends. The winner is the person who has covered the most numbers. Play the game for several rounds. Performance Some students are uncomfortable giving answers in large groups but will enjoy the opportunity to play this game with a partner. Teacher can observe small groups and listen as students discuss strategies used in this game. Prepare a set of flash cards for each pair of students containing multiplication facts with products to 81 (e.g. 5 x 4, etc.) Hole punch the cards and place each set of facts on a metal shower ring. Students work in pairs and, one at a time, flashes the cards for their partner. The partner gives the answer and tells which strategy he/she used to figure it out. Both students are encouraged to think of the answer and in some cases may need to discuss how they both arrived at an answer but by using a different strategy.

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Lesson 8 (Cont’d): Division and Multiplication N4 (4.3) N5 (5.1) N7 (7.1/ 7.3/ 7.5) PR6 (6.1/ 6.4/ 6.6) TG pp. 46-49 SB pp. 192-195) Lesson 9 (Cont’d): Patterns in a Multiplication Table N4 (4.1/ 4.2/ 4.3) N5 (5.1) PR1 (1.1/ 1.2) TG pp. 50-52 SB pp. 196 It might be useful to work with lesson 8 and 9 together. Outcome N4 appears in both lessons but is addressed more overtly in lesson 9.

Math Game (optional): Matching Pairs N5 (5.1) PR6 (6.4) TG pp. 53-54 SB pp. 197






Educators have identified several problem solving strategies that prove to be useful in a variety of situations. Each strategy can be discussed with students, preferably after it has arisen naturally and the student has used it. There is some value in naming the strategies so students can easily recall and use them. Throughout the grade levels students probably have encountered strategies such as: • Act it out • Use a model • Draw a picture • Guess and Test • Look for a pattern • Use an open sentence • Make chart/table or graph • Solve a simpler problem • Make an organized list • Use logical reasoning

N6 Solve one-step equations involving a symbol to represent an unknown number. [C, CN, PS, R, V]

Achievement Indicators: N6.2 Solve a given one-step equation, using guess and test. N6.6 Represent and solve a given multiplication or division problem involving equal grouping or partitioning (equal sharing), using a symbol to represent the unknown.

Another strategy that students should learn in the elementary grades is called working backwards. Sometimes, by starting with the result, a student can work backwards to determine information about the original situation. Working Backwards is a more complex problem solving strategy and is covered in this unit but applies to other strands as well. You may need to start with smaller numbers The resource uses a diagram to represent each step in the process. Some students require this type of visual while others may find it confusing. Example: I doubled a number added 10 divided by 3 and then subtracted 2. The result is 18. What number did I start with? Start with the end result of 18: The last step was to subtract 2, so I do the opposite and add 2 (18 + 2 = 20) The second last step was to divide by 3 so I multiply by 3 (20 x 3 = 60) The step before that says add 10 so I subtracted 10 (60 – 10 =50) The last step tells me to double the number but I am doing the opposite so I find half (1/2 of 50 = 25). The start number was 25.


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Resources/Notes

Performance Carrie worked at the bakery. Every morning she baked a lot of cakes. One day she noticed that during the first hour the store was open, ½ of the cakes were sold; during the second hour ½ of the cakes that were left were sold. During the third hour, ½ of the remaining cakes were sold and in the fourth hour the same thing happened – ½ of the remaining cakes were sold. Now there were 3 cakes left. How many cakes were sold that day?

Lesson 10: Solving Problems by Working Backwards N5 (5.1) N7 (7.1/ 7.3/ 7.5) PR6(6.2/ 6.6) TG pp. 55-58 SB pp. 198-199

(Answer: The information needed to begin solving this problem is found at the end. How many cakes were left at the end of the day? (3). If 3 is ½, then in the last hour 6 cakes must have been on the shelf. In the hour before that, 12 cakes must have been available. In the second hour (if 12 was half) 24 cakes must have been on the shelf. In the first hour (if 24 was half) then 48 cakes were baked that morning. There are 3 cakes left at the end of the day so 45 were actually sold). (Teachers may suggest using a T-chart since students often use more than one strategy. A T-chart would help organize the data as well as create a link to the previous unit.) Chapter Review: N4 (4.1/ 4.2) N5 (5.1) N7 (7.3) PR6(6.2/ 6.4/ 6.6) TG pp. 59-64 SB pp. 200-202

Chapter Task: Planning a Bone Puzzle Game N5 (5.1) N7 (7.3) TG pp. 65-67 SB p.203 Other Unit Assessment Be selective.

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FRACTIONS AND DECIMALS Suggested Time: approx 3½-4 weeks October



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FRACTIONS AND DECIMALS

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Unit Overview Big Idea

In everyday life, we often have measures that are less than one. In Grade Four, the focus is on students initially developing a firm understanding of these numbers. At first, students will learn that fractions are one way to represent numbers less than 1. Students need to understand that a fraction represents one idea although it uses 2 numbers. The important part is the relationship between these two numbers. Teachers should provide situations in which students should compare fractions using concrete representations such as: • Area models (part of a whole area) • Length models (part of a length measurement) • Set models (part of a set like objects) “The ability to tell which of two fractions is greater is another aspect of number sense with fraction. That ability is built around concepts of fractions, not on an algorithmic skill or symbolic trick.” (Walle and Lovin, 2006) In the second part of this unit, students will be introduced to another way to represent numbers less than one - through decimals. An introduction to decimals requires familiarity with the concept of fractional tenths (lesson 7). Some students will be comfortable with the concept of tenths and will be ready to move into a study of decimal hundredths fairly quickly (lesson 8). Students will learn that decimals allow for calculations that are consistent with whole number calculations.




3


Strand: Number Specific Outcome It is expected that students will: 4N8 Demonstrate an understanding of fractions less than or equal to one by using concrete, pictorial and symbolic representations to: • name and record fractions for the parts of a whole or a set • compare and order fractions • model and explain that for different wholes, two identical fractions may not represent the same quantity • provide examples of where fractions are used [C, CN, PS, R, V]


4N8.4 Name and record the shaded and non-shaded parts of a given whole.

4N8.6 Represent a given fraction pictorially by shading parts of a given whole.

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Suggestions for Teaching and Learning The first goal in the development of fractions should be to help children construct the idea of fractional parts of the whole- the parts that result when the whole or unit has been partitioned into equalsized portions or fair-shares. Children’s familiarity with sharing amongst friends and separating a quantity into two or more equal parts is a real life connection to draw upon when beginning the concept development of fractions. Even though a fraction has two numbers, it is one idea – the relationship between the two numbers. This can sometimes be confusing for students. The denominator tells how many parts the whole is divided into and the numerator tells how many there are of those equal parts. The first fraction that students meet is usually 12 . Students will be familiar with the concept of

1 2

as they frequently share things into two

equal groups. Later students will use comparing fractions.

1 2

as a benchmark when

To strengthen their fraction number sense, it is also recommended that the size of the whole be changed regularly. In Grade Four, the focus is on students initially developing a firm understanding of fractions less than one. Students should be encouraged to develop visual images for fractions and be able to tell about how much a particular fraction represents. Representing a given fraction pictorially or by shading parts of a given whole, will help conceptualize this understanding. Provide opportunities for students to name a fraction from a given picture, such as:

Colour pictures to show a fraction. E.g. 82 is green, 18 is blue, and 58 is red. Possible answer:




Resources/Notes

Performance Have students summarize their understanding of fractions of a whole by completing a Frayer Model (Sample answers included below, however other answers are possible). (N8.4)

Math Focus 4 Chapter Opener TR. p 9 Getting Started TR. pp.10-11

Lesson 1 Fractions of a Whole N8 (8.4/8.6/8.13/ 8.14) TR pp. 12 – 15 SB pp. 208 - 211 Additional reading: Teaching Student-Centered Mathematics, Van de Walle and Lovin, 2006 p. 252) Teaching Children Mathematics, NCTM

Performance Given the following diagram:

(a) write a fraction representing the shaded part of the diagram (b) write a fraction representing the unshaded part of the diagram (N8.4) Student-Teacher Dialogue Show a strip of 9 squares. Ask students to indicate explain how they know. (N8.4)


3 9

of the strip and

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It is expected that students will: 4N8 Continued Achievement Indicators: 4N8.4 Name and record the shaded and non-shaded parts of a given whole.

4N8.6 Represent a given fraction pictorially by shading parts of a given whole.

4N8.13 Provide examples of when two identical fractions may not represent the same quantity; e.g., half of a large apple is not equivalent to half of a small apple, half of ten blueberries is not equivalent to half of sixteen blueberries.

Invite students to show the same fractional part of a whole in as many ways as they can. E.g. The shaded part of the rectangle in these grids below shows onethird

“A key idea about fractions that students must come to understand is that a fraction does not say anything about the size of the whole or the size of the parts. A fraction tells us only about the relationship between the part and the whole. ” (Van De Walle, 2006, p.267) Consider this example: Both Alex and Jennifer attend a pizza party. They decide that they both want 14 of a pizza. They go to different areas to pick up their pizza. Alex takes 14 of a pepperoni pizza, and Jennifer takes 14 of a veggie pizza. When they meet back at their table, they realize that they do not have the same amount of pizza, but that Jennifer’s is larger. They come to the realization that Jennifer’s slice came from a larger pizza, and that they did not check the size of the wholes before selecting their choice. Van de Walle (2006, p 267) refers to this as the “pizza fallacy” in that whenever two or more fractions are discussed in the same context, the correct assumption (the one that Jennifer and Mark made) is that the fractions are all parts of the same size whole.

It is important for students to be able to explain why two identical fractions do not represent the same amount (when the wholes are different sizes). By providing everyday contexts in which the whole region varies in size, you stimulate the students’ thinking to generalize that when comparing fractions, the whole must be the same size for each fraction. Ask: Are halves always the same? Discuss student responses and demonstrate by cutting different kinds of fruit in half. For example, show students an orange and a watermelon and cut them in half. Discuss that the halves are different sizes even though they both represent the fraction 12 . This can also be demonstrated by using different sized glasses of water.

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General Outcome: Develop number sense Suggested Assessment Strategies Journal Sam ate 34 of his pizza and Sara ate 34 of her pizza. Sam said that he ate more pizza than Sara. Use pictures and words to explain how Sam could be correct. (N8.13/8.14)

Resources/Notes Math Focus 4 Chapter Opener TR. p 9 Getting Started TR. pp.10-11

Lesson 1 (Cont’d) Fractions of a Whole N8 (8.4/8.6/8.13/ 8.14) TR pp. 12 – 15 SB pp. 208 - 211 Additional reading: Teaching Student-Centered Mathematics, Van de Walle and Lovin, 2006 p. 252) Teaching Children Mathematics, NCTM


7





In previous grades, students have worked only with wholes or regions and have had little experience working with parts of sets. The part of a set model is new to Grade Four, and therefore opportunities should be provided to allow this concept to be developed carefully.

4N8 Continued 4N8.3 Name and record the shaded and non-shaded parts of a given set.

An important point relating to fractions of a set is that the equal parts into which the whole is divided are equal but do not have to be identical. Students may be easily confused by sets that contain different items or are different shapes. For example:

What fraction of the set are female? What fraction of the set are children? What fraction of the set are wearing glasses? 4N8.1Represent a given fraction, using concrete materials.

Concrete materials must be used to develop fractional concepts adequately, therefore a variety of materials are effective. Pattern blocks are very useful models. Using pattern blocks as concrete representations for either fractions of a whole or fractions of a set can help students make connections between the two models. For example:

The triangle is

1 3

of the trapezoid (fractions of a whole)

The triangle is 14 of this set of 4 blocks (fractions of a set). Other appropriate manipulatives when working with fractions include fraction circles, paper (for folding), fraction pieces, square tiles, egg cartons, Cuisenaire rods, counters, fraction bars or strips, money, number lines, geoboards, and grid/dot paper. 4N8.2 Identify a fraction from its given concrete representation.

8

Show students a set of counters such as 8, 12, 16, or 20 and four paper plates. Share the set of counters into four equal groups, placing each group on a separate paper plate to show quarters. Point to one plate and ask students to identify the fraction. Repeat for a different number of plates. Extension: Have students draw diagrams to match each concrete representation.




Resources/Notes

Using the pattern blocks identified in column 2, students create the whole (the hexagon) shown in column 1. Complete a table such as the one below, telling what fractional part of the whole each shape represents. (Possible answers: a trapezoid is 12 of the hexagon, a

Math Focus 4

rhombus is

1 3

of the hexagon and a triangle is

1 6

of the hexagon.

(N8.2)

Lesson 2 Fractions of a Group N8 (8.1/8.2/8.3/8.5/ 8.9/ 8.14) TR. pp. 16 – 20 SB pp. 212 - 214

Student-Teacher Dialogue Ask the students to “shake and spill” a number of colored counters and tell what fraction of the set is green, blue, yellow and red. (N8.1) Journal Have students create a design with at least two different types of pattern blocks and draw the design in their journal. What fraction of the design is Red? Blue? Yellow? Green? (N8.2) Ask students to explain why both pictures show

Student-Teacher Dialogue In this set of shapes, how many are grey? (N8.3)


3 10

(N8.2)

Teachers should supplement with additional practice as needed.

9




It is expected that students will: 4N8 (Continued)

4N8.9 Order a given set of fractions that have the same denominator, and explain the ordering.

N8.9 An important fraction principle for students to understand, when comparing fractions, is that they can only be compared if the whole is known in each situation. When fractions have the same denominator, the one with the larger numerator is greater. E.g. 23 is greater then 13 because both fractions are thirds and you have more thirds in two-thirds than in one-third. Choose seven students and provide them each with a yarn necklace displaying a fraction, all having the same denominator. Instruct students to wear their necklaces so that the fractions are visible. Choose two other classmates to order the students from least to greatest. Invite discussions as to how they conducted the ordering. Repeat, using different fractions, ordering them from greatest to least. Ask three students to build the following towers using only red and yellow multi-link cubes: Tower #1- represents 102 yellow cubes 5 10 represents 108

Tower #2- represents

yellow cubes

Tower #3 -

yellow cubes

First using the , 105 , 108 , ask students to secretly order the fractions from greatest to least and record their answer. Next, stand the towers side by side to check the ordering. This may be repeated with more fraction towers and ordering from least to greatest. fractions 102

4N8.14 Provide, from everyday contexts, an example of a fraction that represents part of a set and an example of a fraction that represents part of a whole.

To help develop understanding of fractions, include everyday contexts for fractions, then use concrete representations and connect them to pictorial and symbolic representations. E.g. Ask a group of 10 students to come to the front of the class. Ask: What fraction of the students are girls? Boys? Wear glasses? Have dark hair? Have students present their findings using pictures (shading parts of a set), and symbols (write fractions). Some other valuable contexts that can be used to teach fractions of a set include: - sharing food (sharing a dozen cookies by giving everyone 1 of a dozen) 4

10

-

teams ( 109 of the volleyball team played last year)

-

music (4 quarter notes in each measure of a

4 4

time signature)



General Outcome: Develop number sense Suggested Assessment Strategies Student-teacher Dialogue Have various concrete materials available. Present pairs of fractions with like denominators to students. Have students: • decide which fraction is greater • explain why they think this is so • test their choice using any model they wish to use (N8.9) Presentation Randomly divide the class into 3 different sized groups, and ask each group to write a fraction that represents the number of boys and girls in their group. Then ask them to present their fractions to the class and tell which fraction is greater, the fraction representing the boys, or the one representing the girls. (e.g. “Our group has 3/8 boys and 5/8 girls. There are more girls than boys”). The groups should then be able to explain their reasoning based on the concept that when the denominators are the same, the numerator determines the larger number. (N8.9)

Resources/Notes Math Focus 4 Lesson 2 (Cont’d) Fractions of a Group N8 (8.1/8.2/8.3/8.5/ 8.9/ 8.14) TR. pp. 16 – 20 SB pp. 212 - 214

Performance Sharing Brownies (N8.5) - This activity will allow students to use higher level thinking skills as they construct their knowledge about fractions of a set. Provide students with several paper squares to represent fudge brownies. Have students experiment by cutting the brownies to solve the following problems: - How can 4 people share 3 brownies? - How can 3 people share 2 brownies? - How can 12 people share 6 brownies? - How can 6 people share 4 brownies? (Differentiate task, using numbers according to individual student ability). Write responses on a recording sheet such as the following: Draw a picture of the divided brownies.

_________ brownies shared by ________ people. One person’s share is ___________ (write the fraction).

Student-Teacher Dialogue (N8.9) What possible denominators could be used in the statement below? (Several answers possible) 1 < 1 ? ?


11




It is expected that students will: 4N8 Continued


Students should be encouraged to think about how they use fractions in their everyday lives. Have students look for fractions in magazines and newspapers and discuss how they display important information about real life. Some valuable contexts that can be used to make connections with fractions of a whole include: - sharing food (dividing a pizza into 8 equal pieces) - measuring time ( 34 of an hour) -

measuring food for recipes ( 12 cup of butter)

-

money ( 14 of a dollar)

-

art ( 34 of the design is red) music (eighth, quarter and half notes)

Write the fractions 4N8.5 Represent a given fraction pictorially by shading parts of a given set.

8 12

and

4 12

on the board and brainstorm real-life

situations involving parts of a set that these fractions could represent. Using student generated stories, have them represent the fractions pictorially by shading parts of the set. E.g. a carton of eggs fell off the shelf in a grocery store, and 8 of them were broken. What fraction of eggs were not broken? Repeat using other fractions. This activity could be reversed whereby teachers could draw the shaded and nonshaded parts of the set, ask students to brainstorm real-life situations, and then write the fractions represented by their models. Ask students to make illustrations to show the following: a. 23 of the bananas are ripe b. c. d.

12

4 5 1 3 4 9

of the floor tiles are striped of the balls are basketballs of the fruit are oranges (N8.5)




Resources/Notes

Fraction Concentration Create fraction cards sets containing fractions of a whole and corresponding fractions of a set. Shuffle the cards, and place them face down on a desk. The first player turns over any two cards and checks to see if the cards match. If a match is made, the player must tell his/her opponent what fraction each card represents. If both agree that the answer is correct, the player keeps the cards and takes another turn. Players continue to take turns until all the possible matches have been made. The player with the most matches wins the game.

Math Focus 4 Lesson 2 (Cont’d) Fractions of a Group N8 (8.1/8.2/8.3/8.5/ 8.9/ 8.14) TR. pp. 16–20 SB pp. 212 - 214

Sample play:

Player says: “My match represents the fraction

3 4

(three-fourths)” (N8.5/8.6)

Performance Fraction Sundaes-Provide a variety of colored construction paper and instruct students to make scoops of ice cream to represent various flavors (e.g. brown for chocolate, green for pistachio, etc.). (You may provide a stencil). On a 12”x 17” paper students should design a dish to hold their sundae. Create the sundae by gluing the scoops on the dish. On the side of the paper or on the sundae dish, students use fractions to represent each flavor of ice cream in their sundaes. (N8.5)


13





Once students have developed a good understanding of naming and recording both fractions of a whole and of a set, they should be able to communicate how the fractions they have modeled are alike and different. They should be encouraged to communicate their findings pictorially, symbolically, and with words. Ask students to draw as many pictures as they can of 46 , then share their pictures with a partner and discuss the following: 1. How are your pictures the same? sample responses: -they both show 4 parts out of 6 -they both show parts of a whole -they both show parts of a set 2. How do your pictures differ? sample response: -one shows parts of a whole, and the other shows parts of a set

4N8.7 Explain how denominators can be used to compare two given unit fractions.

A unit fraction has a numerator 1, as in 15 . Therefore with a unit fraction, the larger the denominator the smaller the fraction part. Ask students: Three girls took part in a skip-a-thon. Sydney, exercised 15 of an hour, Paula exercised 13 of an hour, and Beth exercised for

1 2

of an hour. Who won the skip-a-thon? (Beth)

Models such a Fraction Strip Kit, described below, will help students compare fractions in symbolic form: Have students make fraction strips by cutting strips of varied colored construction paper and folding them to represent fractions as parts of a whole. Ask students to take a strip of a particular color, and label it 1, or 11 . This strip represents the whole. Then take a second strip of the same length, fold it in half, and label each section 12 . Next, choose a third strip and have the students fold it into four equal pieces, and label each section 14 . Continue this process with eighths and sixteenths. Discuss the meaning of each fraction as students fold the strips. Having students cut and label the pieces helps them relate the fraction paper and compare the sizes of fractional parts. For example, the students can see that: 18 is larger than 161 and that two 14 pieces are equal to a 12 piece. (Tip: Have students label the back of the pieces with their initials)

14




Resources/Notes

Performance Fraction Game - “Cover Up”- This game requires using the cut up fraction pieces from the fraction kit made on the previous page, and involves two or more players. Each player starts with a whole strip from the fraction kit. The goal is to be the first to cover the whole strip completely with the other pieces of the fraction kit. No overlapping pieces are allowed.

Math Focus 4 Lesson 3 Sorting Fractions N8 (8.1/8.5/8.6) TR pp. 21 – 23 SB p. 215

Rules for play: 1. Children take turns rolling a cube labeled with fractions: ( 12 , 14 , 18 , 18 , 161 , 161 ) 2. The fraction face up on the cube tells what size piece to place the whole strip. 3. When the games nears the end and a student needs only a small piece, such as 18 or 161 , rolling 12 or 14 won’t do. The student must roll exactly what is needed.(Burns, About Teaching Mathematics, 2000 p. 227) (N8.1/8.7) Student-Teacher Dialogue Pose the following question: What would you rather have,

1 4

of a

pizza, or 13 of a pizza? Explain the reason for your choice using pictures and words. (N8.7) Journal When can 14 give you a bigger piece of something than 12 ? Draw diagrams to help explain your answer. (N8.7)


Lesson 4 Comparing and Ordering Fractions N8 (8.1/8.2/8.6/8.7/8.8) TR pp. 24 – 28 SB p. 216 - 218 Lessons 3 and 4 can be addressed together. Lesson 3 is brief but helps to set the stage for lesson 4.

15





4N8.8 Order a given set of fractions that have the same numerator, and explain the ordering.

Children have a strong mindset about numbers that may cause them difficulties with the relative size of fractions. In their experience, larger numbers mean “more”. A common misconception is for students to transfer previously learned whole-number concepts to fractions, thinking seven is more than four, so sevenths should be larger than fourths. The inverse relationship between number of parts and size of parts is better understood by students when they explore and discover this on their own rather than be told. Provide students with 4 strips of ribbon of equal length. Instruct them to fold and cut the ribbons to represent the fractions below, and then order them from least to greatest. Keep one strip “whole” for comparison. Ask students to explain their thinking. (1) 46 (2) (3)

4 8 4 10

Lead students into a discussion on the concept that fractions with different denominators have the whole divided into the different sized parts. If you have the same number of parts (like numerators) in two situations but the parts in one fraction are smaller than the parts in another fraction, than the fraction with the smaller denominator (showing larger parts) is greater. E.g. Sixths are greater than tenths, therefore four-sixths is greater than four tenths.

Review with students the concept discussed in earlier lessons that the whole must be the same when comparing fractions.

16




Resources/Notes Math Focus 4

Performance Pose the following problem: Matthew, Chris and Peter recorded their batting scores at the batting cage: Matthew- 82 2 6

Chris Peter-

Lesson 4 Comparing and Ordering Fractions N8 (8.1/8.2/8.6/8.7/8.8) TR pp. 24 – 28 SB p. 216 - 218

2 5

Matthew put the scores in order from greatest to least, and said that his batting average was the highest, Chris’ second, and Peter’s third. Was Matthew’s ordering correct? Explain your thinking. (N8.8)

Curious Math TR p. 29 SB p 219


17




It is expected that students will: 4N8 (Continued) 4N8.10 Identify which of the benchmarks, 0 or 1, is closer to a given fraction.

The most important reference points for fractions are 0, 12 and 1, and are referred to as benchmarks. Comparing fractions to these three benchmarks can provide students with a lot of information. Understanding why a fraction is close to 0, 12 , or 1 is a good beginning for fraction number sense. It begins to focus on the relative size of fractions in an important, yet similar manner. Invite 3 students to represent the 3 benchmarks by holding a skipping rope at the beginning, middle and end. Two students, each holding one end, represent endpoints, 0 and 1 and have a third student stands in the middle to represent 12 . Give several students fraction cards and ask them to stand in front of the person representing the benchmark closest to their fraction. Example: A student might say, “ 102 is closer to 0, so I’ll stand in front of Amy who is holding the zero end of the rope.” Identifying which of the benchmarks 0, 12 , or 1 is closer to a given fraction can be done by using the following strategies: (1) with paper fraction strips Provide the students with fraction strips showing halves and other fractions, such as thirds, quarters, fifths and tenths. Using the fraction strips, have the students order two fractions by comparing each fraction to one-half, such as one-quarter and two-thirds or three-fifths and eight-tenths. Through discussion, have the students generalize that some fractions can be ordered by deciding if they are greater than or less than one-half. (2) by looking at the denominator and numerator Ask the students to explain how they would know if a fraction was greater than or less than one-half without using the paper fraction strips. Guide students to explore and conclude, on their own, that if the numerator is less than half the denominator, then the fraction is less than one-half. Similarly, if the numerator is greater than half the denominator, then the fraction is greater than one-half. If the numerator is half the denominator, then the fraction shows another name for one-half.

18





Performance Prepare a lunch bag with several fraction cards inside. Divide the class into groups of three and assign each player a benchmark title of 0, 12 , or 1. Have players draw a fraction card from the bag and decide which benchmark the fraction is closest to. The player with that benchmark name receives a point. Play continues until a player has reached a predetermined number of points. (N8.10) Journal A mother instructed her two children Gregory and Brandon, to eat all of the broccoli on their plates. Brandon ate 6 out of his 8 pieces of broccoli, and Gregory ate 5 out of his 10 pieces. Which child better followed the mother’s instructions? Draw pictures to help explain your answer. (N8.10)

Lesson 5 Using Benchmarks to Order Fractions N8 ( 8.10/8.11/8.12) TR pp. 31 -34 SB p. 220 - 222

Math Game Pot of Gold TR pp. 24 – 28 SB pp. 216 - 218

Michael wanted to run all the way home but only got 83 of the way when he became tired and walked the rest of the way. Did he run more/less than half the way home or all the way home? Draw a number line to show your thinking. (N8.10) For each situation, decide whether the best estimate is more or less than 12 . 1. When pitching, Dan struck out 6 out of 16 batters. 2. Lauryn couldn’t finish 3 out of 10 math problems. 3. Nick fouled out 5 times in 9 basketball games. 4. Jane sold 4 out of 12 boxes of Girl Guide cookies. 5. Adrian won 9 out of 16 cup stacking games. Create your own situation and trade with a partner to solve. (N8.10)

Performance Look at these fractions: 34 or 78 Which fraction is closer to 12 ? Explain using words and pictures. (N8.10)


19




It is expected that students will: 4N8 Continued 4N8.11 Name fractions between two given benchmarks on a number line (horizontal and vertical).

To place the fractions on a number line using benchmarks of 0, 12 and 1, students must also make estimates of fraction size in addition to simply ordering the fractions. Ask students to name fractions between two given benchmarks on a number line. For example, when asked to name a fraction between 0 and 12 , encourage students to think of as many possibilities as they can, using a set of fractions with like denominators, ( 15 , 25 , 53 , 45 ), or unlike denominators ( 13 , 103 , 125 , etc.)

4N8.12 Order a given set of fractions by placing them on a number line (horizontal and vertical) with given benchmarks.

20

When ordering fractions, fractions strips can be placed against a number line to help mark the fractions. A good introduction to this concept is to provide students with fraction strips of fourths, eighths, twelfths, and sixteenths, and ask students to identify the fractions that are equal 12 . This benchmark is the most familiar with students as they frequently share things into two equal groups. Students can then extend their understanding by ordering other fractions by using words such as ‘closer to’, or ‘less than’ half. Consider using and overhead transparency cut into fraction strips resembling student sets. Using an overhead projector to order fractions will help confirm students’ individual responses to ordering of their own fraction strips.




Resources/Notes

Performance Fractions Marching in Order - Have the students draw a number line indicating 0, 12 and 1. Provide the students with five or six fractions and have them place them in order on the number line, explaining how they know where each fraction should be placed. Examples of fractions to place on the number line using the benchmarks might include: 3 1 7 3 2 1 , , , , , 4 5 8 6 5 3

Math Focus 4 Lesson 5 Using Benchmarks to Order Fractions N8 ( 8.10/8.11/8.12) TR pp. 31 -34 SB pp. 220 - 222

(N8.10/8.11/8.12)

Performance Play a game with the whole class in which students use tiles or pattern blocks to illustrate fractional amounts. Place fractions on a number line labeled 0, 12 and 1. Place some in correct places and some in incorrect places (e.g., place 109 between 0 and 12 ). Have students illustrate the specified amount with their manipulatives. Then have them close their eyes and respond by showing thumbs up to indicate agreement with your placement or thumbs down to show disagreement. Students can then play this game in pairs taking turns placing fractions on the number line and responding. (N8.10/8.11/8.12)


21





Below are three types of problems using fractions of a whole and set that can help children develop their understanding of fractional parts. Students can draw diagrams and pictures to help visualize their understanding.

4N8 Continued 4N8.2 Identify a fraction from its given concrete representation. 4N8.5 Represent a given fraction pictorially by shading parts of a given set. 4N8.6 Represent a given fraction pictorially by shading parts of a given whole.


1.Find the Part (given the whole and the fraction): Mr. Hann is building a patio and wants to partition one-fourth of it for a BBQ. If the whole patio looks like this:

(given the set and the fraction) Michael purchased a set of 40 golf balls and wants to take one-fourth of them to his golf tournament. How many did he take?

2.Find the Whole: (given the part and the fraction) Mr. Hann has finished one-third of his patio. It looks like this:

Draw a picture that might be the shape of the finished patio. (given the part of the set and the fraction) If 12 cookies make up 34 of a batch of cookies, how many cookies are in the entire batch?

3. Find the Fraction: (given the whole and the part) Shawn ran in a cross- country race. What fraction of the race did he complete?

(given the whole of the set and the part) Tyler bought a dozen eggs to make pancakes. If the recipe requires 4 eggs, what fraction of the carton will Tyler use?

22




Resources/Notes

Journal Give an example of how drawing a diagram could help you solve a fraction problem. Use words, pictures and numbers to help you explain.

Math Focus 4

Pencil-Paper If adults get the daily-recommended amount of sleep - 8 hrs per night - what fraction of our day do we spend sleeping? Awake? Draw a diagram to help explain your thinking. Pencil-Paper Jane gave 124 of her hockey cards to her brother, and 123 of the cards to her friend. What fraction of the cards did she keep for herself? Paper-Pencil Twenty students in the class were surveyed on their favourite sport. 1 4 2 of the class preferred basketball, 20 of the class preferred soccer,

Lesson 6 Solving Problems by Drawing Diagrams N8 (8.2/ 8.5/ 8.6/ 8.14) TR pp. 37 - 40 SB pp. 224 - 226

Curious Math Drawing With Fractions N8 (8.4/ 8.6/ 8.14) TR pp.41 - 42 SB pp. 227

and 205 preferred volleyball. Did all twenty students in the class vote? Draw a diagram to help explain your answer. Sample of student possible student response:

Mid Chapter Review Be Selective


23




It is expected that students will: 4N8 Continued 4N8.3 Name and record the shaded and non-shaded parts of a given set.

It is expected that students will: 4N9 Represent and describe decimals (tenths and hundredths), concretely, pictorially and symbolically. [C, CN, R, V]

N8.3 was previously addressed with work related to fractions; however, it reappears here in the context of decimals. Students have had experience learning about fractions and have an understanding that can be transferred to incorporating the symbols to a new notation - decimal numbers. Most importantly, students will learn to make sense of decimal numerals which they will use later. They will also learn that decimals allow for calculations that are consistent with whole number calculations.

According to Small (2009, p.62), students will learn important decimal principles, through the use of concrete materials, pictorial representations, and modeling. Using decimals extends the place value system to represent parts of a whole. The use of a decimal point must be taught as a symbol that separates the tenths from the ones, or in other words, the ‘part from the whole’. Some principles are: 1. The base ten place value system is built on symmetry around the ones place and the decimal. 2. Decimals can represent parts of a whole, as well as mixed numbers. 3. Decimals can be interpreted and read in more than one way. Students should become familiar and comfortable renaming and reading decimals in several ways. E.g. 4.3 may be renamed 43 tenths. 4. Decimals can be renamed as other decimals or fractions. E.g. 60 may be represented as 0.60, 6 , or 0.6. 10

100

We often have measures that are less than one. Previously in this unit students learned that fractions are one way to represent these numbers. Now students will be introduced to another way to represent numbers less than one - through decimals. An introduction to decimals requires familiarity with the concept of fractional tenths (lesson 7). Some students will be comfortable with the concept of tenths and will be ready to move into a study of decimal hundredths fairly quickly (lesson 8). Focus on the need to continue the pattern in our base ten number system, so that the unit (or the whole) is divided into ten equal parts (or tenths) and another place value is included to the right of the ones place, separated by a dot (referred to as a decimal) to show that it is a fractional part. This shows the connection between fractions and decimals, as well as the connection between whole numbers and decimals; e.g. 2 = 0.2. Explain that we often write 0.2 rather than 10

the fractional notation.

24

(Continued)




Resources/Notes Math Focus 4 Lesson 7 Decimal Tenths N8 (8.3) N9 (9.1/ 9.2/ 9.6) N10 (10.1/ 10.2/ 10.3/ / 10.10.4/ 10.5) TR pp.46 – 49 SB pp. 230 - 232

These elaborations, while specifically referring to tenths in lesson 7, are also applicable to lesson 8 with extensions to hundredths.

It is important to foster understanding of decimals by ensuring that they can be read correctly. If cases arise in which there is a numeral in the ones place and well as numerals in the tenths and/or hundredth places, avoid using the term ‘point’. It has no mathematical meaning to students. Rather, use the word “and” to represent the decimal when reading a number aloud. For example 3.4 should be read as ‘3 and 4 tenths’ not ‘3 point 4’ or ‘3 decimal 4’. Saying decimal numbers correctly will assist students in gaining an understanding of how decimals relate to fractions.


25





Throughout the study of decimals, there are different concrete materials that will aid students in the understanding of decimal concepts: • Ten frames (tenths)

•

Number lines (tenths and hundredths)

•

Money (hundredths) - dollar as the whole, dimes as tenths, and pennies as hundredths

•

Meter Stick (hundredths)

•

Hundredths Disks (tenths and hundredths) – copy circular discs such as these shown below on two different colored card. Each disk is marked with 100 equal intervals around the edge and cut along one radius. The two discs are slipped together and can be used to represent a fraction or a decimal less than 100.

(Source: Van de Walle, Teaching Student Centered Mathematics Grades 3-5. Page 182)

(Continued)

26





Performance This activity could be repeated frequently as a part of the daily routines. Supply each student with a small whiteboard and a dry erase marker (This can be also done with paper and marker). Display a large number line. Place a removable pointer such as a magnet, clothespin or Post-It arrow on one of the tenth divisions. Students write the decimal to represent the number indicated and hold up the answer. Repeat for other numbers. (N9.1/10.4)


Lesson 7 (Cont’d) Decimal Tenths N8 (8.3) N9 (9.1/ 9.2/ 9.6) N10 (10.1/ 10.2/ 10.3/ 10.4/ 10.5) TR pp.46 – 49 SB pp. 230 - 232

27


Strand: Number Specific Outcome It is expected that students will:

Suggestions for Teaching and Learning •

4N9 Continued

Base Ten Materials and/ or Decimal Grids (10 x 10 Paper squares - hundredths and tenths)

Previously in the Numeration unit, students used base ten materials to represent whole numbers. Using the same materials, to now represent decimals, can be challenging for some students. When using whole numbers previously, the flat represented 100, the rod represented 10, and the small unit cube represented 1. Throughout this unit, the flat is 1, the rod is 0.1, and the small unit cube is 0.01. Ensure that students do not refer to the flat as ‘100’ but as a ‘whole’. It would be helpful if the flat were related to everyday items such as a rectangular cake, one whole, then the rod could represent a slice that is one tenth of the whole cake. The small unit cube would represent a bite of the slice or one hundredth of the whole cake. Refer to the N1 elaboration (beginning of Numeration unit) for review of appropriate use of terms relating to the Base Ten materials. Note that the overhead base ten materials are very useful as a means of showing various base ten representations to the whole class for discussion. Some students struggle with this model change, from whole numbers to decimals. As you discuss this different way of thinking about using Base Ten materials, in the teaching of decimals, you may have students work with paper copies of a hundredths grid. The cutting of the paper model may clarify students’ understanding and better suit various learning styles. •

28

Place Value Mat (used with money and base ten blocks)






29




It is expected that students will: 4N9 Continued 4N9.1 Write the decimal for a given concrete or pictorial representation of part of a set, part of a region or part of a unit of measure

4N9.2 Represent a given decimal, using concrete materials or a pictorial representation.

4N9.6 Provide examples of everyday contexts in which tenths and hundredths are used.

Give students various shapes cut from Bristol board or drawn on large paper. When you prepare the shapes be sure to use the flats, rods and units to guide your drawings so that the materials actually fit inside the shapes. Have them cover the shape using base ten materials and then proceed to find the value of the shapes if a flat represented a whole, a rod represented tenths and a unit represented hundredths. Name the shapes with a letter for easy reference as students record the decimal numeral represented. Conversely, give the students a decimal number and have them use that amount of base ten materials to create a 2-D or 3-D object. Children’s Literature Link - Use children’s literature to provide contexts for dealing with decimal tenths. For example, 10 for Dinner by Ellen Bogart (1989) is a funny counting book about a girl, Margo, who invites ten friends for dinner on her birthday. One guest stands out more than the others. He arrives early wearing his Halloween costume, asks for a peanut butter sandwich with olives and sauerkraut, makes a hat like the Loch Ness monster, sings a solo, wants to play "dirty-double no-hands blindfolded marbles," and brings the most interesting present. Use this story to provide a context when asking students to describe the subgroups of children.

Make a set of cards showing decimal tenths (for example 0.5 or 0.9) Have students choose a card and in their journals, illustrate the decimal with a picture or show the decimal using concrete materials (see previous pages for suggested materials). Teaching decimals, through meaningful contexts such as those below, will strengthen student understanding: • Fingers and toes • Items that are packaged in tens, such as pencils, stickers, sticks of gum • Food that can be shared among ten people, such as pizza or cake • Metre stick: metre stick as whole, centimetres as hundredths • Scores and times for various sporting events e.g. hundred meter dash was completed in 13.9 sec. • Statistics of athletes (e.g. points per game, etc.) e.g. NBA’s Chris Paul averages 11.8 assists per game or NHL’s Sidney Crosby averages 1.5 points per game • Gas prices on signs show price to the nearest tenth (89.9 cents per litre)

30



General Outcome: Develop number sense Suggested Assessment Strategies Presentation Ask students where they would find decimal numbers in their daily lives. Working in groups, have students find as many decimal numbers as they can in newspapers. Next, have students tell classmates what the number is and the context in which the number is used (e.g. price of an item, average rainfall or temperature in an area, price of gas, etc.). Students present their findings to the class. Performance Using a number line from zero to one (a paper copy, a piece of string, ribbon or wool, or a string displayed at the front of the room), ask individual students to plot numbers such as the following: 0.5, ten tenths, seven tenths, etc.

Resources/Notes Math Focus 4 Lesson 7 (Cont’d) Decimal Tenths N8 (8.3) N9 (9.1/ 9.2/ 9.6) N10 (10.1/ 10.2/ 10.3/ / 10.10.4/ 10.5) TR pp.46 – 49 SB pp. 230 - 232

Performance Present the following poem - Why the Elephant Painted Its Toenails Red by Wayne Edwards as a lead-in to this task about decimal tenths: I asked the little elephant, “Why are your toenails red?” “It’s really very simple Is what I think he said. It seems the little elephant Was playing hide-and-seek, With all the other animals In our back yard last week. Elephants are kind of big, It’s hard to hide, you see. So he painted all his toenails red And hid up in our cherry tree.

Provide students with an illustration of two bare feet. Ask students to color or place red counters on toenails based on fractions that you present to them orally or in writing.


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4N10.Relate decimals to fractions and fractions to decimals (to hundredths). [C, CN, R, V]

Reinforce the connection between decimals and fractions by having the students write the fraction and the decimal for the shaded part. Conversely, provide the students with decimals or fractions (tenths only) and have them shade the appropriate amounts on the hundredth grids. Encourage them to write the decimal and fraction for the unshaded part and compare the numbers they wrote for the shaded and unshaded parts. For example, if 0.4 is shaded then 0.6 is unshaded. The connection between these two decimals provides the foundation for adding and subtracting decimals.

4N10.1 Express, orally and in written form, a given fraction with a denominator of 10 as a decimal.

4N10.2 Read decimals as fractions; e.g., 0.5 is zero and five tenths.

4N10.3 Express, orally and in written form, a given decimal in fraction form.

4N10.4 Express a given pictorial or concrete representation as a fraction or decimal.

4N10.5 Express, orally and in written form, the decimal equivalent for a given fraction; 50 can be expressed as 0.50. e.g., 100

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As stated earlier, fostering understanding of decimals by ensuring that they can be read correctly. Avoid using the term ‘point’ when reading a decimal number as it has no mathematical meaning to students. Rather, use the word “and” to represent the decimal when reading a number aloud. For example 3.4 should be read as ‘3 and 4 tenths’ not ‘3 point 4’ or ‘3 decimal 4’. Saying decimal numbers correctly will assist students in gaining an understanding of how decimals relate to fractions. When writing a decimal less than 1, use a 0 in the ones place to emphasize that the decimal number is less than 1. (i.e. 0.3 rather than .3) Students observe ten classmates using criteria similar to those below, and record their results as decimal numbers. • Number of classmates with brown hair • Wearing black • Wearing glasses • Wearing jewelry • Etc. Provide groups of students with counters and egg cartons with the last two compartments cut off. Instruct students to take turns adding counters to compartments, and posing the question “How many?” to other group members. For example, a student may fill 7 compartments of the carton (0.7). Group members may answer “seven tenths”. To reinforce the connections between the concrete (egg cartons and counters) and the symbolic representation for decimals, supply students with sheets of tenths grids to record numbers created. To show the connection between decimals and fractions, students may be instructed to first record the number as a fraction then as a decimal number.






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It is expected that students will: 4N9 Continued 4N9.1 Write the decimal for a given concrete or pictorial representation of part of a set, part of a region or part of a unit of measure. 4N9.2 Represent a given decimal, using concrete materials or a pictorial representation.

Morning Routine using Decimal Circles – Have each student construct decimal circles as previously described on page 26. Prepare a set of large cards showing decimal amounts less than 1 and using both tenths and hundredths (e.g. 0.7, 0.23, 0.90. 0.4, 0.65 etc). Pick a card and ask students to manipulate the circles to represent the given decimal. Students should hold it up for you to see as soon as they have finished. Roarin’ Representations: Students should be comfortable reading decimal numbers, naming and renaming decimal numbers in alternate ways. Using a hundreds grid and counters, student A should create any decimal number:

4N9.6 Provide examples of everyday contexts in which tenths and hundredths are used. 4N10 Continued 4N10.2 Read decimals as fractions; e.g., 0.5 is zero and five tenths. N10.3 Express, orally and in written form, a given decimal in fraction form. 4N10.1 Express, orally and in written form, a given fraction with a denominator of 10 or 100 as a decimal. 4N10.4 Express a given pictorial or concrete representation as a fraction or decimal; e.g., 15 shaded squares on a hundredth grid can be expressed as 0.15 15 . or 100 4N10.5 Express, orally and in written form, the decimal equivalent for a given fraction. 50 e.g. 100 can be expressed as 0.50

Student B names the decimal number of the covered parts of the hundred grid and then represents the same number using a meter stick or a number line.

Student C names the decimal number of the uncovered parts of the hundreds grid and is then challenged to show that number using base ten blocks.

Ask each student to then record the 2 decimal numbers as well as the 43 equivalent fractions. 100 and 0.43 57 100

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and 0.57




Resources/Notes

Performance Name that Number: Students may work together to locate 0.5 and 0.6 on a metre stick or measuring tape and decide how to name the points between the two numbers.

Math Focus 4

Performance To help students recognize a decimal value, they may colour squares on a 100’s grid to create an animal or design. Give decimal values for each color used. Student-Teacher Dialogue/Journal Ask “Why are decimals important?” (Possible responses may include “Because they indicate part of a whole”, “Because they make an answer more exact”, etc.) Student-Teacher Dialogue Ask students to solve this problem: You go to the store to buy sugar and find two different brands. Your call your parent but find that they have gone out and are not answering the phone. You have to make a decision about which one is the better buy. One package is marked 0.8kg for $0.78 and the other is marked 0.80kg for $0.87. Which one is the better buy? Using any manipulatives or pictures, explain how you know.


Lesson 8 Decimal Hundredths N8 (8.3) N9 (9.1/ 9.2/ 9.6) N10 (10.1/ 10.2/ 10.3/ 10.4/ 10.5) TR pp.50-53 SB pp 233-235 Lesson 7 focused on decimal tenths. Lesson 8 extends focus to include decimal hundredths. Elaborations in lesson 7 can be applicable to both with minor modifications.

Math Game Race to 1 TR p 58 SB p. 239

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It is expected that students will: 4N9 Continued 4N9.1 Write the decimal for a given concrete or pictorial representation of part of a set, part of a region or part of a unit of measure. 4N9.2 Represent a given decimal, using concrete materials or a pictorial representation. 4N9.6 Provide examples of everyday contexts in which tenths and hundredths are used. 4N10 Continued

Provide 10 x 10 grids and ask students to shade the part representing four tenths. Identify the decimal number that represents the same amount and be prepared to explain their reasoning. Repeat for other fractions with 10 or 100 in the denominator. A Grade 4 class went on a hike and made various stops along the route. Jordan and his friend stopped for a break after walking 102 of the route. Jill and Allanah stopped after break at

6 10

of the route. At

55 100

35 100

of the route. Sue took a rest

, they stopped for lunch. Their final

95 100

stop was at where the students waited for the others to catch up. Ask students to write decimal equivalents to represent the stops made during the hike. Extension: Using a skipping rope or a piece of string to represent a number line, and paper clips or clothespins, ask students to plot the decimal equivalents.

4N10.2 Read decimals as fractions; e.g., 0.5 is zero and five tenths. 4N10.3 Express, orally and in written form, a given decimal in fraction form. 4N10.1 Express, orally and in written form, a given fraction with a denominator of 10 or 100 as a decimal. 4N10.4 Express a given pictorial or concrete representation as a fraction or decimal; e.g., 15 shaded squares on a hundredth grid can be expressed as 0.15 15 . or 100 4N10.5 Express, orally and in written form, the decimal equivalent for a given fraction. 50 e.g. 100 can be expressed as 0.50

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Resources/Notes

Portfolio Using base ten materials, model, then sketch the following decimal numbers: -three hundredths -three tenths -0.33 -0.03

Math Focus 4

Performance Tell students that your Grandmother had a button can containing 100 buttons. 60 of them were buttons with two holes, 5 were buttons with four holes and 35 buttons were one holed buttons. Write decimal numbers to show: • number of buttons with four holes • number of buttons with two holes • number of buttons with one hole Provide the students with several hundred grids. Say various decimal numbers (in context, if possible) for students to represent by shading their grids. Showing cards containing decimal numbers and ask students to read them allowed to you and represent them on a grid, as well. Performance Using Decimal Circles as described on page 26, have students turn over the Decimal Circle to see the back and make by estimation a friendly fraction such as 12 , 34 and 14 . Next, they turn over the Decimal Circle and record how many tenths and how many hundredths were in the section they estimated. Note the colors reverse when the circle is turned over.


Lesson 8 Decimal Hundredths N8 (8.3) N9 (9.1/ 9.2/ 9.6) N10 (10.1/ 10.2/ 10.3/ 10.4/ 10.5) TR pp.50-53 SB pp 233-235 Lesson 7 focused on decimal tenths. Lesson 8 extends focus to include decimal hundredths. Elaborations in lesson 7 can be applicable to both with minor modifications.

Math Game Race to 1 TR p 58 SB p. 239

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It is expected that students will: 4N9 (Continued) 4N9.3 Explain the meaning of each digit in a given decimal with all digits the same.

4N9.4 Represent a given decimal, using money values (dimes and pennies).

4N9.5 Record a given money value, using decimals.

4N9.7 Model, using manipulatives or pictures, that a given tenth can be expressed as a hundredth; e.g., 0.9 is equivalent to 0.90, or 9 dimes is equivalent to 90 pennies.

Same is Different: Provide groups of students with base ten blocks. Show students a decimal numeral with all digits the same (e.g. 3.33). Instruct them to model it using base ten materials and to show and explain to the group how much each 3 represents. Repeat several times showing different decimal numbers so that each group member has an opportunity to model a number. Students will have seen decimals used in dollar amounts but will probably not have thought about their meaning beyond “how many dollars and how many cents?” It is important that students view a dollar/loonie as a whole with a dime representing 0.1 (one tenth) of a dollar, and a penny representing 0.01(one hundredth) of a dollar. Referring to a penny as ‘one hundredth of a dollar’ instead of a ‘cent’ will help students understand the fractional part of the whole that the penny represents. Provide the students with loonies, dimes and pennies. Review the relationship among the coins and focus on groups of ten. Relate these groups of ten to the Base Ten number system. Have the students write symbols for whole number amounts of money, such as $15. Then focus on the necessity of writing values for money less than one loonie or one whole dollar. Explain that the whole number system is extended to accommodate the need to write numbers smaller than one by dividing the whole (dollar) into ten equal parts, called tenths (dimes). Have the students continue this pattern using their understanding of money; i.e., ten pennies make a dime and one hundred pennies make a dollar. Explain that the decimal symbol separates the whole number from the fractional parts called tenths and hundredths. Have the students suggest how they might write 20 cents as a fraction of one dollar, using fractions and then decimals. Guide 20 but it is usually written as them to see that it can be written as $ 100 $0.20, meaning that there are no dollars, but rather two-tenths of a dollar (two dimes) or twenty-hundredths of a dollar (20 cents). To help students understand that decimals represent a fractional part of a whole, instruct students to fill a 10 by 10 grid with pennies. They should recognize that dimes are worth 0.1 or 1 of a dollar (one row) 10

and that pennies are worth 0.01 or 1 of a dollar (one grid square is 100

worth a penny). Through discussion and observation, students will also see that 0.20 is equivalent to 0.2 or 2 , since 20 pennies (0.20) 10

fill two columns.( Small, 2006)

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(Continued)





Math Portfolio/Journal: Using pictures, numbers and words, show the value of each digit in the number 7.77 (or $7.77). (N9.3) Performance: Identify the number in the tenths place (or hundredths place). If you have… • …$2.38 • …$92.29 • …4 loonies, 5 dimes, 6 nickels and 2 pennies • …2 dimes, 3 nickels and 19 pennies • …9 dimes, 1 nickel, and 108 pennies

Lesson 9 Representing Decimals with Coins N9 (9.1/ 9.2/ 9.3/ 9.4/ 9.5/ 9.7) N10 (10.2/ 10.4) TR pp.54-57 SB pp 236-238

Journal If we did not use decimals for money, what would happen to prices? (They would be rounded up and we would pay more for things.) Student-Teacher Dialogue: Use the 10 x 10 grid and pennies as described on the opposite page and pose questions such as: • How much are 3 columns? • If two and a half columns of pennies were removed, how much would be left? • If the first 4 columns were covered and only six blocks were covered in column 5, what would the value be? Ask the student to show 0.64 on the grid using any combination of pennies and dimes. Ask if there is a different way to show this amount. (Answers include 6 dimes and 4 pennies or 64 pennies)


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Students may use place value mats to model decimal amounts using both base ten materials, as well as coins.

4N10.Relate decimals to fractions and fractions to decimals (to hundredths). [C, CN, R, V] 4N10.2 Read decimals as fractions; e.g., 0.5 is zero and five tenths.

4N10.4 Express a given pictorial or concrete representation as a fraction or decimal; e.g., 15 shaded squares on a hundredth grid can be expressed as 0.15 or 15 . 100

It is important that students understand and recognize the relationship between tenths and hundredths, as well as between pennies, nickels, dimes and loonies. To help consolidate these connections, use the following activity: Provide students with sets of play money. (If commercial sets of plastic money are not available, teachers may photocopy sheets from page 29 of Masters Booklet.) Using grocery ads from a newspaper, ask students to show and record at least two ways they could pay for the item. For example, a box of cereal may cost $3.29 and may be paid for using 3 loonies, 1 quarter and 4 pennies. Another possibility may be 2 loonies, 4 quarters, 2 dimes and 9 pennies. The emphasis in this lesson is on representing parts of a dollar as decimals, therefore most of the grocery items students choose should be less than one dollar. If students have difficulty locating products priced less than one dollar, the teacher may create a master list of items and prices that would fit the activity. Make overhead transparencies of pictures of items as well as pictures of the money used to pay for those items. Instruct students to record how much each item costs using a decimal number.

Likewise, show various decimal values and ask students to model the amount with their play money. To show that tenths can be expressed as hundredths, ask students to model two ways to show the following decimal numbers as money amounts, using play money manipulatives or pictures of coins: • 0.3 • 2.9 • 0.7 • 8.5

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Resources/Notes Math Focus 4 Lesson 9 Representing Decimals with Coins N9 (9.1/ 9.2/ 9.3/ 9.4/ 9.5/ 9.7) N10 (10.2/ 10.4) TR pp.54-57 SB pp 236-238


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It is expected that students will: 4N11 Demonstrate an understanding of addition and subtraction of decimals (limited to hundredths) by: • using compatible numbers • estimating sums and differences • using mental mathematics strategies to solve problems [C, ME, PS, R, V]

By now, most students realize that an exact sum or difference is not always required and an estimate is sometimes sufficient. This is particularly true when adding or subtracting decimal numbers that represent money and distances. When adding and subtracting decimal numbers, students should always estimate first as this requires them to focus on the relationships between numbers and the effect of number operations, rather than simply applying a memorized rule to compute. After estimating, students can add / subtract the decimals to get the exact answer and explain the strategy used to the class or to a partner. Base ten blocks should be available to students at all times during addition and subtraction practice to help them arrive at a solution, as well as model how they found their solutions.

4N11.1 Predict sums and differences of decimals, using estimation strategies.

By providing students with many opportunities to estimate sums and differences in meaningful contexts, students will learn to assess which strategy works best, based on the decimal numbers they are working with. They should also recognize the usefulness of these strategies in everyday life, and in doing so, further develop their number sense. When estimating, students will often use mental computation strategies. A number of these strategies were explored in N3 and can also be used in the context of decimals. Students may choose to use strategies such as: • compatible numbers: e.g., 0.72 + 0.23 is close to 0.75 + 0.25 • front-end addition: e.g., 32.3 + 24.5 may be thought of as 30 + 20 (for a more accurate estimate students may add the tenths to get an estimate of 50.8, or even 51 if they realize that 0.8 is close to 1) • front-end subtraction: e.g., 4.47 – 3.48 may be thought of as “4 ones minus 3 ones is 1, and 7 tenths subtract 4 tenths is 3 tenths, for a difference of approximately 1 and 3 tenths” • rounding: e.g., 4.39 + 5.2 is about 4 + 5 for an estimate of 9

4N11.3 Solve problems, including money problems, which involve addition and subtraction of decimals, limited to hundredths. 4N11.4 Determine the approximate solution of a given problem not requiring an exact answer. 4N11.5 Estimate a sum or difference using compatible numbers.

Let’s Go Shopping! Students may enjoy and benefit from a class ‘store’ where they may use play money to make purchases. Students may bring in items to ‘sell’ with a price tag attached, or bring catalogue pictures. Tell students that exact change is not required at this store as long as the clerk and shopper agree on the approximate total price of items purchased. This activity encourages the use of both estimation and mental math strategies.

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Resources/Notes

Portfolio Create addition or subtraction equations using decimal numbers that would result in an answer close to 50.

Math Focus 4

Performance/Interview/Paper Pencil Pose questions such as the following: • Becky earned $127 in one week of babysitting and has $248 in the bank. She wants to buy a $400 bicycle. Will she have enough money? • Jason's favorite comics cost $2.17 on sale. He wants to buy two of them. About how much will they cost? • Beth has $153 in the bank. She has to pay her mom back for a $49.98 pair of sneakers, and wants to buy her brother a $28.38 logo t-shirt for his birthday. She wants to go to Science camp, which will cost $65. Does Beth have enough money?

Lesson 10 Estimating Decimal Sums and Differences N11 (11.1/ 11.3/ 11.4/ 11.5) TR pp.59 -62 SB pp.240 - 242

Lesson 11 Using Mental Math N11 (11.3) TR pp.63-66 SB pp.243-245

Student-Teacher Dialogue • Show: 26.5 + 53.5 Ask: “How can you know that the sum is less than 100 without actually completing the addition?” •

How would you calculate 4.97 + 6.99 mentally?

Journal • Tell students that to calculate 9.7 – 8.6, Bethany thought ‘86 + 11 = 97’. Explain her thinking.


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It is expected that students will: 4N11 Continued 4N11.3 Solve problems, including money problems, which involve addition and subtraction of decimals, limited to hundredths. 4N11.6 Count back change for a given purchase.

Focus on the value of estimating to determine how much change one would receive after a purchase, as well as determining the exact amount of change. Students will have many opportunities to calculate mentally with decimals, with the goal of arriving at an estimate and being able to explain why the answer is reasonable. To find an exact answer, students may choose to “count on” to calculate change and may use a number line to help them record the jumps taken when counting on. The number line in this case will probably not resemble the traditional number line with even intervals. For example, if a student were to use a number line to help calculate change from $20 for a purchase of $18.65, it may look like this:

To help students further develop their own personal strategies for calculating change, and to reinforce the strategy of “counting on”, teachers could supply groups of students with a catalogue and play money. Ask them to make three separate purchases from the catalogue, pay for each and check that the ‘clerk’ gave them the correct change, by counting on aloud for group members to hear. Highlight with the class any strategy that may be unfamiliar or different than those the majority would use. Let’s Go Shopping! If the class ‘store’ was established, encourage students who visit there to estimate first what their change will be, then calculate the exact amount. The ‘clerk’ will determine whether or not the change is correct. Build It to Show It: Supply group members with price tags (cards) showing decimal amounts such as $0.03, $0.40, $1.12, $2.49, $4.99, etc. Cards are placed face down in a pile. The first player turns over a card and uses base ten blocks to build the number shown. Group members discuss what would be ‘added on’ to get to a loonie, toonie, $5.00, $10.00 and so on, depending on the amount shown on the card. The first member then builds that amount with base ten blocks to show the change one would receive if making a purchase of an item priced like the one on the card. Each group member could then write an equation to represent the round.

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Performance Provide students with the following problem: Tess bought a CD for $22.35. She paid for it with two twenty-dollar bills. What was her change? Using models, number lines, words and pictures, explain how you solved the problem. What other bills might Tess have used to pay for her purchase? What would her change have been then?

Lesson 12 Making Change N11 (11.3/ 11.6) TR pp. 67-70 SB pp. 246-247

Interview Ask students to calculate the change from $5.00, if the bill totaled $3.59. About How Much? Show students a short list of grocery items. Ask students to estimate about how much the list will cost. This may lead to an interesting discussion about the cost of groceries! Next, students check their predictions. Ask: How did you determine the total cost? Encourage discussion and explanation of the strategies students used to estimate and to add mentally. Ask: Will I have enough money if I only have $20? How do you know? About how much change will I have if I pay with $20? $50? $100? Show how you know.


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It is expected that students will: 4N11 Continued 4N11.3 Solve problems, including money problems, which involve addition and subtraction of decimals, limited to hundredths. 4N11.5 Estimate a sum or difference using compatible numbers.

It is important that students recognize that the properties and techniques established for the addition and subtraction of whole numbers also apply to decimals. Students should recognize that adding or subtracting tenths (e.g., 3 tenths and 4 tenths are 7 tenths) is similar to adding or subtracting quantities of other items (e.g., 3 apples and four apples are 7 apples). The same is true with hundredths. Rather than simply telling students to line up decimals vertically, or suggesting that they “add zeroes”, they should be directed to think about what each digit represents and what parts go together. For example: 1.62 + 0.3 may be thought of as 1 whole, 9 (6 + 3) tenths and 2 hundredths, or 1.92. Base ten blocks and hundredths grids continue to be useful models. Once students recognize these similarities, addition and subtraction equations should be given to students in horizontal form so they can practice aligning the decimals vertically when calculating (e.g., 2.5 + 17.36). It is particularly important that students estimate to determine the reasonableness of their answer (e.g., 2.5 + 17.36 may be thought of as 3 + 17 = 20). Place a strip of wide masking tape on the floor to represent a number line. With endpoints of 14 and 16, divide the line into tenths. Pose this problem: The city record for the 100-metre run is 15.9 seconds. Derek ran the race in 14.6 seconds. By how much did Derek beat the record? Let’s Go Camping: Ask students to make a list of supplies they would need to take on a camping trip. Research the prices using catalogues and calculate the cost. Supply students with a list of average temperatures for towns/cities they may study as part of Social Studies: Average temperature in May in Place degrees Celsius Great Barrier Reef, 25.1 Australia Edmonton, Alberta 10.7 Nairobi, Kenya 19.5 Oslo, Norway 12.3 Ask questions such as: • In May, how much warmer is it in Australia than in Alberta? • Would the combined temperatures of Nairobi and Oslo be greater or less than 30? How do you know? • How much colder is Edmonton than Oslo?

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Performance Luci loves to hike. Her goal is to hike 15 kilometres on weekends. She keeps track of her weekend hiking distances: Saturday Sunday First Weekend 4.9 km 3.81 km Second Weekend 7.19 km 5.8 km Third Weekend 9.3 km 5.9 km Fourth Weekend 8.42 km 6.6 km Did Luci reach her goal? On which day(s)? Show how you solved the problem. Portfolio/Journal Ken’s solution to an addition question is shown below. Write to Ken and explain what he did wrong and show him through numbers, pictures and words how to solve the problem. 0.78 + 1 2.3 2.01

Lesson 13 Adding and Subtracting Decimals N11 (11.3/ 11.5) TR pp. 71-74 SB pp. 248-250

Chapter Review Chapter Task

Interview How is calculating 0.3 + 0.8 like calculating 3 + 8? Journal Is the difference between 1.8 and 0.52 greater than 1 or less than 1? Show how you know. Paper-Pencil Provide short grocery receipts. Students can sort items on the receipt into categories of their choice and then figure out how much was spent in each category. This may be used as a cross curricular activity with a discussion of the food groups or nutrition in Health. Paper-Pencil Make several sport cards available and ask students to compare sports statistics among two players. E.g. compare batting averages using addition and subtraction.


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48


MEASUREMENT Suggested Time: approx 3 weeks October


March


September

April

May

June

MEASUREMENT

Big Idea

Measurement is a fundamental mathematical process that is an essential link to many areas of the mathematics curriculum. Measurement permeates many areas of life including careers and everyday living. In its simplest form it merely attaches a number to some attribute of an object and it increases in it’s breadth and depth as students move on through the curriculum. Students also must learn proficiency in choosing and using measurement tools. Although measurement is the focus of this unit, it should be continued throughout the school year as it relevant in other areas of the mathematics curriculum as well as everyday experiences. In this unit, measurement is addressed first as it relates to measuring time. “Standard units of time should always be introduced to children in a way that relates to their own experiences.” (Small, 2008, p. 444). Then measurement is taught as it relates to the measurement of area. Measurement skills are best learned by students if they have plenty of opportunities to complete activies focusing on measurement. They most often learn best by first hand experience. Although students have not had any explicit teaching related to reading and recording time using clocks by grade four, they have had opportunities in previous grades to explore the passage of time and have an understanding that there are 60 minutes in an hour. As well, students will have had many opportunities to use time through their own experiences with the real world. The units, “minutes and hours”, are usually introduced before the unit “seconds” because children use them more often throughout their daily lives. Students will need to learn these standard units of time and then have many opportunities to explore the relationship among the units. Throughout the school day, teachers need to provide numerous opportunities for students to read and record time using a variety of clocks such as digital, analog and 24 hour clocks. Embedded throughout this unit is a focus on how the measurement of area can be used to solve problems. Students should read times on clocks to provide information about relevant situations such as: • comparing start and finish times to determine how much time has passed • estimate how long before an event begins, e.g., How long until lunchtime? • planning events • reading schedules The concept of area measurement for regular and irregular shapes is introduced in this unit as well. Students’ previous experience with measurement units would have dealt mainly with linear measurements, including perimeter. Van de Walle and Lovin, 2006 define area as “a measure of the space inside a region or how much it takes to cover a region. (p. 234)

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MEASUREMENT

In using any type of measurement, such as length, area or volume, it is important to discuss the similarities in developing understanding of the different measures. First, identify the attribute to be measured, then choose an appropriate unit and finally, compare that unit to the object being measured (NCTM 2000, p. 17). As with other attributes, it is important to understand the attribute of area before measuring. Key ideas in understanding the attribute of area include the following: conservation – an object retains its size when the orientation is changed or it is rearranged by subdividing it in any way • iteration – the repetitive use of a identical nonstandard or standard units of area to entirely cover the entire surface of the region • tiling—the units used to measure the area of a region must not overlap and must completely cover the region, leaving no gaps • additivity—add the measures of the area for each part of a region to obtain the measure of the entire region • proportionality—there is an inverse relationship between the size of the unit used to measure area and the number of units needed to measure the area of a given region; i.e., the smaller the unit, the more you need to measure the area of a given region • congruence—comparison of the area of two regions can be done by superimposing one region on the other region, subdividing and rearrangement as necessary • transitivity—when direct comparison of two areas is not possible, use a third item that allows comparison; e.g., to compare the area of two windows, find the area of one window using nonstandard or standard units and compare that measure with the area of the other window; i.e., if A = B and B = C, then A = C • standardization—using standard units for measuring area such as cm² and m² facilitates communication of measures globally • unit/unit-attribute relations—units used for measuring area must relate to area; e.g., cm² must be used to measure area and not cm or mL. Adapted from Alberta Education, Teaching Measurement Concepts, Grades 4–6 (unpublished workshop handout) (Edmonton, AB: Alberta Education, 2006), Research section, pp. 2–4. •




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MEASUREMENT

Strand: Shape and Space (Measurement) Specific Outcome



Time can be represented using a linear model such as a timeline because time is actually linear however our ‘descriptions’ of time with words such as days, weeks, months, years, etc. describes cycles. This is a common misconception. Although time is linear because nothing actually repeats, we still can use a ‘time circle’ when teaching time to show the cyclical nature of the words that are ‘descriptions’ of time.

4SS1 Read and record time, using digital and analog clocks, including 24-hour clocks [C, CN, V] Achievement Indicators: 4SS1.1 State the number of hours in a day.

4SS1.2 Express time orally and numerically from a 12-hour analogue clock.

Students will learn that there are 24 hours in a day; however, time is often described using the 12-hour clock. Although the world has become increasingly digital, there are still may analogue clocks in use and students must learn to tell time on both analogue and digital clocks as well as on a 24 hour analogue clock which will be discussed later. On a 12 hour analogue clock, the hours go from 1:00 in the morning until 12:00 noon and then it repeats the cycle from 1:00 in the afternoon until midnight. Consider setting an alarm clock to alarm every hour on the hour. Count how many hours are in a school day and use that information to conclude that there are 24 hours in one day. Van de Walle & Lovin K-3 (2006 page 244) suggests using a “onehanded clock” to help students understand and read analogue clocks. Discuss what happens to the long hand as the short hand goes from one hour to the next. Often students do not position the hour hand to reflect the number of minutes after the hour. Discuss with them how the hour hand moves over the course of each hour. If available, use a geared demonstration clock. Also, break the long hand from an old clock and set the short hand in varying places as shown below and use approximate language such as:

“It’s about 8 o’clock”

“It’s halfway between 3 o’clock and 4 o’clock”

“A little bit past 9 o’clock”

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General Outcome: Use Direct or Indirect Measurement to Solve Problems Suggested Assessment Strategies

Resources/Notes

Performance Have students track events throughout a specific day by means of a time line. Students should record the time of the activity or event and note it at the appropriate spot on a time line.

Math Focus4

Encourage the children to use different colours for the a.m. and p.m. hours.

Lesson 1 Telling Time to the Hour SS1 (1.1/ 1.2/ 1.7/ 1.8) SB p.262-264 TR p.12-15

(SS.1) Ask students how many hours are in a day and a half? Two days? (SS1.1) Example: 2:10 a.m. is “Two ten” or “Ten minutes past two.” Note that a.m. and p.m. still need to be indicated. Consider: - 5:03 might be said as “3 minutes after 5” - Can students express 7:40 as “twenty minutes to eight”? - How would “10 minutes to 8” be written? - Would students know that 7:57 means “nearly 8:00?” (SS1.2) Performance Using masking tape, form a large clock on the floor. Use a metre stick for the long hand and a ruler for the short hand. Have students take turns and work in pairs to make the clock show a time you call out. Involve all the students by having them make the time on individual paper plate clocks at the same time. Variation: Use two children, one taller than the other, to stand in the center of the clock to represent the hands on the clock. Ask: Who should be the minute hand? Why? (the taller child because the minute hand is longer – be sensitive to children who might be embarrassed about being tall or short). Where should the minute hand point to show 1:00? (to 12). Where should the hour hand point to show 1:00? (to 12). Pass out index cards showing various times. Children hold up the time cards one at a time and tell the children representing the hands where they should point to show the time. (SS1.2/ SS1.2)


Sources: Teaching Student-Centered Mathematics K-3, Van de Walle and Lovin, 2006, pp.242 - 245 Making Math Meaningful to Canadian Students K-8, Marion Small, 2008, pp. 441 - 448

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It is expected that students will: 4SS1 Continued Achievement Indicators: 4SS1.7 Explain the meaning of a.m. and p.m., and provide an example of an activity that occurs during the a.m., and another that occurs during the p.m.

Students may want to investigate the meaning of the terminology a.m. and p.m. [a.m. is the abbreviation for the Latin ante meridiem meaning before noon and p.m. is the abbreviation for post meridiem meaning after noon]. Students will likely encounter other representations of a.m. and p.m., such as AM and PM; A M and P M [with or without periods/ capitals/lower case]. Students can use the terms “noon” and “midnight” for those precise times since the use of 12:00 a.m. and 12:00 p.m. may be a source of confusion. A good way to investigate a.m. and p.m. is to use a full-day timeline. This helps students become familiar with using a.m. and p.m. notation correctly. It also helps students who confuse a.m. with p.m. (Small, 2008)

Students would benefit from ongoing practice telling time during morning/ daily routines through the use of a demonstration clock. If you prefer, make an overhead clock by photocopying a clock template and separate hands on transparencies. Cut out the hands and attach them with a fastener so that they are moveable.

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Resources/Notes Math Focus4

Ask students to name an activity they would typically do in the a.m.? p.m.?

Lesson 1 (Cont’d)

Curious Math SB p.265 TR p.16

Authorized Resources: Van de Walle, Teaching StudentCentered Mathematics, Grades K3, pp. 242-244 Other Resources: Marian Small, Making Math Meaningful to Canadian Students K-8, pp. 441-452 NCTM, Navigating through Geometry in Grades 3-5 (2003) Van de Walle, Teaching StudentCentered Mathematics, Grades 3-5 pp. 269-271


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It is expected that students will: 4SS1 Continued 4SS1.2 Express the time orally and in writing from a 12-hour analog clock.

4SS1.6 Express the time orally and in writing “minutes to” or “minutes after” the hour.

Students need to learn how to express time using a 12-hour analog clock. Students have had experiences telling time to the hour but time that falls between the hours is more challenging for students; therefore, it is important to take time to analyze, with students, how the analog clock shows the passage of the standard units of time. Building student’s conceptual understanding of analog time helps students make sense of time related terms. e.g. 8:15 can also be read as “a quarter past 8” Expressing time orally offers an excellent opportunity for students to use their bodily-kinesthetics to visualize and solve problems relating to time on an analog clock. Introduce the terms “half past”, “quarter after” and “quarter to” using an analog clock. Provide an open space for creative movement where children can sit on the floor and then arrange themselves to represent the numbers and hands on a clock. Once they have physically arranged themselves, ask them to show various times on a clock. For a student to relate 8:15 to a quarter past 8, they would need to know that 15 minutes is 14 of 60 minutes. If the time falls more than halfway through the hour, we read it as a number of “minutes to” one hour, or as a number of “minutes after” the previous hour. Students need to be very attentive to the hand lengths, as well as the numbers on an analog clock. They need to learn that whether they read the number as an hour or as minutes will depend on which hand is pointing to the number. Students should not be limited to reading time to the nearest five minutes if they show a good understanding. They can be encouraged to read time to the minute. Have students explore that the minute hand and hour hand on an analog clock are different lengths and that the minute hand is at 6 for the _:30 and at 12 for the _:00. Students need to be aware that the hour hand moves during the course of the hour, and when it is at the _:30, it is halfway between two numbers. Time after the hour and time before the hour should be addressed. Examples: 8:45 might be spoken as “eight forty-five”; “forty-five minutes past eight”; “quarter to nine”; or “fifteen minutes to nine” Have students work in pairs. Have one student say a time and the other student make the time on the clock. (Continued)

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Resources/Notes

Paper-Pencil - One-Handed Clocks: Prepare a page of clock faces and on each analog clock draw on an hour hand (short hand) only. Include times that are approximately a quarter past the hour, a quarter to the hour, half past the hour and some that are close to but not on the hour. Ask students to write the possible digital time and then draw a minute hand on the analog clock where they think it would be so both clocks will be displaying the same time. Practice this activity of reading approximate time daily during morning/daily routines many times before having students attempt it independently. If students have difficulty, try pairing the one handed clock with a two handed clock and a digital clock. (SS1.2/SS1.7) For example:

Lesson 2 Time to the Hour and Quarter Hour SS1 (1.2/ 1.6 1.8)) SB p.266-267 TR p.17-19

Performance Ask students to predict the reading on a digital clock when shown an analog clock and vice versa, set an analog clock when shown a digital clock. At various times throughout the day, uncover one clock and have students predict, orally or in writing, what’s happening on the other clock.

Lesson 3 Telling Time to 5 Minutes SS1 (1.2/ 1.6/ 1.8) SB p.268-270 TR p.20-22

Lessons 2 and 3 may be addressed together.

Student-Teacher Dialogue Using two real clocks, one with only an hour hand (break off the minute hand from an old clock) and one with two hands, cover the two handed clock and periodically throughout the day, direct attention to the other one-handed clock. Have students predict where the minute hand should be and then uncover the two-handed clock and check.


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It is expected that students will: 4SS1 Continued It is essential for teachers to provide many opportunities for students to manipulate the hands of an analog clock to help them visualize how the hour hand moves in relation to the minute hand. Ask students what time it might be if: • the minute hand is pointed to an area between 4 and 5 • the hour hand and minute hand are both pointed directly at a number

Using a clock that shows not only the numbers from 1-12, but also the minute amounts from 5-55 beside the numbers from 1-11 may be useful. As students establish a comfort level with skip counting by 5, this will enable them to read time to the nearest five minutes. This provides students with an opportunity to relate the numbers on a clock to time. Students may have already become familiar with the ‘clock strategy’ as one mental math strategy for learning multiplication of 5x facts. This would be an appropriate time for teachers to make students aware that there are 5 minutes between numbers on a clock. The long hand on the 2 represents 10 minutes, so two one minute spaces past the 2 is 12 minutes. Show students a standard analog wall clock, drawing attention to the hour and minute hands and how they relate to the previous activities. As time telling skills develop, continually suggest to students to look first to the hour hand to predict an approximate time and then to the minute hand for precision.

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Resources/Notes

Performance Initially have students read the time from a one-handed analog hour clock. Ask him/her to read the time to the nearest hour and to name an event/activity that often happens at about that time of day. This approach helps students to attend to the placement of the hour hand and its meaning. Encourage students to verbalize phrases such as: - “before eight o’clock” - “ between eight and nine o’clock” - “a few minutes after eight o’clock” - “precisely eight o’clock” - “nearly eight o’clock” - “about half past eight” - “about eight thirty” Note misconceptions to guide teaching by listening closely to student responses. (SS.1.2) Performance • Ask the student to show, on an analog clock, the time (to the nearest half hour) at which they arrive at school, have lunch, go to bed, etc. • Have the student make a list of the times when the minute hand and the hour hand just about line up as well as other patterns, such as all of the times that include a 4 in a 24-hour period. • Ask students what time might it be if the minute hand and hour hand are opposite one another. • Provide students with many opportunities to use hands-on manipulatives such as play clocks and clock stamps to express time orally and in writing. Then listen as students read the times on the various clock cards. Student Teacher Dialogue Display clock as shown:

• Where is the hour hand? (A little past 3) • What does that tell you? (It’s after 3:00) • How much after 3:00 is it? (It’s twenty five minutes after 3) • How do you know? (The minute hand is pointing at the 5)


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Strand: Shape and Space (Measurement) Specific Outcome It is expected that students will: 4SS1 Continued 4SS1.4 Express the time orally and in writing form, from a 12-hour digital clock

4SS1.8 Solve problems related to time including elapsed time

Suggestions for Teaching and Learning Ask students to think about some different kinds of clocks or watches they have seen. Students find it easier to ‘read’ times from a digital clock but it is important to talk about the meaning of the times they are reading. Using an analog and a digital clock together may help with this. Having a real digital clock available for students to manipulate would be beneficial. Reading a clock, or ‘telling time’ is more about reading an instrument. Time as a measurement; however, encompasses duration. For students to make sense of the concept of time, they need to understand that time, as a measurement, is about how long an event takes from beginning to end. This is called Elapsed Time.. Elapsed time can be found only by counting the hours and minutes between the start and the end times. Much of the learning that students are to attain can be assessed on an on-going basis through daily conversations and activities in which time is naturally included Initiate a discussion about a recent birthday invitation. Show students a birthday invitation that you have received. Ask the students: What information did the birthday invitation give? Focus on the start and end time for the party. Provide a 4-column chart and model how to use it to record each event, its start and end times and the elapsed time Next, brainstorm a list of student events and have them complete the chart..

Provide the students with clock manipulatives to use. Invite children to share their charts and then discuss the strategies they used to determine the elapsed time. By making conscious efforts throughout the school day, teachers can provide numerous opportunities for students to learn duration of short and long events that can be measured in second, minutes and hours. Students should read times on various clocks to provide information about relevant situations, such as: - comparing start and finish times to determine how much time has passed - estimating how long before an event begins [e.g. How long until lunch time?] - planning events - reading schedules [bus, travel itinerary, school timetable, television, etc.]

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Resources/Notes

Performance/Presentation This task encourages exercise and can be done as a class or can be modified to be done at home (with parent permission). Have students brainstorm to pick a reasonable location within walking distance from the school. Choose one clock to be the official clock. Students record the time, to the minute, they leave school. They walk to their destination and record the time they get there, to the minute. After arriving back at school use this information to figure out how much time it took them to walk to their destination (how much time had elapsed). Hint: Leaving before noon and returning after noon, would provide an opportunity to assess students’ appropriate use of a.m. and p.m.! For example, leave school at 11:30 a.m. and return at 12:06 p.m. As a follow up, if you have access to an inexpensive wrist watch, assign each student a day to wear the watch home. The next morning during daily routines, the student should present recorded times that he/she left school the previous day, the time he/she arrived home, tell how much time had elapsed and show or tell how they knew. You may use a digital clock, an analogue clock or a 24 hour clock (after working on this in class) depending on the needs of your class or individual student. This task provides you with an opportunity to assess students’ understanding of time as students read different types of clocks and compute elapsed time. (1.2/ 1.4/ 1.6/ 1.8) Performance Using an index card, ask students to create a story problem involving elapsed time. Then trade their cards with other students to have them solve the problem. (1.8)

Lesson 4 Telling Time to 1 Minute SS1 (1.2/ 1.4/ 1.6/ 1.8) SB p. 272-275 TR p. 24-27

Performance Refer to the class clock frequently throughout the day and ask questions such as, “What time will it be in 20 minutes?” or, “Our math class started at 9:30. How long did it last?” Make clocks available (students may make their own using a paper plate). At various times during the day, preferably on the hour or half hour ask questions such as the following. It is 10:00 now. What time will it be: • 2 hours from now • 4 12 hours from now • • • •

12 hours from now 10 minutes ago 2 hours ago 1 12 hours ago (1.2/1.6/1.8)


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It is expected that students will: 4SS1 Read and record time, using digital and analog clocks, including 24-hour clocks (Cont’d) [C, CN, V]

4SS1.3 Express the time orally and in numerically from a 24-hour analogue clock.

4SS1.5 Express time orally and numerically from a 24-hour digital clock.

It is expected that students will: 4SS2Read and record calendar dates in a variety of formats [C, V]

It is important for students to learn about the 24 hour time system. The 24 hour clock is a system used for telling time in which the day runs from midnight to midnight and is numbered from 0 to 23. It is often used when it is very important that times are not confused. The 24-hour clock eliminates uncertainty as there is only one 11:32, for instance, during the day. Students may have encountered everyday life situations where the 24 hour time system is used if they have traveled on flights and ferries. It is also used in the practice of medicine because it helps prevent ambiguity about important events in a patient’s medical history. This system is sometimes referred to as ‘military time’. Ask students why using a 24-hour clock in the military would be important. Some students who struggle with the 12 hour clock can be exposed but may not be ready to work on the 24 hour clock. Once students are comfortable reading the 24 hour clock, they may observe that subtracting 12 is a convenient way to “tell” the more familiar 12 hour time. In the 24-hour notation, a time of day is written in the form hh:mm (for example 22:30) where 22 means 22 full hours have passed since midnight and 30 full minutes have passed since the last full hour. Note that digital time is always expressed with 4 digits with a “0” being placed at the beginning of times less than 10. For example, 8:00 is expressed as 08:00 hours. Using a calendar throughout the school year will strengthen the students’ sense of time. Each month brings a new calendar to explore. Students need to become aware of the variety of ways dates can be recorded. Students would have previous experience relating the number of days to a week; number of months to a year; days to a month. The calendar also provides rich opportunities to explore: - number sense [ See also N 5] - number patterns [See also PR 1.4 & PR 3.2]

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General Outcome: Use Direct or Indirect Measurement to Solve Problems Suggested Assessment Strategies Performance Provide a chart such as the one shown on the right of the page, with some of the times missing and ask students to fill in the empty spaces. (SS1.5)

Resources/Notes Lesson 5 Writing Dates and Times SS1 (1.1/1.3/1.5/1.7) SS2 (2.1/2.2/ 2.3) SB p. 276-278 TR p. 28-31

Performance Ask the student to move the hands of an analog clock to match the time shown on a digital clock. Ask students to express the time orally and numerically that has been created on a 12-hour analog clock, 24-hour analog clock, and 12-hour (SS1.3) digital clock. Student-Teacher Dialogue Discuss when a 24-hour clock would be more appropriate to use than a 12-hour clock. (SS1.3) Portfolio Have students work individually to create a number line. Provide ribbon, clips and the following index cards and have students clip cards to the ribbon in appropriate places. -

00:00 hrs 12 a.m. Recess Lunch

-24:00 hrs - 12 p.m. -school ends (SS1.3)

Student-Teacher Dialogue Ask students what they might be doing when the clock reads 3 p.m.? 2 a.m.? (SS1.7) Performance Ask students to work in pairs to set up a schedule, using the 24-hour system, in which every student will get 30 minutes on the computer, starting at 10:00 hours. Ask: • Can all students in our class have computer time before noon, and if not, how long will it take to finish after lunch? • What time will the last student finish? (Remind them to leave time for recess) (1.8)


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Achievement Indicators: 4SS2 Continued 4SS2.1 Write dates in a variety of format .E.g: • yyyy/mm/dd • dd/mm/yyyy • dd/mm/yy • March 28, 2007

Writing dates in a numeric representation is faster than writing words and is often in use today. Have students explore newspapers, receipts, school registration forms, etc. to see how the dates are written. Consider numeric dates, as they appear in different forms, such as registration for school, newspapers, receipts, etc. and how many have different interpretations. It is important to note the confusion that the different combinations of numerals might cause. On an ongoing basis, encourage students to date their work in different formats.

4SS2.2 Relate dates written in the format yyyy/mm/dd to dates on a calendar.

Example: Show 2008/06/23 – Have students read the numeric date and circle the corresponding date on a calendar. Repeat for other dates.

4SS2.3 Identify possible interpretations of a given date; e.g., 06/03/04.

Example: 06/03/04 might mean 6th March, 2004 or June 3rd, 2004 or March 4th, 2006 While discussing numeric dates, it might be worthwhile for students to learn the words to "Thirty days hath September" which is still used by many adults to prompt them into recalling how many days there are in each month! The origin of the lyrics to “Thirty days hath September” is obscure but the use of olde English can date this poem back to at least the 16th century. There are many slightly differing variations. Here is one: Thirty days hath September, April, June and November; All the rest have thirty-one except February alone which hath 28 and sometimes 29

Or students might enjoy the “Knuckle Method” for remembering the number of days in each month: Make a fist showing four knuckles; start by pointing to the first knuckle and saying, “January.” The space between knuckles is February; the second knuckle is March, and so on. After saying, “July,” go back to the beginning, making August land on the first knuckle and continuing until year end. The months that land on the knuckles each has 31 days.

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General Outcome: Use Direct or Indirect Measurement to Solve Problems Suggested Assessment Strategies Performance Ask students to play a concentration game to match calendar dates that are written in a variety of formats e.g. yyyy/mm/dd; 06-03-09; dd/mm/yyyy; June 3, 2009 (2.1/2.1/2.3) Journal Send students on a scavenger hunt and ask them to bring in different dates from magazines, posters, items printed from the internet, cheques, and newspapers. Then share, discuss and display the variety of formats as a class. Ask students to explore what calendar dates can be confused with other dates when they are interpreted using various formats.

Resources/Notes

Lesson 5 Writing Dates and Times SS1 ( 1.1/ 1.3/ 1.5/ 1.7) SS2 (2.1/2.2/ 2.3) SB p. 276-278 TR p. 28-31

Math Game It’s About Time SS1 TR p. 32 SB p. 279

Journal • Ask students to write about their favourite format for recording a calendar date and explain their choice. Performance • Show the student a calendar for the year. Ask him/her to point out the day’s date and have them record it using one of the formats: - dd/mm/yy - yyyy/mm/dd - dd/mm/yyyy - May 23, 2009 Pencil-paper Ask students to write their birth date using the different formats.


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At the grade 4 level, students should participate in explorations that serve to deepen and expand upon previously learned measurement ideas and skills. Through investigations, students should come to understand that area refers to: “a measure of the space inside a region or how much it takes to cover a region.” (Van de Walle & Lovin, 2006, p. 234)

4SS3 Demonstrate an understanding of area of regular and irregular 2-D shapes by: • recognizing that area is measured in square units • selecting and justifying referents for the units cm2 or m2 • estimating area, using referents for cm2 or m2 • determining and recording area (cm2 or m 2) • constructing different rectangles for a given area (cm2 or m2) in order to demonstrate that many different rectangles may have the same area. [C, CN, ME, PS, R, V]

Achievement Indicators: 4SS3.1 Describe area as the measure of surface recorded in square units.

Select and design activities which help students to make sense of what the real life applications of finding the area measurement of something is. Have the students share some everyday contexts in which people need to know the amount of surface covered, such as painting a wall or tiling a floor. Encourage students’ development of ways to determine area by providing concrete experiences. In learning how to measure, students should be actively involved. Have squares of paper, tiles, cubes, base ten blocks and other suitable objects available. Talk briefly about how area differs from perimeter as students would have already explored perimeter concepts prior to grade 4. Review, with students, which unit is usually used to measure the perimeter of 2-D shapes. Ask if these units would be useful in measuring the area (or amount of surface covered) of 2-D shapes. As students explore area concepts, reinforce the importance of naming the measurement unit each time a measurement is said because the units communicate how big the measurement is. Without the unit name, there is no way of knowing what the numbers mean. It is also important that students learn that the units used to measure the area of an object (or to compare the areas of two objects) must be the same size. Length is one-dimensional measurement whereas area describes how many units (square units) are required to measure 2-dimensional surfaces. Sometimes square units refer to the space inside a region (inside the perimeter) such as the area of a field. Other times, square units measure how much it takes to cover a region, as in the number of tiles needed to cover a floor. Area is most often expressed in square units, such as square centimeters (cm²) and square meters (m²). Area measurements are often thought of as being “flat” and at this time, students will mainly investigate area of flat surfaces. Be aware, however, that there are instances in our environment where this might not necessarily be the case (e.g. farm acreage or a golf course which might include hills).

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Resources/Notes

Performance Using Circles, Squares and Triangles to Measure Area Cut out congruent shapes of circles, squares and equilateral triangles. Provide the students with packages of each shape as well as rectangular mat to measure the area. Have the students use the different shapes to measure the area of the mat and discuss the advantages and/or disadvantages of using each shape. (3.1/3.2)

Lesson 6 Measuring with Area Units SS3 (3.1/ 3.2) TR pp.36-39 SB pp.283-284

Performance Give students a shape made by tracing four yellow hexagon pattern blocks:

Be sure to include questions #5 and #6 on page on page 284 since they are important questions to consider.

Have students measure the shape using yellow pattern blocks, then try to cover the shape again using a different pattern block. Repeat with other pattern blocks. Ask students to record the areas according to the type of unit used to measure. E.g. “I used 8 trapezoid units” (3.1)

Sources: Teaching Student-Centered Mathematics K-3, Van de Walle and Lovin, 2006, pp.234-238 Making Math Meaningful to Canadien Students K-8, Marion Small, 2002008, pp. 388 - 411


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It is expected that students will: 4SS3 Continued 4SS3.2 Identify and explain why the square is the most efficient unit for measuring area.

During introductory work, students can explore measuring area with different types of non-standard units and then transition into using standard units to measure area. To help students identify the square as the most efficient unit for measuring area, have them measure a rectangle (such as a book cover) using pennies that obviously do not ‘tile’ or fit tightly together. When using units such as pennies, students will see that there are spaces not covered and are therefore not counted in the measurement. Consequently they end up with an inaccurate measurement. Students will come to understand that any object that fills a space can be used, but squares are most commonly used because they fit together on any side and because they make rows which are easy to count. It is important to point out, however that any units, that fit together with no spaces in between and are not overlapping, can also be used. Use Children’s literature such as A Cloak for the Dreamer by Aileen Friedman to reinforce this concept. Provide bristol board templates of various shapes (big square, small square, big triangle, small triangle, hexagon, rectangle, circle) and construction paper in different colors. After reading the book, explain that students have been asked to create a cloak and can illustrate their design on 12 x 18 white paper. After deciding on ONE shape to be used in the design (a variety of colors may be used) for the 12 x 18 cloak, students should trace and cut out the pieces to cover the cloak. Students are to arrange the pieces so that the sides of the same length match. These are okay:

These are not okay:

(Continued)

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Resources/Notes

Journal: Show students a rectangular placemat and 3 different shapes of approximately the same area:

Students may explore measuring using a cut out of these shapes until they come to a conclusion about which shape was the most efficient for measuring. Ask which they would choose to measure the area of a placemat and explain why. (3.2)

Lesson 6 (cont’d) Measuring with Area Units SS3 (3.1/ 3.2) TR pp.36-39 SB pp.283-284 Be sure to include questions #5 and #6 on page on page 284 since they are important questions to consider.

Performance: Iteration (repeating) with Pattern Blocks - Provide the students with rectangular papers that each measure 10 cm by 13 cm. Have them estimate how many copies of each shape of pattern block it would take to cover the rectangle. Then have the students measure the area using each of the shapes in turn.

(3.1/3.2) Performance Show students a variety of surfaces which have been partially covered with smaller units. Example: Teacher’s desk covered with 3 or 4 adjoining math texts. Estimate and measure other items around the room. Surface Shelf

Estimate 50 sticky notes


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It is expected that students will: 4SS3 Continued 4SS3.2 Identify and explain why the square is the most efficient unit for measuring area.

(Cloak for the Dreamer task continued) Suggest that they explore different arrangements before gluing them onto the paper. The shapes must touch each other without overlapping or leaving spaces in between. We want the Archduke to be really warm without holes in his cloak! Students will present their cloak designs to their classmates upon completion and tell the measurement of the area and tell which unit they used to make and measure the cloak. By allowing students to make choices of which unit to use to measure an area, they are also afforded the opportunity to explore area concepts. Some students will use squares and others may select circles. As the discussion evolves take opportunities to discuss and compare the effectiveness of each area unit. Ask: • Which area unit gives a more accurate measure? • Which area unit is easier to count? • Why would leaving gaps not give an accurate measurement? (all spaces are not covered) • Why would overlapping not give an accurate measurement? Some spaces are counted twice) Following this activity elicit student responses as to why squares are the most efficient unit for measuring. Answers might include: • “squares do not leave gaps” • “you can count squares by rows” • “squares fit no matter which way to turn them” As students begin to find areas they will often have available multiple copies of the measuring tool, for example cubes or square tiles. However, some may want to iterate (move one tile from one location to another) if there are not enough square tiles available. Provide modeling clay for students to roll out to form a rectangle of any size. Give them one interlocking cube and instruct them to use it to find the area of their rectangle. Beginning iteration this way makes it easier for students to keep track of the iterations and total number of units. Although the area formula (length x width) is not expected at this grade level, student will eventually use their knowledge of multiplication to make iterating easier. If you prefer you may have students use a stamp pad to complete the same task.

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Resources/Notes

Performance Provide paper and scissors. Have the students make a nonstandard unit for area that they could use to measure their desktop. Remind them that the desk must be completely covered with the unit they choose and that they must repeat their unit at least four times to measure the area of their desk. They should be ready to justify their choice of unit.

Lesson 6 (cont’d) Measuring with Area Units SS3 (3.1/ 3.2) TR pp.36-39 SB pp.283-284

Performance Provide the students with square units to measure their desks or measure congruent mats. Give some groups large squares and other groups small squares. Compare the answers for the areas found. Generalize: to compare areas, the same size unit of measure must be used; i.e., either small squares or large squares. Review the fact that the smaller the unit used to measure area the more of these units are needed. (Adapted from Alberta Education, Diagnostic Mathematics Program, Elementary: Measurement, Division II (Edmonton, AB: Alberta Education, 1990), pp. 142–143.)

Be sure to include questions #5 and #6 on page on page 284 since they are important questions to consider.

Curious Math Pattern-Block Areas TR p. 40


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It is essential to provide students with opportunities to measure irregular shapes, since the real-life applications of area measurement apply to all 2-dimensional shapes or regions, not just rectangular ones. The measurement of irregular 2 dimensional shapes can be explored in the following ways: • students drawings on grid paper • transparent grid overlays - Opportunities should be provided for students to estimate and calculate the area of various surfaces. Laying an acetate centimetre grid over objects is helpful when determining the area of a surface. • students might investigate the area of shapes drawn on centimetre dot paper. Strategies for doing this include adding squares and half squares within the figure; placing a rectangle around the shape, determining its area, and subtracting the area of the "extra" pieces • using geoboards • cut and reassemble – students can be said to have ‘conservation of area if, when shapes are cut and rearranged they realize that the same amount of space is covered. Provide tangram sets to pairs or individual students. Have students assemble the tangram shapes into a square. Ask: Do all the tangram squares have the same area measurement? Next, students are instructed to reassemble the tangram pieces into a design of their own choosing, using all pieces. Ask: Does you design have the same area measurement as the square you started with? Discuss with students how we might determine whether all of their designs had the same area or not. (Reassemble design into original tangram square)

4SS3 Continued Achievement Indicators: 4SS3.8 Determine the area of an irregular 2-D shape, and explain the strategy.

Pentominoes may also be used to illustrate this concept. Pentominoes are shapes each made up of 5 squares, all of which must have at least one side matching up with a side of another.

Comparison activities should be designed to help students discover that it is necessary to apply the same unit of measure when comparing 2 different areas. A common misconception is for students to rely on number alone, without considering the size of the units.

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General Outcome: Use Direct or Indirect Measurement to Solve Problems Suggested Assessment Strategies Performance Provide students with square grid paper. Instruct them to make a quilt design, using 3 colours, that covers an area of 200 square units. Find the area of each colour. Ask: How does the area of each colour compare to the total area of the design? e.g. green – 125 square units yellow – 50 square units blue – 25 square units 200 square units Performance Use triangles to create and display a simple design on the overhead projector. [Note: The triangles used in the design should be able to form a square when reassembled.] Ask students to determine the area of the design in square units and record the answer. Invite a student to come to the overhead projector and reassemble the design squares before counting the square units. Students compare the answer found to their own answer. Repeat with different numbers of triangles and/or squares. (3.8)

Resources/Notes Lesson 7 Counting Square Units SS3 (3.1/ 3.8) SB p. 286-288 TR p. 41-44 Be selective when choosing tasks in the resource.

Math Game: Area Logic SB p. 289 TR p. 45

Performance Provide each group of students with a copy of three shapes, such as the following, that could represent different garden plots. Have lima beans, tiles, buttons, pattern blocks and grid paper available for the students to use. Present the following problem: Mr. McGregor wants the largest possible garden plot to plant his carrots. He knows that he has to share some of them with the rabbits. Which garden plot should he choose? Estimate first, and then find the area of each garden plot. Explain your thinking. Examples:

• Have the students explore different ways to find the area of the garden plots, such as covering the surface with objects such as lima beans or tiles, a grid and counting the squares, and drawing a grid on the paper. Ask the students to share the different ways that they might find the area of the garden plots. Then have them choose a method to solve the problem. Have the students share their answers to the problem and discuss which method they think is the most accurate in finding the areas.


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It is expected that students will: 4SS3 Continued

4SS3.3 Provide a referent for a square centimetre, and explain the choice.

4SS3.6 Estimate the area of a given 2-D shape, using personal referents.

Review the linear units (centimetre and metre) used to find the perimeter of 2-D shapes. Explain that these standard units of measure were used to find perimeter so that the perimeters of shapes could be compared and communicated clearly. Connect the need for standard units in finding perimeter to the need for standard units in finding area. Remind students that measuring with different units makes a difference to the answer. This is a good lead-in to why we use standard units. Rather than measuring with non-standard units, which can mean different things to different people, we use standard units that everyone can understand. The first standard unit students encounter is the square centimeter. A square centimeter is an area equivalent to the area of a square with a side length of 1 cm. (Small 2008. Write the symbols and explain that 1 cm2 is read as "one square centimetre," not "one centimetre squared." Note: One square centimetre is a measure for the area of a variety of shapes, whereas one centimetre squared is a square that is 1 cm on each side. Ask students to carefully trace one face of a Base 10 unit cube. Ask: What are the side lengths of the resulting square? [Answer: 1 cm] Ask: How can we tell how much space lies inside the square? [They might respond with “1 cm of space”] Their responses will give clues to misconceptions. Use their responses to elicit the idea that the area inside the square is represented as 1 cm2. Referents are familiar objects that students can refer to, or visualize, to help them have a strong understanding of a unit of measurement. Review the referents used for centimetre (the width of the pinky finger). Ask the students to suggest a suitable referent for 1 cm² and explain why they think it would work. Have the students use their referent for 1 cm² to estimate the area of a book cover in square centimetres. Then, have them check their estimate by finding the area of the book cover by overlaying a transparency of a centimetre grid. Follow up: Ask what objects around their desks/classroom might we use the standard unit of 1 cm2 to estimate the area of? Would you try to measure the area of the floor using your personal referent (pinkie fingernail? Why or why not? The following are discussion starters: - Using your referent, estimate the area of a napkin/desk top/door/white board, exercise, etc. Explain your strategy. How many fingernails could cover a crayon box?

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Resources/Notes

Performance HIDE THAT STAIN! Present class with a pair of old jeans or t-shirt that has an obvious stain on it. Have students estimate the area of a patch that would cover the stain. Next, instruct students to cut a patch from centimeter grid paper to match their estimate. Have students check the effectiveness of their “patch” by using it to cover the stain.

Lesson 8 Using Square Centimeters SS3 (3.1/ 3.3/ 3.6/ 3.7/ 3.8) SB p. 290-293 TR p. 46-49

Ask: What is the area of the smallest patch that might be used to cover the stain? Are there any patches of different shapes with the same area? How could we order our patches according to their area measurement? (3.6) Performance Ask students to estimate how many math texts [units] would be needed to measure the area of the desk surface. Ask: What might the area of the desktop be if 5” x 7” file cards were the unit of measure? How would the area measure change if we used a sheet of Bristol board to measure? Next, distribute 5 sticky notes to pairs of students. Have them chart estimates for area measurements of different objects found through the classroom using sticky notes as the unit of measure. Pair share estimates and discuss reasons for similarities or differences. Variation – some pairs of students can use sticky tabs twice or four times bigger than other pairs. Performance Provide the students with centicubes or the units of the base ten materials and explain that the area of one face is 1 cm2. Also, provide them with centimetre grid paper on which they may place the centicubes or base ten units. Have the students use their centimetre rulers to measure the side of each square on the centimetre grid paper to verify that each square is 1 cm².


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It is expected that students will: Small (2009) suggests a sequence for the introduction of square centimetres. First, the faces of centimetre cubes (from Base 10 materials) can be used as models. Then, centimetre grid paper and 4SS3.7 Determine the area of a transparent centimetre grids can be used to illustrate square regular 2-D shape, and explain the centimetres pictorially. Both grids and cubes are good transition tools strategy. from non-standard units because they can also be used to cover and count, even before they are recognized as standard units. It is 4SS3.8 Determine the area of an important that students realize that the squares can be cut and irregular 2-D shape, and explain rearranged to form many different shapes. Ask students to make as the strategy. many different shapes as they can with 5 cm². Show them to the class to help students see that although they all have an area of 5 square units, they look different. E.g. 4SS3 Continued

When recording and reporting area measurements in standard units, students need to realize that it is important to state the square unit of measure, usually square centimeters (or square metres). It is recommended that the use of words precede the use of abbreviated form in order to facilitate conceptual understanding. Students will be comfortable, eventually, with the understanding that an area measuring 14 square centimetres can be written as 14 cm2. Provide students with multiple opportunities to determine area of a surface. It is important to have students estimate first and then choose a method to calculate the area of different surfaces. Provide students with shapes drawn on grid paper, transparent grid paper and shapes (both regular and irregular shapes) to determine the area. E.g.

Area = 7 squares (7cm²)

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Area= 7 full squares + 2 more halves which is 8 squares (8 cm²)


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General Outcome: Use Direct or Indirect Measurement to Solve Problems Suggested Assessment Strategies Performance Provide envelopes or box lids (e.g. shoebox lids) of varying sizes. Ask students to use base ten blocks to measure the area of each. E.g.

Resources/Notes Lesson 8 (Cont’d) Using Square Centimeters SS3 (3.1/ 3.3/ 3.6/ 3.7/ 3.8) SB p. 290-293 TR p. 46-49

“I used 2 hundred centimeters and 40 centimeters to cover the entire envelope, so the area is about 240 square centimeters.” Performance Provide students with pairs of paper rectangles, such as the following: First pair: 1 cm by 9 cm, 3 cm by 6 cm Second pair: 1 cm by 10 cm, 3 cm by 5 cm Provide the students with scissors, transparent centimeter grid paper overlays, squares that are each 1 cm2 and centimeter rulers. Have the students decide which rectangle in each pair has the greater area and explain their thinking. Encourage the students to share their ideas and critique which strategy, for finding area, works best for them. (3.7) Performance Provide congruent shapes such as the following. Decide if Part R has the same area as Part S. Explain your thinking.

(3.8) Performance Make the design below on an overhead geoboard and ask a student to explain to the class how to find the area. Have the students alter the shape on their geoboards to increase the area by 1 cm2.

Performance Ask students to circle the letters of the shapes that have the same area as the one on the left:


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Review the referents used for metre (the distance from the teacher's finger tip to his or her opposite shoulder). Ask the students to suggest a suitable referent for one square metre and explain why they think it would work. Discuss possible referents for one square metre after the students make a square on the floor that is one square metre, using masking tape or other suitable materials. Have the students use their referents and estimate the area of a large tabletop or a section of the classroom floor. Use a square piece of paper that is 1 m by 1m to measure the area and check the estimates. The ability to visualize standard units in different configurations is useful when estimating area of some objects. For instance, the area of the surface of a shelf or long, narrow countertop would take a long time to estimate in square centimetres. Square metres would give a faster estimate, but the student would first need to understand that an area of 1 m2 can take on different shapes.

4SS3 (Continued) 4SS3.4 Provide a referent for a square metre, and explain the choice.

Ask students to use their arms to show you how much area a square metre would cover. Hold up a metre stick and ask how this measuring tool might help us determine a more precise representation. Make 4 metre sticks available and guide students to the discovery that the area inside a square with 1 metre sides represents the standard unit called 1 m2. Provide students with a metre stick, newspaper, gift wrap or paper; scissors and tape. Ask them to make a model of a square metre. This model can be now used as a referent to measure larger surfaces. Display it in the classroom for several days during your focus on measurement. Ask: How can we find the area of our whiteboard? Discuss if it would be a good idea to find the area using square centimeters and students should conclude that this activity would not be practical since it would take a very long time to find an answer. To extend on students’ understanding that the area of a square metre can take on different shapes, provide groups with a pre-cut newsprint square measuring 1 m2. Pose problem: What would a triangle with an area of 1 m2 look like? How many rectangles can you make with an area of 1 m2? What irregular shapes have an area of 1 m2? Have students cut and reassemble shapes with tape.

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General Outcome: Use Direct or Indirect Measurement to Solve Problems Suggested Assessment Strategies Performance Playground Makeover. Distribute paper grids to represent the area of an abandoned lot. Each square on the grid represents 1 m2. Tell the students that their job is to draw the plans that will be used to turn the lot into a functional and pleasant play area for children. Each plan must include: • a 16 m2 sandbox • a 100 m2 grassy field • a 16 m2 rectangular flower bed • a 8 m2 area for every swing set • three 2 m2 picnic tables Instruct students to draw or cut and paste the features onto the grid paper in a way they feel is most pleasing. Label each feature.


Resources/Notes Lesson 9 Using Square Meters SS3 (3.1/ 3.4/ 3.5/ 3.6/ 3.7/ 3.8) SB p. 294-295 TR p. 50-52

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It is expected that students will: 4SS3 Continued Achievement Indicators: 4SS3.6 Estimate the area of a given 2-D shape, using personal referents

Allow students to work in pairs to find the area. Students are likely to come up with differing answers for the same region. Discuss why groups might have different answers and avoid saying there is one ‘right’ answer. Students should share their strategies so others are exposed to many different strategies. Personal referents help students to estimate. The various techniques are useful for estimating area: • referents – use a referent for the single unit of measure and iterate to obtain the estimate; e.g., use the size of the fingernail on your small finger as a referent for 1 cm2 • chunking – estimate the area of a smaller portion of a shape initially and use this estimate to estimate the entire area of the shape; e.g., estimate the area of the smaller section of the floor and then multiply that answer by the number of these sections in the entire floor. Have students use their referents to estimate the area of a napkin, a desktop, a door, a white board, an exercise book, etc. Explain their strategy. Next challenge students to find out how many fingernails would cover a crayon box? Once students have developed personal referents for standard units to measure area, they need ongoing opportunities to apply their understanding to problem solving situations. It is important to allow enough time for students to share solutions. Ensuing discussion will likely provoke thinking as to how shapes with the same perimeter can have different areas. Students will benefit from sharing strategies with one another. Talking about thinking serves to clarify thought for the speaker and it is also in this setting that students sometimes discover their own misconceptions. Listeners benefit from hearing the ideas of others because they get to see other approaches and strategies besides their own. Listening to the thoughts of others may elicit new ideas or questions in the group.

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General Outcome: Use Direct or Indirect Measurement to Solve Problems. Suggested Assessment Strategies

Resources/Notes

Performance Using newspaper, have students make area models for one square metre. These should be used to explore the area of larger regions in the activities below:

Lesson 10 Estimating Areas SS3 (3.4/ 3.6/ 3.8) SB p. 296 TR p. 53-55

Performance Gymnasium Math - Ask students to choose a personal referent to estimate the area inside a hula-hoop. Is it greater, less or about the same as 1 m2? Divide the class in half. Ask each group to come to a concession as to the approximate area of half of the gymnasium. Students elect someone from their team to record predictions and someone to explain their estimation strategy to the class. Following sharing, have teams each measure the area of one half of the gymnasium using their newsprint models. Record results and compare with predictions. Both groups will come together to share findings. Were the results similar? Should they be? Performance Divide class into pairs. Supply pairs with sheets of newsprint, tape and scissors. Ask them to make a rectangle that is 2 m2and to explain their strategy to the group. As students to generate a list of objects that might represent a given measurement such as 1 m2. As each student contributes to the list, he/she should explain why they think their contribution is reasonable.

[Additional practice may be required]

Curious Math (optional) SB p. 297 TR p. 56 This is a good activity if time permits.

Performance Divide class into groups. Distribute to each group: 6 m2 lengths of string, old newspapers, scissors, tape and student made 1 m2models prepared in previous activity. Pose this problem: Imagine that some milk has spilled from a 2L container. Instruct students to tie the string together at both ends and next to form the loop of string into an irregular shape on the floor. Say: The shape inside the string represents how much milk has been spilled. Can you find the area of spilled milk?


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It is expected that students will: 4SS3 Continued 4SS3.9 Construct a rectangle for a given area.

Provide opportunities for students to explore different strategies to accomplish the following task: Examples: Give groups of students a select number of tiles to construct different rectangles and then compare and discuss results. Some students might relate this activity to the construction of arrays used to represent multiplication number sentences. [See also N 6] - Students can use geoboards to construct rectangles of a given area. Objects of different shapes can have the same area. Figures A & B have the same area even though the perimeter of A is greater.

4SS3.10 Demonstrate that many rectangles are possible for a given area by drawing at least two different rectangles for the same given area.

Students may not realize that an area which is rearranged into different shapes will still have the same area measurement as the original shape. Provide students with geopaper or squared dot paper to draw or construct rectangles. Ask: How many rectangles can you find with an area of 12 square units. Examine student work to find all of the possibilities. Cut and paste a sample of each type of rectangle to a coloured background to display on bulletin board. Note that some students, through their exploration of tiling and arrays, might discover the multiplication formula for finding area on their own, even though this would not be an expectation at this level. Ask students how we would know if we’ve come up with all the possibilities for rectangles with an area of 12 cm2? This may be an opportune time to discuss how organized lists can help us to keep track of information.

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Resources/Notes

Performance Lesson 11: Have students use colour tiles or grid paper to investigate the numbers Solving Problems using Organized Lists from 1 to 30 to see how many different rectangles can be made for each. Students should record their results and look for patterns. (3.10) SB p. 298-299 TR p. 57-59 Performance Invite the students to select 16 pieces of the same pattern block (the blue rhombus, for example). Using the rule that at least one side must match up exactly with one side of another block, have them make different shapes, all of which have an area of 16 units. Ask them to find, among others, the most compact shape and the longest shape (3.10) Performance Provide the students with tiles and centimetre grid paper. Give them the following instructions: For each of the areas from 1 cm2 to 20 cm2, find all the possible rectangular arrays using whole numbers. For example, the possible arrays for an area of 6 cm2 would be as follows: (3.10) Performance Divide class into groups of 4. Each student is given tiles and grid paper. Student A creates a rectangle on grid paper which the others cannot see, and records the area in cm2. Students B, C, and D are asked to create a rectangle of the same area. The team then analyzes the various responses. Ask: What do your responses tell us about area? Can you make any connections or see any patterns that relate to the area of rectangles? (3.9)


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Suggestions for Teaching and Learning .

It is expected that students will: 4SS3 Continued 4SS3.9 Construct a rectangle for a given area.

4SS3.10 Demonstrate that many rectangles are possible for a given area by drawing at least two different rectangles for the same given area.

Present the following scenario to the students: Grandma is making a quilt with three patches of blue cloth on a white background. She needs help deciding where to put the patches. If grandma wants the patches to cover the same amount of space on the white background, should she scatter them about or put them together in a group? Explain your answer. Provide the students with three pieces of blue paper to represent the patches of blue cloth. Ask them to move the papers around on their desk, which can be the white background for the quilt, and decide which arrangement would be best for the quilt. If necessary, provide scaffolding by moving three pieces of paper on the overhead projector into different positions. Example:

Through discussion, have the students verbalize that grandma can put her three patches of blue cloth anywhere on the white background and they will always cover the same amount of surface. Changing Shapes - Provide each student with at least four congruent squares each of a different colour. Have the students compare the squares by superimposing one on the other to show that they are congruent – same size and shape. Reaffirm that these squares each cover the same amount of surface or have the same area. Have the students cut one of their squares along the diagonal to make two triangles. Instruct them to rearrange the two triangles to make as many different shapes as possible with two sides aligned. Encourage the students to make other designs by cutting a square in different ways and rearranging the pieces. Reinforce that the pieces of one square must be rearranged to make one design – the pieces are the same colour. The students may glue their designs on newspaper and place them into groups on the floor, justifying the categories. Ask the students to describe the differences and similarities among the designs. Guide the discussions to generalize that all designs cover the same amount of surface or have the same area.

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General Outcome: Use Direct or Indirect Measurement to Solve Problems Suggested Assessment Strategies Performance Sammy wants a dog pen for his dog. He wants the pen to have an area of 32 m2. Draw all the possible rectangular pens that have the area of 32 m2 on centimetre grid paper. The sides of the rectangles must be measured in whole numbers. Explain how you know that you have drawn all the possible rectangular pens. Which pen would you advise Sammy to use? Explain why. (3.10)

Resources/Notes Lesson 11 Solving Problems Using Organized Lists SB p. 298-299 TR p. 57-59

Pencil Paper Provide the students with a template for the Frayer Model and have them fill in the sections individually or as a group to consolidate their understanding of area. A sample of a Frayer Model is provided below:


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It is expected that students will: 4SS3 Continued 4SS3.6 Estimate the area of a given 2-D shape, using personal referents.

As students gain experience with area measurement, they may extend on their base of personal referents. The Base 10 flat, for example, is a handy tool to visualize the surface area covered by 100 cm2. The largest face of a standard white vinyl eraser may be used as a referent for about 10 cm2.

4SS3.8 Determine the area of an irregular 2-D shape, and explain the strategy.

Standard unit tools that students may find useful in determining the area of irregular 2-D shapes, such as hand or foot prints, include but are not limited to: • transparent centimetre grid paper that can be used to overlay a shape in order to find the area in square centimeters. Laying an acetate or transparent centimeter grid placed on the top of an object provides a pictorial model for measuring area. It allows students to count the number of units that cover or partially cover the shape. • centimetre grid paper upon which an object is laid and the area to be measured in square centimetres is traced. Area measurement is then determined by counting the squares and part squares. Ask students to estimate and determine the area of familiar objects such as mittens, leaves and fancy cut sticky note sheet. Have student explain their chosen strategies. In measuring irregular shapes there may be lots of units that only partially fit. Students will count full units and can visually put together parts of units to count as one unit. Students may use colored marks to represent whole units and use a different colored mark to represent ½ units. Give students centimeter grid paper and ask them to use centimeters and half centimeters to make a design. Have them record the area of their design and pass it to the teacher. Assign a letter to each design and display all designs in an area where students have easy access. This activity can be used in a center, during non-instructional times or when students finish other tasks. Ask students to find the area for each shape, write their answers on paper (e.g. A is 21 square centimeters, B is 45 square centimeters, etc.) and deposit their answer sheet into a nearby box. At the end of the week, reward students for “Measuring up”!

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Resources/Notes

.

Lesson 12 Estimating Areas on Grids SB p. 300-301 TR p. 60-62

Pencil-Paper Students use small cardboard boxes to design a house for a small toy figure. Then students write a description of the house in their journal using area estimates or measurements to describe the floor, roof, windows, etc. Performance Provide students with equal size pieces of modeling clay. Set a time limit for students to roll out the largest surface area they can make (without holes). Have students choose and explain a strategy to determine the area measurement of their clay “blob”. Performance Have students make paint blotches by dropping a spoonful of paint onto centimetre grid paper. Allow to dry. Choose a method to determine area of blotches and explain procedure used. Post blotches and measurements for comparison purposes.

Chapter Task Making a Photo Display TR pp. 68-69 SB p. 305

Performance Use the partial floor plan for a school as shown below to complete the instructions that follow.

Find the area of the computer room. Explain your thinking. Find the area of the office. Explain your thinking. Performance The area of the entire design below is 12 m². Find the area of the shaded part.


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It is expected that students will: 4SS3 Continued 4SS3.6 Estimate the area of a given 2-D shape, using personal referents.

Have students work in pairs to trace around odd shaped items, for example their own hand with fingers together, a leaf, etc. Next, cut the shapes out. Students then estimate the area of each shape. After finding the approximate area of each shape, have students place the items in order from least to greatest. Pick some students to present their findings and describe their strategy for finding the area, to the rest of the class.

4SS3.8 Determine the area of an irregular 2-D shape, and explain the strategy.

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Resources/Notes

The Lake and Island Board uses one square centimetre as the basic unit for measuring area. Provide the student with copies of the Lake and Island Board shown below without the centimetre grid. Make a transparency of the board without the centimetre grid to use for discussion with the whole class.

Lesson 12 Estimating Areas on Grids SB p. 300-301 TR p. 60-62

Present the students with a set of problems, such as the following: • Suppose you and your family are moving to Lake and Island Country. You wish to purchase the largest island available so that there is plenty of room for the family. Which island would you purchase? Estimate first and then find the areas of the islands to determine your answer. Do you have a choice? If so, what would be the advantages and disadvantages of either choice? • Some friends of yours, the Jones, also wish to purchase an island. They want one that is only one-third as large as yours. Which island would you suggest that they buy? Do they have a choice? If so, what is it? • As time goes by, members of your family grow up. Two of the grown children, Lauren and Kevin, each wish to buy an island the same size. Which two islands would you suggest they purchase? Estimate first and then find the areas to determine your answer. • A plant disease epidemic breaks out and all the islands have to be sprayed. What is the total area of all the islands? • The Jones' family wants to double the amount of land they now own. Which set of islands could they buy to satisfy this need? • Name a pair of islands in which the area of one island is three times the area of the other island. • Name a pair of islands in which the area of one island is double the area of the other island. • Draw as many different rectangular islands with whole number dimensions that are the same area as Island D. Explain your thinking. (3.10)


Chapter Task Making a Photo Display TR pp. 68-69 SB p. 305

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Unit 9 MULTIPLYING MULTI-DIGIT NUMBERS Suggested Time: approx 3-4 weeks

MULTIPLYING MULTI-DIGIT NUMBERS

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This unit is a continuation of unit 6. In this unit, students will review strategies that learned early and now apply them to multiply two and three digit numbers. Students will experience greater success with applying these strategies to larger numbers if they can automatically recall the multiplication facts. The outcomes addressed in this unit prepares students for multiplication of two-digit by two-digit numbers which will be studied in Grade 5.



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Strand: Number Specific Outcome It is expected that students will: 4N5 Describe and apply mental mathematics strategies, such as: • skip counting from a known fact • using doubling or halving • using doubling or halving and adding or subtracting one more group • using patterns in the 9s facts • using repeated doubling to determine basic multiplication facts to 9 × 9 and related division facts. [C, CN, ME, R] 4N6. Demonstrate an understanding of multiplication (2 or 3-digit by 1-digit) to solve problems by: • using personal strategies for multiplication with and without concrete materials • using arrays to represent multiplication • connecting concrete representations to symbolic representations • estimating products • applying the distributive property. (Cont’d) [C, CN, ME, PS, R, V] Achievement Indicators: 4N6.4 Refine personal strategies to increase their efficiency.

Suggestions for Teaching and Learning By solving problems in contexts that relate to the children’s own lives, the students use their prior knowledge to make sense out of the problem and then use computational strategies that they are able to explain and justify. The student’s understanding of multiplication is enhanced as they develop their own methods and share them with one another, explaining why their strategies work and are efficient to use (Principles and Standards for School Mathematics, 2000, p. 220).

When solving a multiplication problem, work with the whole group initially and have the students paraphrase the problem to enhance understanding (Willis et al. 2006) before they record their process. Provide a variety of materials such as base ten blocks, counters, chart paper and markers.

There are many good reasons why students should be encouraged to use personal and varying strategies for multiplication. Sometimes one strategy or algorithm makes more sense to a student than another one or works better for a particular set of numbers. Sometimes students may get help from a parent who exposes them to a strategy that is different than those he/she learned at school. It is helpful for students to be ‘open’ to both. Also, strategies that students ‘invent’ are usually more meaningful because they ‘created’ it and they are better able to apply them because it ‘makes sense’ to them. Another benefit of students’ hearing varying strategies is that it means a student can use one strategy to solve the computation and another one to check it. Allow plenty of time for students to explore their own strategies and to develop a range of strategies. This understanding will help students become more efficient and have a better understanding of the traditional algorithm. As children are working using their invented strategies encourage movement towards more efficient strategies.

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Resources/Notes

Journal Writing/Self Assessment Book ~ Classroom Assessment Model, P. 72)

Lesson 1: Exploring Multiplication N5 N6 (6.7) SB p. 310 TR p. 13-15

Additional reading: Big Ideas from Dr. Small (Small, 2009) pp. 25 - 41

Teaching Student-Centered Mathematics Grades 3-5, (Van de Walle & Lovin, 2006)pp.113120

Math Game – Twenty-Four N5 SB p. 311 TB p. 16-17


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Strand: Number Specific Outcome It is expected that students will: 4N6 Continued 46.7 Solve a given multiplication problem, and record the process.

Suggestions for Teaching and Learning In problem solving situations where multiplication is involved, you will need to draw on prior knowledge by reviewing multiplication facts. Emphasize the connections among the story problems, the models/diagrams, the number sentences and the personal strategies used in calculations. Present the students with a problem or a number sentence involving multiplication, such as the following: • Laura wanted to make beaded bracelets for her friends. She has 6 friends and each bracelet will require 45 beads. How many beads will Laura need to purchase at Michael’s craft store? Have the students divide their page into four sections to make graphic organizers and label them as follows:

Have the students complete the graphic organizer by writing the story problem or the number sentence in one corner and filling in the other corners appropriately. This will help the students organize their thoughts before recording their process. Provide the students with a Multiplication problem and then ask them to prepare an infomercial for a fictitious TV show, Math News. They will need to explain their personal strategies for solving the given multiplication problem. Students may use concrete materials or pictures to demonstrate the personal strategies they used to solve the problem. Have students present their infomercial. While the students are presenting their commercial, look for evidence that personal strategies were used, personal strategies were effective and solved the problem accurately and that students included models, illustrations, symbolic representations in their descriptions of personal strategies

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Resources/Notes

Student/ Teacher Dialogue Conduct interviews with the students to determine their abilities to use personal strategies when solving given multiplication problems. During the interview record observations about how each student applies personal strategies and also their confidence in solving the problem

Lesson 1 (Cont’d)

Pencil/Paper Solve the following problems. Record the process: • Your class is celebrating the end of your Australian unit in Social Studies. If there were 24 people in your class, and each person wanted two Anzac cookies, how many Anzac cookies would you need to make for the celebration? (48) If the Anzac cookie recipe makes 12 cookies, how would you make 48 cookies? • You have 15 cookies to share equally among 3 children. How many cookies will each child receive? • Hannah walks her dog for 48 minutes each day. How many minutes does she walk in 3 days? • Last year you saved $32.00. This year you saved 4 times as much money as last year. How much money did you save this year? • How many different single-scoop ice cream cones can be made with 4 different kinds of cones and 28 different flavours of ice cream? (N6.7) Presentation Have students create a picture book for the class library explaining the personal strategies they used to solve a given multiplication problem. Students could also design a cover for their Math picture book indicating a title, the author and illustrator and share them with other classrooms. When reviewing the book that each student has created, look for evidence that they clearly described the process (in words and pictorially) they used to solve the multiplication problem using their personal strategies. (N6.7) Have students assess each other’s work by completing a peer assessment sheet, using criteria established as a class such as the following examples: • The presentation and explanations were clear. I understood what ________ was trying to say. • Here is what I think ________ said. • This group used graphic presentations that were clear and had something important to show. • _______’s strategy of _________ solved the problem correctly. • This group used appropriate mathematical vocabulary. • Something that _______ did really well was _____. • A question I would like to ask _______ about is _____.


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It is expected that students will: 4N6. Continued

There is value to exposing students to products involving multiples of 10’s and 100’s. Students will use this understanding to help them compute multiplication situations.

Achievement Indicators: 4N6.2 Use concrete materials, such as base ten blocks or their pictorial representations, to represent multiplication; and record the process symbolically

The use of manipulatives or models helps the students to understand the structure of the story problem and also connects the meaning of the problem to the number sentence (Van de Walle, 2001, p. 108). To develop understanding of the meaning of operations, the student needs to connect the story problem to the manipulatives, create a number sentence and then use personal strategies to solve the problem. Computational strategies for multiplication are more complex and it is important that students think about number meaning and not ‘digits’. For multiplication the ability to break numbers apart is important. Give students time to practice breaking apart numbers and to make sense of them. For example

4N6.6 Model and solve a given multiplication problem, using an array, and record the process.

Students can be taught to multiply by breaking up the multiplier.An array is a good way to show this. For Example, 3 x 12 can be viewed as 3 x 10 and 3 x 2;

You can easily separate 6 rows of 5 squares into 2 groups, each with 3 rows of 5, without changing the total number of squares (Small, Prime, p. 58, 2005).

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Resources/Notes

Lesson 2: Multiplying 10s and 100s N6 (6.2/ 6.7) SB p. 312-313 TB p. 18-21

Lesson 3: Multiplying Using Arrays N6 (6.1 / 6.6 / 6.7) SB p. 314-317 TB p. 22-25


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It is expected that students will: 4N6. Continued 4N6.1 Model a given multiplication problem, using the distributive property; e.g., 8 × 365 = (8 × 300) + (8 × 60) + (8 × 5).

Students can learn that they can multiply in parts i.e. the Distributive Property. This property lets you separate numbers into parts so that the numbers are easier to work with. For Example: How many weeks are there in 3 years? To solve, you can find the product of 3 and 52. To multiply in your head, you can use the Distributive Property: 3 x 53 = 3 x (50 + 2) (3 x 50) + (3 x 2) 150 + 6 = 156 There are 156 weeks in 3 years. The Distributive Property also tells you that you can multiply a difference by multiplying each part separately and then subtracting the products. For Example: 6 x 19 = 6 x (20 – 1) (6 x 20) – (6 x 1) 120 – 6 = 114

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Resources/Notes

Student/Teacher Dialogue Explain why the following solution is correct or not: 6 x 128 = (6 x 100) + (6 x 20) + (6 x 5) = 600 + 12 + 30 = 642

Lesson 4: Multiplying Using Expanded Form

Model the following multiplication problem using the distributive property: 6 x 256 = (N6.1)

N6 (6.1 / 6.2 / 6.7)


SB p. 318-321 TB p. 26-29

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It is expected that students will: 4N6. Continued 4N6.5 Estimate a product, using a personal strategy; e.g., 2 x 243 is close to or a little more than 2 x 200, or close to or a little less than 2 x 250.

Students should be encouraged to use ‘reason’ when estimating the answer to a computation (Small, Prime, 2005, p. 139). To estimate products, students need to know the multiplication facts and how to multiply with multiples of 10, 100, 1000, and so on. Ask students if they have lived closer to 300, 3000, or 30 000 days, and then have them explain how they know. For example, one year has 365 days, so it can’t be 300 days. Ten years is just a bit less than 10 x 400 days or 4000 days, so 3000 makes the most sense because 30 000 would be much too high. A number of factors come into play when making decisions about estimating such as the context, and the numbers and operations involved. Students should be aware that estimates which involve multiplication and division with greater values often tend to be further from the actual values than is the case when estimating with addition and subtraction. To estimate products students might: • round one or both numbers to the nearest multiply of 10, 100 0r 1000 … for example, 25 x 52 is about 25 x 50 = 1250 • round one factor up and one factor down. For example, 65 x 12 is about 60 x 20 = 1200 • round numbers such that familiar multiplication facts can be used. For example, 4 x 51 ( I know 4 x 5 - think 40x 5 =200 + 4 x 1 = 4 so the answer is 204) (Small, 2009)

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Resources/Notes

Student/Teacher Dialogue Ben estimated 47 x 7 as 500. Ask the student to explain Ben’s thinking, and if he/she would estimate it differently. (N6.5)

Lesson 5: Estimating Products N6 (6.5 / 6.7)

Student/Teacher Dialogue Ask the students to give an estimate for each of the following and explain the strategy they chose. 79 x 6 = 215 x 7 = (N6.5)

SB p. 322-324 TB p. 30-33

Student/Teacher Dialogue Estimate the following product using a personal strategy: 2 x 243 = (e.g. 2 x 243 is close to or a little more than 2 x 200, or close to or a little less than 2 x 250) (N6.5) Student/Teacher Dialogue You travel 375 km each day for 3 days. Will you reach the cabin that is 1200 km away by the end of the third day? Explain. (N6.5) Journal Writing Ask students to explain in writing the estimation strategies for each equation below, and to decide which estimation is closer to the actual product. 79 x 9 as 80 x 10 or 80 x 9 17 x 15 as 8 x 30 or 20 x 10 (N6.5)


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It is expected that students will: 4N6. Continued

Please Note: Students need to be aware that when multiplying, rounding one factor has a different effect than rounding the other. In the following example, rounding the 8 to a 10 has a greater effect on the estimated product than rounding the 68 to a 70, even though it is an increase of 2 for each. This is because two extra 68s in 68 x 10 is more than 8 extra 2s in 70 x 8 (Small, Big Ideas, 2009, p. 34).

4N6.5 Estimate a product, using a personal strategy; e.g., 2 x 243 is close to or a little more than 2 x 200, or close to or a little less than 2 x 250.

Numbers that are easier to work with in estimation are sometimes referred to as “Friendly Numbers”. Students should be familiar with this term, so continue to use it when covering this outcome. Present the students with the following problem: You have 3 pieces of licorice, each 27 cm long. About how many centimeters of licorice do you have? Have the students paraphrase the problem. Draw attention to the word about, which indicates that an estimated answer is needed and no calculation has to be done. They may use the Four Corner Strategy (discussed in lesson 1) to help them solve the problem. Through discussion, have the students verbalize that 3 x 27 is close to or a little more than 3 x 20, or close to or a little less than 3 x 30: however, it is closest to the latter. Explain that you will be use 30 to represent 27 because 30 is closer to 27 than 20. Use the number sentence, 3 x 30 = 90, to show the estimated product. Stimulate the students’ thinking by asking whether 90 cm would be a good estimate for the answer. With the base ten blocks, the students should see readily that the blocks show a number less than 90. Therefore, a good estimate would be 90 – 10 = 80cm. Explain that the 10 is subtracted to compensate for the value that was added on when choosing the nearest multiple of ten and multiplying it by 3. Estimate: Three licorices would be a little less than 90 cm long or about 80 cm long. Have the students apply this estimation strategy and any other strategies that make sense to them in solving problems that require the multiplication of a 3-digit number by a 1-digit number. In these examples, the students would take the 3-digit number to the nearest multiple of 100, estimate the product and then compensate their answer appropriately.

46.7 Solve a given multiplication problem, and record the process

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When students are encouraged to think and communicate their reasoning to others, they learn to be clear and to make connections. Likewise, as others listen, they themselves, develop their own understandings. Written communication has to be nurtured from pictures to writing words and symbols in grade 4 that should become more elaborate with a sense of giving a sequence and some details to the reader.




Resources/Notes

Lesson 5 – Estimating Products (Cont’d)

Math Game: Greatest Product N6 SB p. 325 TR p. 34-35

Lesson 6: Communicating About Solving Problems N6 (6.7) SB p. 328-329 TR p. 39-42


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It is expected that students will: 4N6. Continued 4N6.3 Create and solve a multiplication problem that is limited to 2- or 3-digits by 1-digit.

Students should use a variety of models to investigate multiplication problems to help develop an understanding of the connection between the model and the symbols. It is important to start with a word problem and then have students use materials to determine the product. Base-ten blocks serve as a tool for understanding the multiplication operation. It is important that the students use language as they manipulate the materials and record the corresponding symbols for the product. It is not expected that students would be explicitly taught all possible algorithms, but provide opportunities to discover which is most efficient for the numbers included in a given problem. After students are comfortable with using base ten blocks when multiplying, they should be encouraged to use the front-end multiplication strategy (multiply showing the number of hundreds first). Student should have many opportunities to solve and create word problems for the purpose of answering real-life question, preferable choosing topics of interest to them. These opportunities provide students with a chance to practice their computational skills and clarify their mathematical thinking. The use of manipulatives or models helps the students to understand the structure of the story problem and also connects the meaning of the problem to the number sentence (Van de Walle 2001, p. 108).

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General Outcome: Develop Number Sense Suggested Assessment Strategies Student-Teacher Dialogue/ Performance Ask student to solve the following problems using three different strategies - concrete materials (base ten blocks), personal strategies, distributive property, arrays or algorithms: •

Here is $60. Ask, “Do you have enough money to buy 3 CDs if each costs $17? How do you know?”

•

For a school assembly, 9 rows of 38 chairs have been placed in the gym. Are there enough chairs for 370 students? Explain your thinking.

•

Ask students to create and solve a realistic problem that includes the factors 6 and 329.

•

Your family has planned a trip to Florida. You save 6 times as much money this year as you saved last year for your trip. If you saved $125 last year, how much spending money did you save this year? 6 x 125 =

Resources/Notes Lesson 7: Multiplying 3-Digit Numbers N6 (6.1 / 6.2 / 6.3 / 6.5 / 6.7) SB p. 330-332 TR p. 43-46

Journal Writing Have students create a multiplication problem for the equation 260 x 5 and then solve it.


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To develop understanding of the meaning of operations, the students connect the story problem to the manipulatives, create a number sentence and then use personal strategies to solve the problem. Provide appropriate time for students to create their personal strategies to solve the problem and then challenge them to solve the problem another way. It’s important that students understand the multiplication operation and not be taught to simply follow a procedure to obtain a product. Students need to understand what is given in a multiplication problem and what is the unknown, using the following terms: number of groups, quantity in each group or whole. Solve the following multiplication problem by estimating first, then modeling the problem using base ten blocks and record using expanded form, then multiply to show the number of hundreds first or by showing the ones place first. For Example: You made 22 oatmeal cookies from one batch of mix. You make 3 batches of cookies. How many cookies did you make? Step 1: Estimate First 3 x 22 is about 3 x 20 = 60 cookies. I predict that I will make a little more than 60 cookies. Step 2: Use base ten blocks to represent your problem

I’ll make 3 groups of 22 with Base ten blocks then I’ll record using expanded form. Step 3: Multiply showing the number of hundreds first or the number of ones first 20 + 2 x3 60 + 6 66

20 + 2 x3 6 + 60 66

Remember to discuss and compare the final product in step 3 with their estimate in step 1.

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Resources/Notes Lesson 7 Continued)

Curious Math: Egyptian Multiplication Sum and Product N6 (6.6 / 6.7) SB p. 333 TR p. 47-48

Lesson 8: Multiplying Another Way N6 (6.2 / 6.3 / 6.5 / 6.7) SB p. 334-337 TR p. 49-52

Lesson 9: Choosing a Method for Multiplying N6 (6.2 / 6.5 / 6.7) SB p. 338-340 TR p. 53-56

Lesson 10: Creating Multiplication Problems N6 (6.3 / 6.7) SB p. 341 TR p. 57-59


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Unit 10 DIVIDING MULTI-DIGIT NUMBERS Suggested Time: approx. 3-3 ½ weeks

DIVIDING MULTI-DIGIT NUMBERS

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Earlier, in unit 6, students worked with smaller numbers in developing the concept of division. In this unit they continue to develop an understanding of division but with larger whole numbers. Through encounters with various problem situations, students will develop fluency in computing division problems that will be an essential skill to use in real life situations. They will also continue to learn the meaning of division and how it relates to multiplication. Focus is on computational fluency with larger numbers so that students learn efficient and accurate methods of computing. This can be developed by solving problems with larger numbers that require calculation and by recording and sharing their strategies with others. Estimation plays an important role in division because it provides a tool for judging the reasonableness of an answer. . . . There are two different concepts of division, depending on which factor is unknown . . . . If a quantity is to be separated evenly into a given number of subsets [i.e., fair sharing] then division expresses the number in each subset. . . . If a quantity is to be measured out into sets of a specified size, then the division expresses the number of such sets that can be made [i.e., how many groups]. (Elementary School Mathematics, pp. 124-25)




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It is expected that students will: 4N7 Demonstrate an understanding of division (1-digit divisor and up to 2digit dividend) to solve problems by: • using personal strategies for dividing with and without concrete materials • estimating quotients • relating division to multiplication [C, CN, ME, PS, R, V] 4N7.2 Solve a given division problem with a remainder, using arrays or base ten materials, and connect this process to the symbolic representation.

4N7.3 Solve a given division problem, using a personal strategy, and record the process.

4N7.4 Refine personal strategies to increase their efficiency.

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Remainders are new to grade 4 students. Although remainders are not formally introduced until a little later, students may experience situations where there are remainders. You may want to introduce the concept when it arises incidentally. Reference to “left-overs” in given problem CAN be a clue that there will be a remainder in the quotient. In cases where students discover a remainder, they should understand that the remainder must be less than the divisor. The use of models in problem solving situations will help clarify their thinking about this. Refer back to page 159 in curriculum guide for the introduction of this indicator. Additional detail on personal strategies is discussed throughout this unit. Provide time for students to explore strategies to use with division of larger numbers, and to use the one that works best for them. Allow them to invent their own strategies before showing them a strategy you prefer. They should also be able to discuss their strategies with classmates. Teachers could help stimulate reasoning and communicating skills by asking the following questions: How did you solve that problem? How else could you have solved it? What did you like best about your partners strategy? Why did your strategy and your partner’s strategy result in the same answer?




Resources/Notes

Performance Tell the students the following story: There were two cartons in the refrigerator with a dozen eggs in each, plus three extra eggs in the holders in the fridge door. Mom liked to eat an omelette each day and used two eggs in each omelette. How many days could she make omelettes before she had to buy more eggs? Ask students to represent the story and solution with the correct mathematical symbols and explain their reasoning. (N7.2)

Chapter Opener TR p. 8 SB pp 346 - 347

Performance An apple tree owner wanted to give away the extra apples that fell from his apple. He offered 3 apples to each child who offered to help collect them. If 50 apples fell to the ground, how many children would get free apples? Will there be any apples left over? Make an array using counters to show your answer. (N7.2) Performance Ask students to use a model to explain to a classmate how to share 86 marbles among 5 people. (N7.2)

Getting Started TR pp. 9 - 11 SB 348 - 349

Lesson 1 Exploring Division N7 (7.2 / 7.3 / 7.4) TR pp. 12-14 SB p. 350

Performance • Roll two dice to create a 2-digit dividend. • Arrange the order of the digits so that when divided by 7, will give you the lowest remainder. • Record the remainder after each play, and total the score after 5 plays. • Lowest score wins the game. (example: if you roll a 5 and a 6, decide if the dividend will be “56” or “65”. 56 ÷ 7 = 8 R0, and 65 ÷ 7 = 9 R2, so I’ll record “0” as my score. (N7.2) Performance A family of 4 has received 60 free text messages each month to share equally. How many texts would each family member receive? Would there be any texts messages left over? Explain your answer. (N7.2) Performance Timothy bought a package of 50 dog treats to be shared equally between his 4 Labrador Retrievers, and Joey bought a package of 40 treats to be shared between his 3 Dalmatians. Which dogs received the most treats - the Labrador Retrievers or the Dalmatians? Will there be treats left over? Use square tiles or draw arrays to solve the problem. (N7.2)


Additional reading: Big Ideas from Dr. Small (Small, 2009) pp. 25 - 41

Teaching Student-Centered Mathematics Grades 3-5, (Van de Walle & Lovin, 2006) pp.121128

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Strand: Number Specific Outcome It is expected that students will: 4N7 Continued 4N7.7 Solve a given division problem by relating division to multiplication

Suggestions for Teaching and Learning Relating division to multiplication was discussed earlier on pages 159 and 161 of the curriculum guide) Solving division problems by multiplying helps students to get a close estimate of a division answer. Students will also connect multiplication to division by using multiplication to check that their quotient is correct. Students may need to review related facts of multiplication and division. Proficiency with multiplication and division should not be judged by students’ ability to perform one particular algorithm but by their ability to find answers to multiplication and division calculations accurately and efficiently using approaches that are appropriate to specific problems. Present the following problem to students: Andre collected $57 in 3 days. If he collected the same amount of money each day, how much was his daily collection? Have students discuss whether the problem could be represented by the number sentence, 3 x___ = 57. Why or why not? Students could complete a Four Corner Strategy like the one below to show their understanding of a division problem. Draw on prior knowledge by reviewing multiplication and division problems involving number facts to 5 × 5 or 25 ÷ 5. Emphasize the connections among the story problems, the models/diagrams, the number sentences and the personal strategies used in calculations. Have students divide their page into four sections to make graphic organizers and label them as follows: Four Corner Strategy Story Problem

Models/Diagrams

Number Sentence

Personal Strategy

Present the students with a problem or a number sentence involving multiplication or division, such as the following: • You paid 95 cents for 5 apples. What is the cost of each apple? • 5 × • = 95 Have the students complete the graphic organizer by writing the story problem or the number sentence in one corner and filling in the other corners appropriately.

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Resources/Notes

Performance Story Problems • You travel 84 km in three days. If you travel the same distance each day, how far do you travel each day? • You have 76 flowers to put into bouquets of 8 flowers each. How many bouquets can you make with these flowers? • 60 students were going on a bus to the museum. If 3 students could fit on each seat, how many seats are needed for the whole group?

Lesson 2 Relating Division To Multiplication N7 (7.3 / 7.7) TR pp. 15-17 SB p 351

Ask students to solve these problems using multiplication, and write the related multiplication and division sentences. (N7.7) Performance Thumbs Up, Thumbs Down, Thumbs Sideways: Present the students with a variety of multiplication and division problems. (Note: Reading the problems orally and also having them displayed on the white board or the overhead projector addresses the different learning styles of the students). For each problem, ask the students to put their thumbs up if multiplication can be used to solve the problem, thumbs down if division can be used, and thumbs sideways if both multiplication and division can be used. Have the students justify their choice, either in small groups or with the entire class. Finally, have the students write number sentences to support their choices. Emphasize the relationship between multiplication and division as the students suggest different number sentences. (N7.7) Performance Write all the possible number sentences that are represented in the following array. Explain how each number sentence relates to the array. ********** ********** ********** * * ********** ********** ********** * * ********** ********** ********** * * (N7.7) Journal How could you use multiplication to help you solve 65÷8?


(N7.7)

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Students need to learn to make sense of the remainder conceptually, as well as how to account for the remainder symbolically, when using a division algorithm. Context is what determines how the reminder should be treated. Exposure to a wide variety of problems using larger numbers with remainders will help support development of number sense.

4N7.2 Solve a given division problem with a remainder, using arrays or base ten materials, and connect this process to the symbolic representation.

In Grade Four, it is not intended that reminders be expressed as decimals or fractions, but as whole numbers in problem solving contexts. Example: Four students wanted to share 46 marbles equally. Students will record the answer as “11 Remainder 2”, because each person would get 11 marbles and there would be 2 marbles left over (46÷ 3 =11 R2).


Students previously solved problems using personal strategies in unit 6. Refer to previous discussion to now extend to larger numbers. Students have had many experiences using number lines and most many will be proficient in using it as a tool for division. They may begin to use repeated subtraction on a number line in this lesson to help them with larger dividends. Students will soon realize that choosing greater numbers to subtract will help them solve the problem more quickly. Encourage students to explain their reasoning for choosing the size of groups to subtract. It is important that students record their steps as they work through the processes in this strategy.

4N7.5 Create and solve a division problem involving a 1- 0r 2- digit dividend, and record the process.

This was discuss in earlier unit (refer to Curriculum Guide p.155). It is important to restate that students should have many opportunities to solve and create word problems for the purpose of answering real-life questions of personal interest. These opportunities provide students with a chance to practice their computational skills and clarify their mathematical thinking. Examples: •solving a problem: 63 people lined up to ride the roller coaster, The Bat, at Canada’s Wonderland. Each car holds 4 people. How many cars were needed for the line up? (16 will be needed because 15x 4 = 60, which would give you 15 full cars, but another car will be needed for the remaining 3 people) •creating a problem: Ask students to make up division problems about situations in the classroom. Have the problems posted and invite the class to circulate around the classroom solving their classmate’s problems.

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Resources/Notes

Performance

Lesson 3 Using Subtraction to Divide N7 (7.2 / 7.3 / 7.5 ) P6 (6.4) TR p. 18-21 SB pp. 352-355

An excellent game for students to practice solving problems with remainders can be found at: http://www.mathsolutions.com/documents/0-941355-42-X_L3.pdf (N7.2 / N7.3)

Student – Teacher Dialogue (Insert name of your school) has collected 85 “VOCM Coats for Kids” and is packaging 5 coats in each bag. How many bags will be needed? Draw number line showing division involving repeated subtraction. Ask students to justify the size of group they chose to subtract, in order to solve the problem. If smaller groups were chosen, ask them if the problem could have been solved using larger groups, and therefore fewer steps. (N7.3) Journal Create a problem that can be represented by the number sentence: 64 ÷ 4 = ___ Use a number line and repeated subtraction to show your workings. (N7.5 / N7.3/ PR 6.4) Pencil-Paper Melanie has 95 dollars in nickels. How many nickels does she have? Use repeated subtraction to help you solve the division problem. (N7.3)


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It is expected that students will: 4PR6 Solve one-step equations involving a symbol to represent an unknown number. [C, CN, PS, R, V] 4PR6.4 Solve a given equation when the unknown is on the left or right side of the equation.

This indicator was discussed in unit 6 for smaller numbers. Again, incorporate opportunities for students to encounter equations with the missing number being in different places.

4N7 Continued 4N7.1 Solve a given division problem without a remainder, using arrays or base ten materials, and connect this process to the symbolic representation.

See CG page 155


Another personal strategy that students may choose when dividing using larger dividends is “renaming” the dividend by splitting it into comfortable parts. Students should be encouraged to choose numbers that relate to multiplication and division facts they already know when renaming numbers to solve a division problem. The usefulness of this strategy depends on the students’ number sense, and may require a lot of practice in renaming numbers that are easily divided by various 1-digit divisors. Modeling using base ten blocks will be an important tool in implementing this personal strategy. Play money can be a substitute for base ten materials for students who have grasped place value money concepts. Example: 93 ÷ 5 can be renamed as 45 + 45 + 3, because I know that 45 is divisible by 5. 45 ÷ 5 + 45 ÷ 5 +3 = 9 + 9 R3 = 18 R3

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Resources/Notes Lesson 3 Continued

Pencil/Paper Show 2 strategies that could be used to calculate 48÷4. (N7.3 / N7.1) Journal How can you use 82= 40+40+2 to calculate 82÷4? (N7.3 / N7.1) Performance Highest Quotient Game: (a pair activity) • Randomly select 3 division fact cards each from a given pile. • Keep only 2 of the cards, and return one to a discard pile. • Calculate the quotient on each fact card • Add quotients together • Player with the highest total receives a point. • Play until one player receives 10 points. • Ask students to choose one play, and share strategies used in how they decided which card to place back in the discard pile.


Lesson 4 Dividing by Renaming N7 (7.1 / 7.2 / 7.3) N6 (6.4 / 6.6) TR p. 22 - 24 SB pp. 356 - 357

More practice may be required.

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It is expected that students will: 4PR6 Continued 4PR6.6 Represent and solve a given multiplication or division problem involving equal grouping or partitioning (equal sharing), using a symbol to represent the unknown.

There are two concepts of division (discussed on page 155 of CG). Both are described by Van de Walle and Lovin (2006): 1) Fair sharing: A bag has 92 jelly beans, and Aidan and her three friends want to share them equally. How many jelly beans will the three girls get?

A rod is traded for 10 units when there are no more rods to pass out. Then the 12 units are distributed, resulting in 23 in each set. Using base ten pieces makes the fair sharing easier to solve. This is digit oriented, as opposed to an approach that helps students think of the whole value of the dividend and is the idea on which the traditional algorithm is built - share the hundreds first, then the tens, then the ones. Present this type of problem as well as the following type, for students to try: 2) Here is the measurement or repeated subtraction concept(discussed on previous page): Jumbo the elephant loves peanuts. His trainer has 625 peanuts. If he gives Jumbo 20 peanuts a day, how many days will the peanuts last? This time students need to find out how many 20’s there are in 625. Initially, they might guess: - Try 20 (peanuts) x 10 (days)→ 200 (not nearly enough) - Try 20 (peanuts) x 30 (days)→ 600 (now there are 25 peanuts left) - Use 20 more peanuts for 1 more day. That makes 620 peanuts used with just 5 peanuts left over – not enough for another full day. The peanuts will last 31 days!

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Resources/Notes Lesson 4 (cont’d) .


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It is expected that students will: 4N7 Continued 4N7.6 Estimate a quotient, using a personal strategy; e.g., 86 ÷ 4 is close to 80 ÷ 4 or close to 80 ÷ 5.

Students make estimates that allow them to judge the reasonableness of their answers. When working on problem situations, students should know when it’s necessary to be accurate, and when an estimate is suffice. Remind students when estimating, to use facts that are close to dividends given. This will make for closer, and more reasonable estimate. Proficiency in recall of multiplication facts, and the use of use predicting skills, will help students provide a reasonable estimate for a division problem. It is important to remember that computational estimation is generally a mental activity; therefore, regular oral practice, accompanied by the sharing of strategies, must be provided and ongoing throughout the year. The goal is for students to become so competent with the skill that they automatically estimate in any problem situation, not only when asked to do so by the text or the teacher. One of the best ways of working on estimation skills seems to be to integrate them with other areas of the mathematics curriculum. (Elementary School Mathematics, p.203) As students explore estimating divisors and dividend they will observe that they have different effects on the problem’s outcome. Rounding the dividend up increases the estimated quotient, but rounding the divisor up decreases the estimated quotient (Small, 2009, p.34) To estimate quotients, students might use the following strategies: Estimation Strategy Example Round numbers such that 574 ÷ 9 is about 560 ÷ 8 = 70 familiar multiplication and 574 ÷ 9 is about 540 ÷ 9 = 60 division facts can be used. When dividing round both 337 ÷ 8 is about 360 ÷ 9 = 40 numbers up or both numbers 337 ÷ 8 is about 280 ÷ 7 = 40 down. Round numbers to the nearest 389 ÷ 27 is about 400 ÷ 25 =16 multiple of 10 when the divisor 612 ÷ 27 is about 600 ÷ 25 = 24 in the problem divides evenly into a multiple of 10…or 25 to be able to divide by 25. (Small, 2009, p.34)

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Resources/Notes

Performance Provide a list of division questions to pairs of students and ask them to do the following: (1) estimate a quotient (2) explain their strategy to their partner (3) tell whether or not the estimate is too high or too low, and why. (N7.6)

Lesson 5 Estimating Quotients N7 (7.3 / 7.6 ) N6 (6.4 / 6.6) TR pp 29-32 SB p. 360-363

Performance • Jason rode his bicycle every day for 8 days. He cycles 68 km in total. About how far did he ride each day? •

Ninety-eight parents were expected to attend the school’s “Volunteer Appreciation” luncheon. About how many packs of 8 muffins should be purchased to ensure there are enough to serve all of the guests? (N7.6)

Journal • Explain how you know that 89 ÷ 9 is about 1 more than 79 ÷ 9. •

Describe a situation in which you might want to estimate 67 ÷ 7. (N7.6)


Math Game 3 Card Quotient N7 TR p. 33 SB p363

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Students have solved problems with personal strategies before but now the focus is on traditional algorithms using base ten blocks. The focus in solving division problems using larger numbers should be on invented strategies and not just the traditional algorithm, as it cannot be taught with a strong conceptual basis. Far too many students learn them as meaningless procedures, develop error patterns, and require an excessive amount of reteaching or remediation. No matter what the division algorithms is, either traditional or otherwise, it is important that students show they understand by being able to explain what they are doing and why.


It is appropriate to initially model the algorithm with manipulatives. Using base ten blocks when choosing to solve division problems using traditional algorithms will help conceptualize students’ understanding of division being “equal groups”. Students will have had many experiences in regrouping using base ten blocks in Chapter Two. Whatever strategy students choose to solve division problems, the emphasis should be on the connections between the concrete representation and the symbolic personal strategies. Have the students apply their personal strategy to solve a variety of division problems that include equal sharing and equal grouping with and without remainders. Note that the traditional division algorithm is built on the process involved with fair-share problems (Van de Walle 2001). Encourage students to use correct mathematical language, such as the following, when describing algorithms: • regroup • trade or exchange • place value terms such as hundreds, tens and ones • product • quotient • remainder Discourage incorrect mathematical language such as “9 doesn’t go into 8” for a division problem such as 87÷9. It is 87, not 8 that you are dividing by 9. Students should be encouraged to discuss whether or not an answer is reasonable by seeing if it is close to an estimate made prior to completing the problem, and then checking the answer by multiplying and adding the remainder

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General Outcome: Develop Number Sense Suggested Assessment Strategies Performance Present the following problems to the students: Andre collected $57 in 3 days. If he collected the same amount of money each day, how much was his daily collection? Guide discussion as to whether the numbers in the problem and the unknown refer to - the whole? the number of groups? or the quantity in each group? Discuss which operation would be used to solve the problem, and what would be a good estimate for the answer. Encourage the students to represent the problem using base ten materials and to record the process.

Resources/Notes Lesson 6 Dividing by Sharing N7 (7.1 / 7.2 / 7.3/ 7.6) N6 (6.4 / 6.6) TR pp. 34-36 SB pp364-366

The school custodian needs to place an order for fluorescent light bulbs for his school. If they come in packages of 6, how many packages will he need to buy if he needs 89 lights? Will he have any light bulbs left over to go towards his next order? If so, how many? Use base ten blocks to help you solve the problem. (N7.2)

This lesson shows an algorithm which is very close to the ‘traditional algorithm’. Consider extending this to the traditional algorithm since the traditional You have 98 beads to make 4 necklaces. If each necklace has the algorithm for division is typically same number of beads, how many beads are on each necklace? Will familiar to parents and allows any beads be left over? If so, how many? Represent the problem using them to be of help to their children base ten blocks. at home. (N7.2) Provide a set of base ten blocks. Ask the students to model 3 different division questions of his/her choice and write the division sentence for each. (N7.1 / N7.2) Present the following problem to the student and have him or her read it orally. Have base ten materials available to use as needed. A bottle contains 76 mL of medicine. Jerry takes 8 mL of medicine each hour. How many hours will pass before all the medicine is gone? Pose the following questions to guide thinking if necessary: • State the problem in your own words. • What do each of the numbers in the problem represent—whole, the number of groups or the quantity in each group? • What is the unknown in the problem—whole, the number of groups or the quantity in each group? • What number sentence could you write to show the meaning of the problem? • Does the problem use multiplication or division or both? Explain. • About how many hours will pass before the medicine is all gone? Explain your thinking. • Will any medicine be left over? Explain. • Use a strategy that makes sense to you to find the answer to the problem. Explain your thinking as you write the numbers. • Explain how you know your answer makes sense and is reasonable. • Would you solve the problem another way? Explain your thinking.


Curious Math (optional) Remainder Magic TR p. 37 SB p. 367

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It is expected that students will: 4N7 Continued 4N7.5 Create and solve a division problem involving a 1- or2- digit dividend, and record the process.

In a “Guess and Test” strategy, a student guesses an answer to a problem, then tests it to see if the “guess” works. If it doesn’t, the student revises the “guess” based on what was learned and tries again. This repetitive process continues until the answer is found. Some students are able to think through several guesses at once; others need to go one step at a time. Although we often talk about guessing as bad, this strategy reinforces the value of taking risks and learning from the information that is garnered. (Small , 2005)

Students should be encouraged to use diagrams, counters, multiplication tables, arrays, and base ten blocks to help them work through possible answers. Organized lists also help students to keep track of guesses so they are not repeated.

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Resources/Notes

Performance All Grade four classes collected the same number of recyclables for the school’s Recycling Blitz. Together, they collected 90 recyclables. How many Grade Four classes could there be in the school? Give 3 possible solutions for your problem. (N7.3)

Lesson 7 Solving Problems by Guessing and Testing N7 (7.3 / 7.5 / 7.6) TR p. 38-39 SB pp. 368-370

Performance Create a problem that you could solve by guessing and testing. Solve your problem. (N7.3 / N7.5) Performance The same digit is missing from each box. What is the digit? 4___ ÷ 8 = 5 R ____ (N7.3) Performance What 2 numbers have a product of 20, and a quotient of 5? (answer: (N7.3) 10 and 2) Performance Christopher had between 60 and 70 hockey cards in his collection. He can package them equally in either groups of 2, 4 or 8. How many cards does he have? Use the Guessing and Testing strategy to help solve your problem. Explain your solution. (answer: 64 hockey cards. 64 is 30 groups of 2, 16 groups of 4, and 8 groups of 8). (N7.3)

Math Game Remainder Hunt N7 TR p. 41 SB p. 371

Chapter Task (Optional) TR pp.47-48 SB p. 375


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Unit 11 3-D GEOMETRY Suggested Time: approx. 1 ½ weeks

3-D GEOMETRY

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3-D GEOMETRY


‘Geometry enables us to describe, analyze and understand our physical world so there is little wonder that it holds a central place in mathematics or that it should be a focus throughout the school mathematics curriculum’ (Navigating through geometry) As students move through school, they should receive instruction that links to, and builds on the foundation of earlier years. They must be continually challenged to apply increasingly more sophisticated spatial thinking to solve problems in all areas of mathematics as well as in all other school home and other life situations. (Navigating through Geometry 3-5, p.8) As students develop mathematically and become more familiar with geometric attributes, they are increasingly able to identify and name a shaped by examining its properties and using reasoning. Through exploration of three dimensional shapes, students develop awareness that there are certain specific attributes that they can use to classify the shapes. They will also be encouraged to develop and communicate mathematical arguments about geometric relationships Spatial sense can be described as an intuition about shapes and the relationship among shapes. This includes the ability to be able to visualize shapes and to be able to turn them around their minds. Many people say that children are born with a spatial sense or not. Van de Walle & Lovin (2006) says “this is simply not true! We know that rich experiences with shape and spatial relationships, when provided consistently over time, can and do develop spatial sense.




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3-D GEOMETRY

Strand: Shape and Space (3-D Objects and 2-D Shapes) Specific Outcome



Students will draw upon their previous knowledge of two dimensional polygons to assist them in their identification and description of prisms. In the earlier grades children will have classified geometric shapes by general characteristics and will now develop more detailed ways to describe objects. They will identify properties of shapes and learn to use proper mathematical vocabulary to describe the shapes.

4SS4 Describe and construct right rectangular and right triangular prisms. [C, CN, R, V]

A good way to explore shapes is to use smaller shapes or tiles to create larger shapes. Different criteria or directions can provide the intended focus to the activity. Pattern blocks are very good for this, but many teacher-made materials can be used. While the pattern block pieces are prisms, they have been treated as 2-D shapes; however, stacking a number of triangles or squares would provide examples of different prisms. This stacking would help students conceptualize the uniform nature of prisms. Also, students can make skeletal models for prisms, using rolled newspapers and tape, straws and string, or toothpicks and miniature marshmallows. Commercial sets of 3-D objects usually have a variety of prisms. Note: All prisms used in Grade 4 are ‘right’ prisms. For clarification purposes, a prism is ‘right’ if the faces form a right angle with the bases (or we can say ‘are perpendicular with the bases’). Below are examples of prisms illustrating the difference:

Although the outcome and indicators use the terminology ‘right’ for purposes of distinguishing between right prisms and other prisms it is not necessary for students to use the term ‘right’ in their descriptions. At this level, ‘rectangular prism’ and ‘triangular prism’ is sufficient for students. All prisms have faces, two of which are customarily referred to as bases. These two bases may take the shape of any polygon. For clarification purposes, prisms can be thought of as having two names. The first name refers to the shape of the bases and a second name, which is prism. Examples: triangular prism, rectangular prism. Some students may be keen to identify other prisms such as hexagonal prisms or square prisms (square prisms fall into the category of rectangular prisms because a square is a rectangle). In Grade Four, exploration is focused on rectangular prisms and triangular prisms only.

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3-D GEOMETRY

General Outcome: Describe the Characteristics of 3-D Objects and 2-D Shapes, and Analyze the Relationships among them Suggested Assessment Strategies

Resources/Notes Chapter Opener

Performance After a thorough treatment of prisms, give students a blank Frayer Model graphic organizer to communicate their understanding of prisms.


Getting Started

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There is a developmental sequence associated with how students think and reason geometrically. Many students in Grade 4 are beginning to develop more sophisticated abilities to identify and name 3-D objects. As levels of geometric thinking develop, students will notice more attributes of three dimensional objects. These attributes are the components that go together to make up the form – edges, vertices and faces, (two which are the bases). In the process of identifying and naming attributes of prisms, it may necessary to review and encourage students to use appropriate vocabulary such as number of faces, number of edges, number of vertices or shapes of the faces/bases.

Achievement Indicators: 4SS4.1 Identify and name common attributes of right rectangular prisms from given sets of right rectangular prisms.

A rectangular prism has 6 faces, 12 edges and 8 vertices. (Note that all square prisms can be called rectangular prisms because a square is a rectangle)

4SS4.2 Identify and name common attributes of right triangular prisms from given sets of right triangular prisms.

A triangular prism has 5 faces, 9 edges and 6 vertices. Allow each student to manipulate concrete models of 3-D shapes so that they are able to touch and count each of the faces, vertices and edges. One way to familiarize students with right rectangular prisms is have several shapes in front of pairs of students for them to examine as you call out clues about the properties of the shape. For example, “This 3D shapes has 8 vertices.” As you give clues, have the children figure out which shape you are thinking of. While some students may be able to think of the shapes visually, it is best to give all student (or pairs of students) the concrete objects to help them.

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Resources/Notes Lesson 1 Recognizing Rectangular Prisms TR pp. 11-13 SB pp. 380

Lesson 2 Recognizing Triangular Prisms TR pp. 16 - 19 SB pp. 382 - 384 Lessons 1 and 2 may be combined or treated separately, depending on teacher preference.


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Sorting requires students to attend to specific attributes of objects. Give students a variety of 3D prisms (real life or commercially made models). Have them sort the prisms according to the attribute: Shape of the Base. Make a class T-chart as shown below:

4SS4 Continued 4SS4.3 Sort a given set of right rectangular and right triangular prisms, using the shape of the base.

Provide 3-D objects (real life or commercially made models) such as spheres, cones, cylinders, pyramids (students will be familiar with these from Grade 3) as well as rectangular prisms and triangular prisms. Place two hula hoops on the floor to represent a large scale Venn diagram such as the one shown below. Provide labels and have students sort the objects according to triangular prisms/ rectangular prisms/ other. As students place their object on the diagram, have them explain, to the class, why they placed objects in certain places.

4SS4.7 Identify examples of rectangular and triangular prisms found in the environment.

Shape Hunt – Display a set of right rectangular prisms on a table (vary the set by including: different sizes of triangular and rectangular prisms; cubes; and prisms positioned in different orientations). Have students go around the room or the school and find objects that match the shapes. Ask individual students to present their findings. Ask how their ‘found’ object is alike or different from prisms on the display table. Listen to student responses and encourage them to name the common attributes. Consider: - What might prisms be used for? [Sample responses: to hold things/to support roofs/to contain things inside] - Why might prisms have different sizes and shapes? [Sample responses: the contents might determine the best shape for a container/some prisms might need to be larger to provide strength or support, such as a table leg/decorative reasons]

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Resources/Notes

Journal Give students two different drawings of prisms. Have them find the model to match from a set in the classroom. Glue the pictures in their journal and answer the question, “How are these prisms different?” (Students may refer to Math Word Wall to write their descriptions)

Lesson 3 Communication About prisms TR pp.21-25 SR pp.386-388

Presentation Riddles - Encourage students to use the attributes of any prism (number of faces, number of edges, number of vertices, or shapes of the faces) to describe prisms . E.g. I have . . .

Performance Have students name the prism that best represents various real-life examples of 3-D objects.

.

Performance Have student work together to sort their collection of 3-D objects into two groups: rectangular and triangular prisms. Ask: what are the attributes of the shapes that made them alike? How are they different? What makes a rectangular prism a cube? What kind of prism would you have if you built from a rectangular base?


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4SS4.4 Construct and describe a model of a right rectangular and a right triangular prism, using materials such as pattern blocks or modeling clay.

Constructing prisms can take many forms. One way to represent shapes is to make concrete models. Have students stack pattern blocks to build a prism such as the following:

Build prisms with other pattern block shapes. Ask which prism can they build from pattern blocks that has the most faces (hexagon-based prism) Have students build rectangle based prisms with interlocking cubes:

4SS4.7 Identify examples of right rectangular and right triangular prisms found in the environment.

140

Another type of representation is a skeleton. This is a model showing only the edges and vertices of a 3-D shape. Have a variety of 3-D shapes available to students. Some may need to touch the edges and vertices in order to construct a skeleton. The process of making a skeleton helps students be able to visualize the shape and remember its properties. Have students build skeletons of prisms using materials such as toothpicks and small balls of modeling clay or straws with pieces of pipe cleaners. E.g.


3-D GEOMETRY


Resources/Notes

Stack pattern blocks to make rectangular prisms and triangular prisms. Describe how they are alike and how they are different.

Lesson 4 Constructing Prisms

Ask students to build skeletal models of two different triangular pyramids. Ask them how they are the same/different?

Listen as students construct nets on geoboards and using grid/isometric paper; are they discussing the attributes of the object, are they using the correct vocabulary (faces, edges, vertices, congruency)?

TR pp.26-28 SB p. 389

Curious Math TR pp. 14-15 SB pp. 381 This activity can be used with Lesson 1 or Lesson 3

Additional reading: Big Ideas from Dr. Small, (Small 2009) pp.106 - 108


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It is expected that students will: 4SS4 Continued 4SS4.6 Construct right triangular prisms from their nets. 4SS4.5 Construct right rectangular prisms from their nets.

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Students should be given copies of nets of rectangular and triangular prisms to cut out and fold up. They should be encouraged to unfold them and examine the 2-D shapes that are connected to make each net. Have them visualize the folding up and unfolding. In addition to cutting out and assembling prepared nets, it is now expected that students will draw their own nets for rectangular and triangular prisms. They will also consider the various possibilities for these nets. Have the students trace on paper the various faces of the different prism to make its net. Have the students cut out the net and fold it up around the shape to see if it works. Ask them to record this net on grid paper. Have them cut one of the faces off and investigate the possible places it could be reattached to make a new net. Have them record each one on grid paper.


3-D GEOMETRY

General Outcome: Describe the Characteristics of 3-D Objects and 2-D Shapes, and Analyze the Relationships among them Suggested Assessment Strategies Performance Provide students with various nets of prisms for them to construct. Have them label each face of their model using the words “face” and “base” as well as identify their 3-D object. Performance Have the students trace on paper the various faces of the different prisms to make its net. Have the students cut out the net and fold it up around the shape to see if it works. Ask them to record this net on grid paper. Have them cut off one of the faces and investigate the possible places it could be reattached to make a new net. Have them record each one on grid paper.

Resources/Notes

Lesson 5 Constructing Prisms from Nets TR pp.29-31 SB p. 390

Performance Provide the students with a pentomino puzzle piece (a 2-D shape made by joining 5 squares along full sides) that would fold to make a box with no top.

Ask them to trace this piece and then add a square for the top of the box. Ask: In how many places can this square be added? (Note: Students may wish to cut this from grid paper.) Performance Tell the students that this diagram is part of a net for a square prism. Ask them to complete the net by drawing the additional faces that would be needed.

Performance Give small groups of students a set of 4 or 5 nets of rectangular or triangular prisms. Each set should consist of one net that can be made into the 3-D object, and 3 or 4 others which cannot be made into the 3-D object. Have the student analyse the nets, without manipulating them, to determine which one of the nets in the group could be used to create the 3-D object. Have them justify, and then test their predictions.


Chapter Review Chapter Task Use selectively

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Appendix A: Outcomes with Achievement Indicators Organized by Strand 96 / Outcomes with Achievement Indicators (Grade 4)


Appendix A

[PS] [R] [T] [V]


Grade 4 Strand: Number

General Outcome: Develop number sense.

Specific Outcomes

Achievement Indicators


The following sets of indicators determine whether students have met the corresponding specific outcome. Other indicators may be added according to teacher preference.

N1. Represent and describe whole numbers to 10 000, pictorially and symbolically. [C, CN, V]

N1.1Read a given four-digit numeral without using the word and; e.g., 5321 is five thousand three hundred twenty-one, NOT five thousand three hundred AND twenty-one. N1.2 Write a given numeral, using proper spacing without commas; e.g., 4567 or 4 567, 10 000. N1.3 Write a given numeral 0–10 000 in words. N1.4 Represent a given numeral, using a place value chart or diagrams. N1.5 Express a given numeral in expanded notation; e.g., 321 = 300 + 20 + 1. N1.6 Write the numeral represented by a given expanded notation. N1.7 Explain the meaning of each digit in a given 4-digit numeral, including numerals with all digits the same; e.g., for the numeral 2222, the first digit represents two thousands, the second digit two hundreds, the third digit two tens and the fourth digit two ones.

N2. Compare and order numbers to 10 000. [C, CN, V]

N2.1Order a given set of numbers in ascending or descending order, and explain the order by making references to place value. N2.2 Create and order three different 4-digit numerals. N2.3Identify the missing numbers in an ordered sequence or on a number line (vertical or horizontal). N2.4 Identify incorrectly placed numbers in an ordered sequence or on a number line (vertical or horizontal).

97 / Outcomes with Achievement Indicators (Grade 4)


[PS] [R] [T] [V]


Grade 4 Strand: Number (continued)


N3. Demonstrate an understanding of addition of numbers with answers to 10 000 and their corresponding subtractions (limited to 3- and 4-digit numerals) by: • using personal strategies for adding and subtracting • estimating sums and differences • solving problems involving addition and subtraction. [C, CN, ME, PS, R]

N3.1Determine the sum of two numbers using a personal strategy (e.g. for 1326 + 548, record 1300 + 500 + 74). N3.2 Determine the difference of two numbers using a personal strategy (e.g. for 4127 – 238, record 238 + 2 + 60 + 700 + 3000 + 127 or 4127 – 27 – 100 – 100 – 11). N3.3 Describe a situation in which an estimate rather than an exact answer is sufficient. N3.4 Estimate sums and differences, using different strategies; (e.g., front-end estimation and compensation.) N3.5Refine personal strategies to increase their efficiency N3.6 Solve problems that involve addition and subtraction of more than 2 numbers.

N4. Explain and apply the properties of 0 and 1 for multiplication and the property of 1 for division. [C, CN, R]

N4.1 Determine the answer to a given question involving the multiplication of a number by 1, and explain the answer. N4.2 Determine the answer to a given question involving the multiplication of a number by 0, and explain the answer. N4.3 Determine the answer to a given question involving the division of a number by 1, and explain the answer.

N5. Describe and apply mental mathematics strategies, such as: • skip counting from a known fact • using doubling or halving • using doubling or halving and adding or subtracting one more group • using patterns in the 9s facts • using repeated doubling to determine basic multiplication facts to 9 × 9 and related division facts. [C, CN, ME, R]

N5.1Provide examples for applying mental mathematics strategies: •

skip counting from a known fact; e.g., for 3 × 6, think 3 × 5 = 15 plus 3 = 18

•

doubling; e.g., for 4 × 3, think 2 × 3 = 6 and 4 × 3 = 6 + 6

•

doubling and adding one more group; e.g., for 3 × 7, think 2 × 7 = 14 and 14 + 7 = 21

•

use ten facts when multiplying by 9; e.g., for 9 × 6, think 10 × 6 = 60 and 60 – 6 = 54; for 7 × 9, think 7 × 10 = 70 and 70 – 7 = 63

•

halving; e.g., if 4 × 6 is equal to 24, then 2 × 6 is equal to 12

•

relating division to multiplication; e.g., for 64 ÷ 8, think 8 × = 64

•

repeated doubling; e.g., for 4 × 6, think 2 × 6 = 12 and 2 × 12 = 24.


Appendix A [C] Communication [CN] Connections [ME] Mental Mathematics and Estimation

[PS] [R] [T] [V]




N6. Demonstrate an understanding of multiplication (2or 3-digit by 1-digit) to solve problems by: • using personal strategies for multiplication with and without concrete materials • using arrays to represent multiplication • connecting concrete representations to symbolic representations • estimating products • applying the distributive property. [C, CN, ME, PS, R, V]

N6.1 Model a given multiplication problem, using the distributive property; e.g., 8 × 365 = (8 × 300) + (8 × 60) + (8 × 5). N6.2 Use concrete materials, such as base ten blocks or their pictorial representations, to represent multiplication; and record the process symbolically. N6.3 Create and solve a multiplication problem that is limited to 2- or 3-digits by 1-digit. N6.4 Refine personal strategies to increase their efficiency.

N7. Demonstrate an understanding of division (1-digit divisor and up to 2-digit dividend) to solve problems by: • using personal strategies for dividing with and without concrete materials • estimating quotients • relating division to multiplication. [C, CN, ME, PS, R, V]

(It is not intended that remainders be expressed as decimals or fractions.) N7.1 Solve a given division problem without a remainder, using arrays or base ten materials, and connect this process to the symbolic representation. N7.2 Solve a given division problem with a remainder, using arrays or base ten materials, and connect this process to the symbolic representation. N7.3 Solve a given division problem, using a personal strategy, and record the process. N7.4 Refine personal strategies to increase their efficiency. N7.5 Create and solve a division problem involving a 1- or 2-digit dividend, and record the process.

N6.5 Estimate a product, using a personal strategy; e.g., 2 × 243 is close to or a little more than 2 × 200, or close to or a little less than 2 × 250. N6.6 Model and solve a given multiplication problem, using an array, and record the process. N6.7 Solve a given multiplication problem, and record the process.

N7.6 Estimate a quotient, using a personal strategy; e.g., 86 ÷ 4 is close to 80 ÷ 4 or close to 80 ÷ 5. N7.7 Solve a given division problem by relating division to multiplication; e.g., for 100 ÷ 4, we know that 4 × 25 = 100, so 100 ÷ 4 = 25.



[PS] [R] [T] [V]




N8. Demonstrate an understanding of fractions less than or equal to one by using concrete, pictorial and symbolic representations to: • name and record fractions for the parts of a whole or a set • compare and order fractions • model and explain that for different wholes, two identical fractions may not represent the same quantity • provide examples of where fractions are used. [C, CN, PS, R, V]

N8.1Represent a given fraction, using concrete materials. N8.2 Identify a fraction from its given concrete representation. N8.3 Name and record the shaded and non-shaded parts of a given set. N8.4 Name and record the shaded and non-shaded parts of a given whole. N8.5 Represent a given fraction pictorially by shading parts of a given set. N8.6 Represent a given fraction pictorially by shading parts of a given whole. N8.7 Explain how denominators can be used to compare two given unit fractions. N8.8 Order a given set of fractions that have the same numerator, and explain the ordering. N8.9 Order a given set of fractions that have the same denominator, and explain the ordering. N8.10 Identify which of the benchmarks 0, 12 or 1 is closer to a given fraction. N8.11 Name fractions between two given benchmarks on a number line (horizontal and vertical). N8.12 Order a given set of fractions by placing them on a number line (horizontal and vertical) with given benchmarks. N8.13 Provide examples of when two identical fractions may not represent the same quantity; e.g., half of a large apple is not equivalent to half of a small apple, half of ten blueberries is not equivalent to half of sixteen blueberries. N8.14 Provide, from everyday contexts, an example of a fraction that represents part of a set and an example of a fraction that represents part of a whole.

N9. Represent and describe decimals (tenths and hundredths), concretely, pictorially and symbolically. [C, CN, R, V]

N9.1 Write the decimal for a given concrete or pictorial representation of part of a set, part of a region or part of a unit of measure. N9.2 Represent a given decimal, using concrete materials or a pictorial representation. N9.3 Explain the meaning of each digit in a given decimal with all digits the same. N9.4 Represent a given decimal, using money values (dimes and pennies). N9.5 Record a given money value, using decimals. N9.6 Provide examples of everyday contexts in which tenths and hundredths are used. N9.7Model, using manipulatives or pictures, that a given tenth can be expressed as a hundredth; e.g., 0.9 is equivalent to 0.90, or 9 dimes is equivalent to 90 pennies.



[PS] [R] [T] [V]




N10.Relate decimals to fractions and fractions to decimals (to hundredths). [C, CN, R, V]

N10.1 Express, orally and in written form, a given fraction with a denominator of 10 or 100 as a decimal. N10.2 Read decimals as fractions; e.g., 0.5 is zero and five tenths. N10.3 Express, orally and in written form, a given decimal in fraction form. N10.4 Express a given pictorial or concrete representation as a fraction or decimal; e.g., 15 shaded 15 . squares on a hundredth grid can be expressed as 0.15 or 100 N10.5 Express, orally and in written form, the decimal equivalent for a given fraction; e.g.,

50 100

be expressed as 0.50. N11.Demonstrate an understanding of addition and subtraction of decimals (limited to hundredths) by: • using compatible numbers • estimating sums and differences • using mental mathematics strategies to solve problems. [C, ME, PS, R, V]

N11.1 Predict sums and differences of decimals, using estimation strategies. N11.2 Refine personal strategies to increase their efficiency. N11.3 Solve problems, including money problems, which involve addition and subtraction of decimals, limited to hundredths. N11.4 Determine the approximate solution of a given problem not requiring an exact answer. N11.5 Estimate a sum or difference using compatible numbers. N11.6 Count back change for a given purchase.


can


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Grade 4 Strand: Patterns and Relations (Patterns)

General Outcome: Use patterns to describe the world and to solve problems.

Specific Outcomes




PR1.Identify and describe patterns found in tables and charts, including a multiplication chart. [C, CN, PS, V]

PR1.1 Identify and describe a variety of patterns in a multiplication chart. PR1.2 Determine the missing element(s) in a given table or chart. PR1.3 Identify the error(s) in a given table or chart. PR1.4 Describe the pattern found in a given table or chart

PR2.Translate among different representations of a pattern, such as a table, a chart or concrete materials. [C, CN, V]

PR2.1 Create a concrete representation of a given pattern displayed in a table or chart. PR2.2 Create a table or chart from a given concrete representation of a pattern.

PR3.Represent, describe and extend patterns and relationships, using charts and tables, to solve problems. [C, CN, PS, R, V]

PR3.1 Translate the information in a given problem into a table or chart. PR3.2 Identify and extend the patterns in a table or chart to solve a given problem.

PR4.Identify and explain mathematical relationships, using charts and diagrams, to solve problems. [CN, PS, R, V]

PR4.1 Complete a Carroll diagram by entering given data into correct squares to solve a given problem. PR4.2 Determine where new elements belong in a given Carroll diagram. PR4.3 Solve a given problem using a Carroll diagram PR4.4 Identify a sorting rule for a given Venn diagram. PR4.5 Describe the relationship shown in a given Venn diagram when the circles intersect, when one circle is contained in the other and when the circles are separate. PR4.6 Determine where new elements belong in a given Venn diagram. PR4.7 Solve a given problem by using a chart or diagram to identify mathematical relationships.



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Grade 4 Strand: Patterns and Relations (Variables and Equations)

General Outcome: Represent algebraic expressions in multiple ways.

Specific Outcomes




PR5.Express a given problem as an equation in which a symbol is used to represent an unknown number. [CN, PS, R]

PR5.1 Explain the purpose of the symbol in a given addition, subtraction, multiplication or division equation with one unknown; e.g., 36 ÷ = 6. PR5.2 Express a given pictorial or concrete representation of an equation in symbolic form. PR5.3 Identify the unknown in a problem; represent the problem with an equation; and solve the problem concretely, pictorially or symbolically. PR5.4 Create a problem for a given equation with one unknown.

PR6.Solve one-step equations involving a symbol to represent an unknown number. [C, CN, PS, R, V]

PR6.1 Solve a given one-step equation using manipulatives. PR6.2 Solve a given one-step equation, using guess and test. PR6.3 Describe, orally, the meaning of a given one-step equation with one unknown. PR6.4 Solve a given equation when the unknown is on the left or right side of the equation. PR6.5 Represent and solve a given addition or subtraction problem involving a “part-part-whole” or comparison context, using a symbol to represent the unknown. PR6.6 Represent and solve a given multiplication or division problem involving equal grouping or partitioning (equal sharing), using a symbol to represent the unknown.



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Grade 4 Strand: Shape and Space (Measurement)

General Outcome: Use direct and indirect measurement to solve problems.

Specific Outcomes




SS1.Read and record time, using digital and analog clocks, including 24-hour clocks. [C, CN, V]

SS1.1 State the number of hours in a day. SS1.2 Express the time orally and in writing from a 12-hour analog clock. SS1.3 Express the time orally and in writing from a 24-hour analog clock. SS1.4 Express the time orally and in writing from a 12-hour digital clock. SS1.5 Express time orally and in writing from a 24-hour digital clock. SS1.6 Express the time orally and in writing “minutes to” or “minutes after” the hour. SS1.7 Explain the meaning of a.m. and p.m., and provide an example of an activity that occurs during the a.m., and another that occurs during the p.m.

SS2.Read and record calendar dates in a variety of formats. [C, V]

SS2.1 Write dates in a variety of formats; e.g., yyyy/mm/dd, dd/mm/yyyy, March 21, 2007, dd/mm/yy. SS2.2 Relate dates written in the format yyyy/mm/dd to dates on a calendar. SS2.3 Identify possible interpretations of a given date; e.g., 06/03/04.

SS3.Demonstrate an understanding of area of regular and irregular 2-D shapes by: • recognizing that area is measured in square units 2 • selecting and justifying referents for the units cm 2 or m 2 2 • estimating area, using referents for cm or m 2 2 • determining and recording area (cm or m ) • constructing different rectangles for a given area (cm2 or m2) in order to demonstrate that many different rectangles may have the same area. [C, CN, ME, PS, R, V]

SS3.1 Describe area as the measure of surface recorded in square units. SS3.2 Identify and explain why the square is the most efficient unit for measuring area. SS3.3 Provide a referent for a square centimetre, and explain the choice. SS3.4 Provide a referent for a square metre, and explain the choice. SS3.5 Determine which standard square unit is represented by a given referent. SS3.6 Estimate the area of a given 2-D shape, using personal referents. SS3.7 Determine the area of a regular 2-D shape, and explain the strategy. SS3.8 Determine the area of an irregular 2-D shape, and explain the strategy. SS3.9 Construct a rectangle for a given area. SS3.10 Demonstrate that many rectangles are possible for a given area by drawing at least two different rectangles for the same given area.



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Grade 4 Strand: Shape and Space (3-D Objects and 2-D Shapes)

General Outcome: Describe the characteristics of 3-D objects and 2-D shapes, and analyze the relationships among them.

Specific Outcomes




SS4.Describe and construct right rectangular and right triangular prisms. [C, CN, R, V]

SS4.1 Identify and name common attributes of right rectangular prisms from given sets of right rectangular prisms. SS4.2 Identify and name common attributes of right triangular prisms from given sets of right triangular prisms. SS4.3 Sort a given set of right rectangular and right triangular prisms, using the shape of the base. SS4.4 Construct and describe a model of a right rectangular and a right triangular prism, using materials such as pattern blocks or modelling clay. SS4.5 Construct right rectangular prisms from their nets. SS4.6 Construct right triangular prisms from their nets. SS4.7 Identify examples of right rectangular and right triangular prisms found in the environment.



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Grade 4 Strand: Shape and Space (Transformations)

General Outcome: Describe and analyze position and motion of objects and shapes.

Specific Outcomes




SS5.Demonstrate an understanding of line symmetry by: • identifying symmetrical 2-D shapes • creating symmetrical 2-D shapes • drawing one or more lines of symmetry in a 2-D shape. [C, CN, V]

SS5.1 Identify the characteristics of given symmetrical and non-symmetrical 2-D shapes. SS5.2 Sort a given set of 2-D shapes as symmetrical and non-symmetrical. SS5.3 Complete a symmetrical 2-D shape, given half the shape and its line of symmetry. SS5.4 Identify lines of symmetry of a given set of 2-D shapes, and explain why each shape is symmetrical. SS5.5 Determine whether or not a given 2-D shape is symmetrical by using an image reflector or by folding and superimposing. SS5.6 Create a symmetrical shape with and without manipulatives. SS5.7 Provide examples of symmetrical shapes found in the environment, and identify the line(s) of symmetry. SS5.8 Sort a given set of 2-D shapes as those that have no lines of symmetry, one line of symmetry or more than one line

SS6. Demonstrate an understanding of congruency, concretely and pictorially. [CN, R, V]

SS6.1 Determine if two given 2-D shapes are congruent, and explain the strategy used. SS6.2 Create a shape that is congruent to a given 2-D shape. SS6.3 Identify congruent 2-D shapes from a given set of shapes shown in different orientations. SS6.4 Identify corresponding vertices and sides of two given congruent shapes.ne of symmetry.



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Grade 4 Strand: Statistics and Probability (Data Analysis)

General Outcome: Collect, display and analyze data to solve problems.

Specific Outcomes




SP1.Demonstrate an understanding of many-to-one correspondence. [C, R, T, V]

SP1.1 Compare graphs in which the same data has been displayed using one-to-one and many-toone correspondences, and explain how they are the same and different. SP1.2 Explain why many-to-one correspondence is sometimes used rather than one-to-one correspondence. SP1.3 Find examples of graphs in which many-to-one correspondence is used in print and electronic media, such as newspapers, magazines and the internet, and describe the correspondence used.

SP2.Construct and interpret pictographs and bar graphs involving many-to-one correspondence to draw conclusions. [C, PS, R, V]

SP2.1 Identify an interval and correspondence for displaying a given set of data in a graph, and justify the choice. SP2.2 Create and label (with categories, title and legend) a pictograph to display a given set of data, using many-to-one correspondence, and justify the choice of correspondence used. SP2.3 Create and label (with axes and title) a bar graph to display a given set of data, using many-to-one correspondence, and justify the choice of interval used. SP2.4 Answer a given question, using a given graph in which data is displayed using many-to-one correspondence.


Appendix B References

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Appendix B

REFERENCES Alberta Education. LearnAlberta.ca: Planning Guides K, 1, 4, and 7, 2005-2008. American Association for the Advancement of Science [AAAS-Benchmarks]. Benchmark for Science Literacy. New York, NY: Oxford University Press, 1993. Banks, J.A. and C.A.M. Banks. Multicultural Education: Issues and Perspectives. Boston: Allyn and Bacon, 1993. Black, Paul and Dylan Wiliam. “Inside the Black Box: Raising Standards Through Classroom Assessment.” Phi Delta Kappan, 20, October 1998, pp.139-148. British Columbia. Ministry of Education. The Primary Program: A Framework for Teaching, 2000. Burns, M. (2000). About teaching mathematics: A K-8 resource. Sausalito, CA: Math Solutions Publications Caine, Renate Numella and Geoffrey Caine. Making Connections: Teaching and the Human Brain. Menlo Park, CA: Addison-Wesley Publishing Company, 1991. Computation, Calculators, and Common Sense. May 2005, NCTM. Davies, Anne. Making Classroom Assessment Work. British Columbia: Classroom Connections International, Inc., 2000. Hope, Jack A. et.al. Mental Math in the Primary Grades (p. v). Dale Seymour Publications, 1988. National Council of Teachers of Mathematics (NCTM). Curriculum Focal Points for Prekindergarten through Grade 8: A Quest for Coherence. Reston, VA: NCTM, 2006. National Council of Teachers of Mathematics. Principals and Standards for School Mathematics. Reston, VA: The National Council of Teachers of Mathematics, 2000. OECD Centre for Educational Research and Innovation. Formative Assessment: Improving Learning in Secondary Classrooms. Paris, France: Organization for Economic Co-operation and Development (OECD) Publishing, 2006. Proulx, Jerome. “Making the Transition to Algebraic Thinking: Taking Students’ Arithmetic Modes of Reasoning into Account.” Selta-K44, 1(2006) Richardson, K.. Developing number concepts addition and subtraction book 2. Pearson Education, Inc. 1999 Richardson, K. Counting comparing and pattern. Pearson Education, Inc. 1999 Rubenstein, Rheta N. Mental Mathematics beyond the Middle School: Why? What? How? September 2001, Vol. 94, Issue 6, p. 442.

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Shaw, J.M. and Cliatt, M.F.P. (1989). “Developing Measurement Sense.” In P.R. Trafton (Ed.), New Directions for Elementary School Mathematics (pp. 149–155). Reston, VA: National Council of Teachers of Mathematics. Small, M. (2008). Making math meaningful to canadian students, K-8. Toronto, Ontario: Nelson Education Ltd. Steen, L.A. (ed.). On the Shoulders of Giants – New Approaches to Numeracy. Washington, DC: National Research Council, 1990. Stenmark, Jean Kerr and William S. Bush, Editor. Mathematics Assessment: A Practical Handbook for Grades 3-5. Reston, VA: National Council of Teachers of Mathematics, Inc., 2001. Van de Walle, John A. and Louann H. Lovin. Teaching Student-Centered Mathematics, Grades K-3. Boston: Pearson Education, Inc. 2006. Van de Walle, John A. and Louann H. Lovin. Teaching Student-Centered Mathematics, Grades 3-5. Boston: Pearson Education, Inc. 2006. Western and Northern Canadian Protocol (WNCP) for Collaboration in Education. The Common Curriculum Framework for K-9 Mathematics, 2006. Reproduced and/or adapted by permission. All rights reserved.

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Appendix C: Strategies for Learning Multiplication Facts

APPENDIX C Strategies for Learning Multiplication Facts (Source: Teaching Student-Centered Mathematics Grade 3 – 5, Van deWalle & Lovin)

The multiplication facts can be mastered by relating new facts to existing knowledge. “Mastery” of a basic fact means that a child can give a quick response (in about 3 seconds). Teachers can help students develop an efficient strategy - one that can be done mentally and quickly. It is also important that students understand the commutative property (turnarounds) since knowledge of this property `will reduce the number of facts they have to learn. There are two approaches to introducing the fact strategies: • Simple story problem designed in such a manner that students are likely to develop a strategy as they solve it. It is recommended that the discussion of these strategies can be done for 5 – 10 minutes at the beginning of every day. • A lesson may revolve around a set of facts for which particular type of strategy is appropriate. You can discuss how these facts might all be alike in some way, or you might suggest an approach and see if students are able to use it on similar facts. Since arrays are powerful thinking tools for teaching the strategies, provide students with copies of ten-by-ten dot arrays.


There are 100 multiplication facts, from 0 x 0 to 9 x 9. The first 4 of the 5 strategies listed below are generally easier and cover 75 out of the 100 facts. These strategies are suggestions and not rules. Listen to students as they discover other ways to help them think of the facts easily. 1. Zeros and Ones (facts with a 0 or 1) • Thirty six facts have at least one factor that is either 0 or 1. Sometimes these facts get confused with the rules children learn about addition facts with 0 or 1. Avoid rules that are without reason such as “Any number multiplies by zero is zero.” Rather, these concepts can be best developed through story problems.

2. Doubles (facts with a 2) • Facts that have 2 as a factor are the same as the addition doubles and are probably already known by students who know their addition facts. Help them to realize that not only is 2 x 7 double 7, but 7 x 2 is also double 7.


3. Clock facts (facts with a 5) • Focus on the minute hand of the clock. When it points to a number, how many minutes past the hour is it? Connect this idea to the multiplication facts with 5 as a factor.

4. Nifty Nines (facts with a 9) • Facts with a factor of 9 include the largest products but can be among the easiest to learn. The 9 row and column of a multiplication table includes some nice patterns and are fun to discover. The following two patterns combined are useful to mastering the nines facts. (1) The tens digit of the product is always 1 less than the “other” factor (the one other than 9), and (2) the sum of the two digits in the product is always 9. These two ideas can be used together to get any nine fact quickly. For 7 x 9, 1 less than 7 is 6, 6 and 3 make 9, so the answer is 63. • An alternative strategy for learning the nine facts is also easy. Students may discover that they can relate the 9 fact to the already known 10 fact. For example, notice that 7 x 9 is the same as 7 x 10 less one set of 7 or 70 – 7.


5. Helping Facts – These 25 facts can be learned by relating each to already know fact or helping fact. Double and double again (facts with a 4) When 4 is one of the factors, students can double and double again. Example, find 4 x 6: double 6 is 12 and double 12 is 24.

Double and one more set (facts with a 3) Example: find 3 x 7: double 7 is 14 and add one more 7 to make 21.


Half then double (facts with an even number) Select the even factor and cut it in half. If the smaller factor is known, that product is doubled to get the new product. Example: find 6 x 7: half the 6 to get 3 x 7. 3 x 7 is 21 and double 21 is 42. Add one more set (any fact). Many children prefer to go to a fact that is “close” and then add one more set to this known fact. Example: Think of 6 x 7 as 6 sevens. Five sevens is close. That’s 35. Six sevens is only one more 7, so that makes 42. The relationship between easy and hard facts is useful. Rather than telling students which strategy is best to use, select a fact from one of the strategies and say, “If you didn’t know … (for example 6 x 8) how could you figure it out by using something else you know?” It would be useful for you to go through each of the 20 “hard facts” and see which strategies from the “Helping Facts” section can be used for each one. “Drill” refers to repetitive non-problem-based activities and it is appropriate ONLY after students understand a strategy but it has not yet become automatic. There is a place for drill of the basic facts but it is critical that it not be used too early. After students have worked on two or three strategies, they should be given opportunities to look at multiplication facts and select a strategy that is most helpful in finding the answer.