PUZZLE !

Ash’s Puzzle This problem gives you the chance to: • find numbers that obey given rules • find rules for sets of numbers

Ash has a book of number puzzles. This is one of the puzzles.

PUZZLE ! Find a two-digit number such that its digits are reversed when 9 is added.

1. Solve this puzzle for Ash.

_________________

Show that your answer works.

Ash wonders if there are other answers to this puzzle. 2. Are there other correct answers to the puzzle?

__________________

If there are more correct answers list them all. If not explain how you know that there is only one correct answer. _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________

Copyright © 2007 by Mathematics Assessment Resource Service. All rights reserved.

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Ash’s Puzzle Test 9

Ash decides to try to find a three-digit number such that its digits are reversed when 99 is added. He finds that there are a lot of numbers that work. 3. Write four three-digit numbers that Ash could have found. ______________

______________

______________

______________

Show your work.

Ash thinks that there must be rules that would make it possible to find all of the three-digit numbers that are reversed when 99 is added to them. 4. Find these rules for Ash. ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________

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Task 3: Ash’s Puzzle

Rubric

The core elements of performance required by this task are: • find numbers which obey given rules • find rules for sets of numbers

points

section points

Based on these, credit for specific aspects of performance should be assigned as follows

1.

Gives a correct answer: 12, 23, 34, 45, 56, 67, 78 or 89 and Gives correct calculation for their answer: such as 12 + 9 = 21

1 1

2.

Gives correct answer: yes and lists the other seven possible answers (ignore their answer to question 1 repeated) 12, 23, 34, 45, 56, 67, 78 and 89 Partial credit:

(1)

An extra 4, 5 or 6 correct answers with no incorrect ones. 3.

2

2

Gives 4 correct answers: any 3 digit numbers with the last digit 1 greater than the first e.g. 152, 798 etc. Shows some correct work for their answers such as:

1

1

152 + 99 = 251 4.

2

Gives correct rules such as: The last digit is one more than the first.

1

The middle digit can be any number.

1 Total Points


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2 7


Ash’s Puzzle Work the task. Look at the rubric. What are the big mathematical ideas being assessed by this task? What types of generalizations do we want students at this level to be able to make?

Look at student work for part 2 of the task, finding additional answers. How many of your students: • Listed all the possibilities?_____________ •

Omitted one possibility?___________

•

Gave 2 or a few possibilities?______________ Gave only one possibility?__________

•

Said yes with no work to back it up?________________

•

Said no, there aren’t other solutions?________________

• Showed evidence of not understanding the constraints of the task?_____________ What opportunities have your students had this year with problem-solving involving making an organized list? What are some of the habits of mind that you want for students in an algebra class? Do they include questions on self-talk like(Taken from Fostering Algebraic Thinking by Mark Driscoll): • Am I able to abstract from computation? • How are things changing? • Do I have all the solutions? Could there be others? How are the solutions related? What are the important properties of the solutions? • Is there information that helps predict what is going to happen? • Can I justify why a rule works? Will it always work? What are the cases for when it is true and when it isn’t true? Have I tested enough different cases to be convinced that I have all the possibilities? • What process reverses the one I am using? • When I do the same thing with different numbers, what still holds true? What changes? • Am I willing to struggle through or persevere when faced with an unfamiliar or nonroutine problem? What do I do when I get stuck? How do these habits of mind relate to the quality of work you saw on students’ papers? How might some of these habits of mind helped students improve their performance? How do you help students to develop these habits of mind in a consistent way throughout the year? How do you help students develop these habits of mind? How can you promote students to use self-talk? Algebra – 2007 Copyright © 2007 by Noyce Foundation Resource Service. All rights reserved.

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Now look at student work in part 3. How many of your students: • Gave 4 numbers that fit the constraints of the task? • Gave numbers that were incorrect and didn’t check their answers to see that they didn’t fit the constraints? • Showed work that did not match the constraints of the task (e.g. answers didn’t reverse the numbers, trying to get answers totaling 99, used operation besides adding)? • Weren’t willing to attempt this part of the task? One of the powerful aspects of algebra is the ability to use it to make and to prove generalizations. In part 4, students were asked to formulate rules for finding solutions to meet the constraints of reversing the digits in a three-digit number by adding 99. Look at student work: How many of your students could give rules relating the hundreds digit and the units digit? How many of those rules gave imprecise language about the digits (first digit, last digit so that the reader had to make assumptions about reading from the left or right of the number)? How many quantified the fact the hundreds digit had to be lower than 9 (quantified the domain)? How many of your students thought the solutions were all in the 100’s and ended in 2? How many students could give a rule about the middle number being equal to any number? How many thought the middle number had to be a specific number, such as 0 (i.e. they didn’t test enough cases to discover all the situations for which the numbers would reverse?) How many of your students thought the middle number had to be a repeat of either the hundreds digit or the units digit? How many tried to give a pattern based a previous correct solution, such as add 20 or add 202 to the last answer? How many had a partially correct idea, but couldn’t articulate the attributes of numbers or think through all the relevant features of the pattern (e.g. split a part any answer from 1)? How many were unwilling to attempt writing a rule? What are other misconceptions that you saw in reading through students’ attempts to find a rule or pattern? What activities or problems do you give students to help them develop the ability to make and articulate generalizations? How often do students in your class conduct investigations with number and number properties?

Algebra – 2007 Copyright © 2007 by Noyce Foundation Resource Service. All rights reserved.

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Looking at Student Work on Ashe’s Puzzle While some students were able to meet the full demands of the task, their language around place value is imprecise. Notice that students A and B talk about first and third digits rather than hundreds and units. This is typical of most student work. Student A shows thinking about the constraints, what does it mean to reverse digits, before giving a solution. The answers for part two are in an organized list. In part 3 the student only tests cases where the middle number is zero, but Student A is still able to generalize about the middle or tens digit. Also notice that the student establishes the domain for the hundreds digit. Student A


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Again, Student B only tests a limited variety of cases in part 3, where the hundreds digit is always 1 and the units digit is always 2. This set allows the student to see and to verify that the middle number can change. The student is also able to make the leap to seeing how the pattern would work for different numbers in the hundreds or units places. Student B


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Student C only tests cases where the tens digit is 0. 8% of all student agreed that the middle number had to be 0. 13% gave rules stating a specific single digit for the tens digit. Student C is starting to see that there is a possibility of other choices, but can’t or doesn’t do enough investigation to really see if he has found all the solutions. He tries to jump to a generalization too quickly. What other questions would you want the student to ask himself to help push his thinking? How do we help students develop skills in investigating and testing a variety of cases or conditions? Student C


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Student D finds a pattern using a limited variety of cases. The student understands the relationship between the hundreds and units digits, but doesn’t see that the tens digit does not effect the solution. 13% of the students thought the tens digit should match either the hundreds or the units digit. Student D


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Student E struggles with language for discussing the mathematics of the task. The student seems to actually connect the hundreds and units digit with the solutions to part 2, which is a powerful idea not noticed by other students in the sample. Like Student D, this student limits the tens digit to a single possibility. The student doesn’t investigate to see what other solutions might work. Student E

Student F, like 23% of the students, did not think there could be multiple solutions to a problem. How do we help students build a productive work ethic, a willingness to persevere when solutions are not immediately obvious? How do you work on this in your classroom? Student F, like 6% of all students, thought the hundreds digit had to be one and the units digit had to be two.


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Student F


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Student G looks at a variety of cases that work, putting solutions into an organized list. This allows the student to then make a generalization from the emerging pattern, that the units digit is the hundreds digit plus one. The student does not address the possibilities for the middle digit, although the student has evidence to help support a correct statement. How do we help students to focus on all the relevant parts of a pattern? In Fostering Algebraic Thinking, Mark Driscoll talks about the importance of abstracting in algebra. “. . . a good case can be made that thinking algebraically involves being able to think about computations freed from the particular numbers to which they are tied in arithmetic.” He also talks about the internal dialog that students can develop to be more productive thinkers, such as “When I do the same thing with different numbers, what still holds true? What changes”? Student G


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Although Student H received no points, the student is starting to think about the number system and why the pattern works in her first bullet. However Student H does not finish the thought by making a rule. The student can’t distinguish between the relevant parts of the information from the irrelevant parts. When do the numbers repeat? When do the numbers not repeat? What does this mean for writing a rule for any number? The student also notices a recursive rule, which only helps when you already know one value in the set and actually doesn’t hold true when moving from a number with 9 in the tens digit to the next solution in the sequence. Again how do we help students develop a set of helpful questions for doing investigations and making generalizations? Does this pattern hold true for all cases? Have I tested enough possibilities? Student H


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Ash’s Puzzle Algebra Student Task Core Idea 3 Algebraic Properties and Representations

Core Idea 2 Mathematical Reasoning

Task 3

Ash’s Puzzle

Find numbers that obey given rules or constraints. Find rules for sets of numbers. Use understanding of place value to solve problems in context. Represent and analyze mathematical situations and structures using algebraic symbols. • Use symbolic expressions to represent relationships arising from various contexts. • Compare and contrast the properties of numbers and number systems including real numbers Employ forms of mathematical reasoning and proof appropriate to the solution of the problem, including deductive and inductive reasoning, making and testing conjectures and using counterexamples and indirect proof. • Show mathematical reasoning in a variety of ways, including words, numbers, symbols, pictures, charts, graphs tables, diagram and models. • Explain the logic inherent in a solution process. • Use induction to make conjectures and use deductive reasoning to prove conclusions. • Draw reasonable conclusions about a situation being modeled.

Mathematics in this task: • Investigating a relationship in number calculations • Identifying relevant information using place value and number theory to discover a pattern in the solution • Generalizing from arithmetic to a pattern for all solutions Based on teacher observations, this is what algebra students knew and were able to do: • Find most solutions which will reverse a two-digit number by adding nine • Give examples of three-digit numbers that will reverse the digits when 99 is added and show supporting evidence Areas of difficulty for algebra students: • Making an organized list or check for “all” solutions that meet a set of constraints • Testing different cases of numbers, investigating enough options or choices before making a generalization • Recognizing all the relevant information needed to make a convincing set of rules for all numbers • Vocabulary for place value


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The maximum score available on this task is 7 points. The minimum score for a level 3 response, meeting standards, is 4 points. Many students could find one solution of adding 9 to a two-digit number that would reverse the digits. More than half the students, 63%, could Find one solution for reversing a two digit number and four examples for reversing a three digit number and give supporting evidence to verify the solutions. Almost half the students could also find at least 5 solutions for reversing the two-digit number. About 16% could find all the solutions for reversing the 2 digit number and could verbalize a rule about the units digit being one larger than the hundreds unit. About 2% of the students could meet all the demands of the task including noticing that the middle digit could be any number from 0 to 9. Almost 20% of the students scored no points on this task. 76% of the students with this score attempted the task. Algebra – 2007 Copyright © 2007 by Noyce Foundation Resource Service. All rights reserved.

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Ash’s Puzzle Points Understandings 76% of the students with this 0 score attempted the task.

1

Students could find 1 solution of a two-digit number that would reverse when 9 was added.

3

Students could find at least one solution for a two-digit number and four solutions with work for reversing the digits in a 3-digit number.

4

Students could find at least five solutions for reversing a twodigit number and four examples for reversing a 3-digit number with supporting work.

6

Students could find all the solutions for reversing a 2-digit number and give 4 examples for reversing a 3-digit number. They also noticed that the units digit was one larger than the hundreds digit.

7

Students could find solutions for reversing 2- and 3-digit numbers, investigate an arithmetic situation and generalize to a rule for finding all solutions. Students generally had a system of making organized lists and testing a variety of cases.


Misunderstandings Some students did not understand the constraints of the task. They tried to find numbers that added to 9, listed multiples of 9, or gave answers that did not reverse the digits, such as 20 + 9=29. 23% thought there were no other solutions. 5.5% only gave one further solution, but did not attempt to find all possible solutions. Another 5.5% gave two or three of the seven additional solutions. 18% of the students didn’t attempt part 3 of the task. Students didn’t have strategies like making an organized list to help them check for all possible solutions in part 2. Students didn’t check enough variety of examples in part 3 to help them make a good generalization in part 4. 25% of the students did not attempt to write a rule for part 4 of the task. About 5.5% of the students thought the hundreds number needed to be one and the units digit needed to be 2. About 5% of the students thought all 3 of the numbers should be consecutive. 7% of the students tried to give a recursive rule, such as add 10 or add 101. Many students did not address the issue of the middle number. They did not connect that feature to describing how to write the number. 8% of the students thought the middle number had to be a 0. 13% gave a specific single value to the middle digit. 13% thought that the middle digit should match one of the other two digits. All students in the sample struggled with using place value language, usually referring to first and last digits rather than hundreds and units digits.

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Ash’s Puzzles Students need more rich tasks that allow them to investigate numbers, number patterns. They need to learn how to organize information from their investigation in a systematic way that will help them to see the relevant information in the pattern and make generalizations. Part of investigating patterns is to think about questions, such as “When does this work? Will this always work? If I change one of the numbers what will change and what will stay the same? Through experience and discussion students develop habits of mind that help them to form generalizations and understand what makes a convincing rule or generalization. There is a whole set of logic skills that develop through productive math talk and discussion. As students get feedback about their ideas or critique the solutions of others, they learn to focus in on the relevant properties and become more specific in the rules they generate. Math talk or discussion also gives them practice in using mathematical vocabulary, thus developing mathematical fluency with academic language that doesn’t happen from just reading and memorizing definitions. Students should be encouraged to try and justify why the patterns work using arguments about place value and algebraic expressions.

Ideas for Action Research Using Problems of the Month Try giving students some longer investigations, like problems of the month. Problem solving is the cornerstone of doing mathematics. George Polya, a famous mathematician from Stanford, once said, “a problem is not a problem if you can solve it in 24 hours.” His point was that a problem that you can solve in less than a day, is usually a problem that is similar to one that you have solved before or at least recognized that a certain approach will lead to the solution. Students need more exposure to problems with extended reasoning chains, that require perseverance, and use a variety of strategies. Give some students some problems, like Courtney’s Collection, which look at number theory and algebra. (taken from the Noyce Website:www.noycefdn.org/math under resources) Encourage your students to restate the problem in their own words. Show their calculations and what they tried. Then write about their process: • How did you get started? What approaches did you try? • Where did you get stuck? • What drawings, charts, graphs, or models did you use? • What was your solution? What did you discover during the investigation? Emphasis should be on the process of solving the problem.


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Courtney’s Collection


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Understanding Struggling Students What are the needs of students who are performing at a low level? How are their needs for support different than those of the students who achieved some success but still need help some specific elements of the task? Sit down with a group of colleagues and look at the student work below. Do these students have any skills to build on that would help them succeed on this task? If not what is the one major target where their thinking breaks down or hole in arithmetic skills that they would need substantial instruction on? Where would you go with this student? As students progress through the grades, they need to develop an understanding of operations with whole numbers and computations with whole numbers. Then, they should start to be able to understand operations and computations with rational numbers. Finally in eighth grade students should start to understand number systems. Do you see evidence of understanding whole numbers? Whole number operations? Understanding of place value and number systems? Is there evidence that these students are ready to start making generalizations about number and number systems? Where does their thinking breaking down? What are the implications of this work for your school across grade levels? What are the implications for you as a teacher of these students? Do you have examples from you own class that would be useful to add to the discuss?


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Agatha


70

Bruce


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Clara


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Dennis

Evelyn


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Evelyn, part 2


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