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CONCEPT DEVELOPMENT

Mathematics Assessment Project

CLASSROOM CHALLENGES A Formative Assessment Lesson

The Difference of Two Squares

Mathematics Assessment Resource Service University of Nottingham & UC Berkeley Draft Version For more details, visit: http://map.mathshell.org © 2012 MARS, Shell Center, University of Nottingham Please do not distribute outside schools participating in the initial trials

Teacher guide The Difference of Two Squares T-1

The Difference of Two Squares

MATHEMATICAL GOALS This lesson unit is intended to help you assess how well students are able, whilst working with square numbers to: • Choose an appropriate, systematic way to collect and organize data. • Examine and look for patterns in the data. • Generalize using numerical, geometrical, graphical and/or algebraic structure. • Describe and explain findings clearly and effectively.

COMMON CORE STATE STANDARDS This lesson relates to the following Standards for Mathematical Content in the Common Core State Standards for Mathematics:

8-EE: Work with radicals and integer exponents. 8-EE: Analyze and solve linear equations and pairs of simultaneous linear equations. 8-F: Define, evaluate, and compare functions. 8-F: Use functions to model relationships between quantities.

This lesson also relates to the following Standards for Mathematical Practice in the Common Core State Standards for Mathematics:

1. Make sense of problems and persevere in solving them. 3. Construct viable arguments and critique the reasoning of others. 7. Look for and make use of structure.

INTRODUCTION This lesson unit is structured in the following way:

• Before the lesson, students work individually on an assessment task designed to reveal their current understanding and difficulties. You then review their responses and create questions for them to consider when improving their work.

• At the start of the lesson, students reflect on their individual responses and use the questions posed to think of ways to improve their work. They then work collaboratively in small groups to produce, in the form of a poster, a better solution to the task than they did individually.

• In a whole-class discussion students compare and evaluate the different methods they have used. • Working in the same small groups, students analyze sample responses to the task. • In a whole-class discussion, students review the methods they have seen, then reflect on their own

work.

MATERIALS REQUIRED • Each student will need a copy of the task: The Difference of Two Squares, some plain paper, a

mini-whiteboard, a pen, an eraser, and a copy of the review questionnaire How Did You Work? • Each small group of students will need a large sheet of paper for making a poster and copies of

the Sample Responses to Discuss. • You will need a supply of graph paper. There is a projector resource to support whole-class

discussions.

TIME NEEDED 15 minutes before the lesson for the assessment task, a 60-minute lesson, and 15 minutes in a follow-up lesson (or for homework). Timings are approximate and will depend on the needs of the class.


BEFORE THE LESSON

Assessment task: The Difference of Two Squares (15 minutes) Have students complete this task, in class or for homework, a few days before the formative assessment lesson. This will give you an opportunity to assess the work, and to find out the kinds of difficulties students have with it. You should then be able to target your help more effectively in the follow-up lesson.

Give each student a copy of the assessment task: The Difference of Two Squares and some plain paper to work on.

Read through the questions and try to answer them as carefully as you can.

It is important that, as far as possible, students are allowed to answer the questions without assistance. If students are struggling to get started then ask questions that help them understand what is required, but make sure you do not do the task for them.

Students who sit together often produce similar answers, and then when they come to compare their work, they have little to discuss. For this reason, we suggest that when students do the task individually, you ask them to move to different seats. Then at the beginning of the formative assessment lesson, allow them to return to their usual seats. Experience has shown that this produces more profitable discussions.

Assessing students’ responses Collect students’ responses to the task. Make some notes on what their work reveals about their current levels of understanding and their different problem solving approaches.

We suggest that you do not score students’ work. The research shows that this will be counterproductive, as it will encourage students to compare their scores, and will distract their attention from what they can do to improve their mathematics.

Instead, help students to make further progress by summarizing their difficulties as a series of questions. Some suggestions for these are given in the Common issues table on the next page. These have been drawn from common difficulties observed in trials of this unit.

We suggest you make a list of your own questions, based on your students’ work. We recommend you either:

• Write one or two questions on each student’s work, or • Give each student a printed version of your list of questions, and highlight the questions for each

individual student. If you do not have time to do this, you could select a few questions that will be of help to the majority of students, and write these on the board when you return the work to the students.

Student Materials The Difference of Two Perfect Squares S-1 © 2012 MARS, Shell Center, University of Nottingham

The Difference of Two Squares Michael writes:

1. Is Michael correct? Explain your answer.

2. What do you notice about the integers that can be expressed as the difference of two square integers? Explain what you notice.


Common issues Suggested questions and prompts

Student answers the question correctly, but does not provide an adequate explanation (Q1) For example: The student figures out that 2 cannot be written as the difference of two squares by trying out just one or two subtractions of square numbers.

• Be systematic when explaining why a number cannot be expressed as the difference of two squares.

• As you increase the first integer in the expression of the difference of the two squares, what do you notice about the answer?

Student notices that all odd numbers can be expressed as the difference of two squares, but does not provide an adequate explanation (Q2)

• Look again at your explanation. Have you made any assumptions?

• When the answer is an odd number, what do you notice about the two numbers that are squared in the expression? Can you use what you notice?

• In terms of a, figure out two expressions for the grey area in the diagram. How does this help explain the value of the difference of two consecutive square numbers?

Student notices that all multiples of four can be expressed as the difference of two squares, but does not provide an adequate explanation (Q2)

• When the answer is a multiple of 4, what do you notice about the two numbers that are squared in the expression? Can you use what you notice?

• In terms of a, figure out two expressions for the grey area in the diagram. How does this help explain the value of the difference of square numbers that differ by two?

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SUGGESTED LESSON OUTLINE

Individual work (10 minutes) Throughout this activity, if students attempt to make generalizations they are likely to make assumptions. Allow students to do this, but through questioning try to make students aware of these assumptions.

Return the assessment task to the students. Give each student a mini-whiteboard, a pen, and an eraser.

Begin the lesson by briefly reintroducing the problem.

If you did not add questions to individual pieces of work, write your list of questions on the board. Students are to select questions appropriate to their own work, and spend a few minutes answering them.

Recall what we were looking at in a previous lesson. What was the task about?

Today we are going to work together to try to improve your initial attempts at this task.

I have had a look at your work, and I have some questions I would like you to think about.

On your own, carefully read through the questions I have written. I would like you to use the questions to help you to think about ways of improving your own work.

Use your mini-whiteboards to make a note of anything you think will help to improve your work.

Collaborative small-group work (10 minutes) Organize the class into small groups of two or three students. Give each group a large piece of paper, and a felt-tipped pen. On request, give students graph paper.

Deciding on a Strategy

Ask students to share their ideas about the task, and plan a joint solution.

I want you to share your work with your group.

Take it in turns to explain how you did the task and how you now think it could be improved.

Listen carefully to any explanation. Ask questions if you don't understand or agree with the method. (You may want to use some of the questions I have written on the board.)

I want you to plan a joint approach that is better than your separate solutions.

Once students have evaluated the relative merits of each approach ask them to write their strategy on the large piece of paper.

Slide P-1 of the projector resource, Planning a Joint Method, summarizes these instructions.

Implementing the Strategy

Students are now to turn their large sheet of paper over and write their joint solution clearly in the form of a poster.

While students work in small groups you have two tasks: to note different student approaches to the task, and to support student problem solving

Listen and watch students carefully. Note different approaches to the task and what assumptions students make. Do students work systematically? How do students explain why certain integers are impossible to express as the difference of two squares? Do students attempt to generalize? Do students use algebra? In particular, note any common mistakes. You can then use this information to focus a whole class discussion towards the end of the lesson.

Try not to make suggestions that move students towards a particular approach to the task. Instead, ask questions that help students to clarify their thinking. In particular focus on the strategies rather than the solution. Encourage students to justify their statements.


You may want to use the questions in the Common issues table to support your own questioning. If the whole class is struggling on the same issue, you could write one or two relevant questions on the board, and hold a brief whole-class discussion. You could also give any struggling students one of the sample responses.

Whole-class discussion (10 minutes) You may want to hold a brief whole-class discussion. Have students solved the problem using a variety of methods? Or have you noticed some interesting ways of working or some incorrect methods, if so, you may want to focus the discussion on these. Equally, if you have noticed different groups using similar strategies but making different assumptions you may want to compare solutions.

Collaborative analysis of Sample Student Responses (15 minutes) Distribute to each group of students, copies of Sample Responses to Discuss. This task gives students an opportunity to evaluate a variety of possible approaches to the task, without providing a complete solution strategy.

There may not be time, and it is not essential, for all groups to look at all the sample responses. If this is the case, be selective about what you hand out. For example, groups that have successfully completed the task using one method will benefit from looking at a different approach. Other groups that have struggled with a particular approach may benefit from seeing a student version of the same strategy.

In your groups you are now going to look at some student work on the task. Notice in what ways this work is similar to yours and in which ways it is different.

There are some questions for you to answer as you look at the work. You may want to add notes to the work to make it easier to follow.

Slide P-2 of the projector resource, Evaluating Sample Student Responses, describes how students should work together:

Encourage students to focus on evaluating the math contained in the student work, not whether the student has neat writing etc.

During the small group work, support the students as in the first collaborative activity. Also, check to see which of the explanations students find more difficult to understand. Note similarities and differences between the sample approaches and those the students took in the collaborative group work.

The Difference of Two Perfect Squares Projector Resources

Evaluating Student Sample Responses

1.  Take it in turns to work through a students’ solution. -  Write your answers on your mini-whiteboards.

3.  Explain your answer to the rest of the group.

4.  Listen carefully to explanations. –  Ask questions if you don't understand.

5.  Once everyone is satisfied with the explanations, write the answers below the students’ solution. -  Make sure the student who writes the answers is not the student who

explained them.

2


Aakar uses a two-way table. He correctly notices patterns along the diagonals. There are some patterns he has missed, the multiples of 4, and odd numbers that go up by 6 each time.

He has not noticed anything special about the numbers within the expressions that make the diagonal numbers.

Aakar has made no attempt to generalize.

For each diagonal there is a number pattern. For each pattern, replace the y in the expression x2 − y2 with an expression in terms of just x. [x2 − (x − 1)2, x2 − (x − 2)2, x2 − (x − 3)2, x2 − (x − 4)2, etc.]

For each of the diagonals, what do you notice about the relationship between the x and the answers in each cell? Use what you notice to write the expression, in terms of x. [ x2 − (x − 1)2 = 2x − 1, x2 − (x − 2)2 = 4x − 4, x2 − (x − 3)2 = 6x − 9, x2 − (x − 4)2 = 8x − 16 etc.]

Students may also notice a pattern in the structure of these expressions: if the difference between two consecutive numbers in a diagonal is n, then the expression can be written in the form nx − (n/2)2. This argument is not a proof as it again assumes the pattern continues.

Aakar has not noticed the number patterns in the rows:

For each row there is a number pattern. Write each pattern in terms of x. [x2, x2 − 1, x2 − 4, x2 − 9, etc.]

These equations make the assumption that the patterns continue.

Tanya provides two examples and then makes a conjecture. She checks her conjecture and realizes it is only correct for consecutive numbers.

Tanya uses algebra to generalize about the difference of the squares of two consecutive numbers. She has assumed the pattern continues. The answer of 2n + 1, for any integer n indicates the answer is always odd (2n must be even so 2n + 1 must be odd.)

Tanya needs now to figure out an expression for numbers that differ by two.

Figure out an expression for the integer answer for the difference of two squares that differ by 2.


The integer answer for the expression (n + 2)2 − n2 is 4n + 4 = 4(n + 1). This shows the answer is a multiple of 4. But again this explanation makes the assumption that the pattern continues.

Tanya could also look at integers that differed by 3, 4 and so on. She could then extend her argument to any numbers n and m. They differ by m − n. So the integer answer is (n + m)(m − n). This argument is not a proof as it assumes the pattern continues.

Heng uses areas to explain his answer. He correctly explains why the answer is always odd when the numbers in the expression are consecutive. He has not attempted to consider any other combination of numbers in the expression, such as when the integers differ by 3, 4 and so on.

Georgina's statement that when the numbers in the expression are consecutive the answer is odd is correct.

She has organized her work into a table. She has drawn a graph of the relationship between the answer and the first number in the expression when the numbers in the expression differ by one.

How does Georgina figure out the equation of the line?

She assumes the relationship between x and y can be generalized.

Georgina could now look at integer answers that are multiples of 4.


Sample Student Response: Georgina

What Math did Georgina do well?

What assumption has Georgina made?

What further work can Georgina do?


Whole-class discussion: comparing different approaches (15 minutes) Hold a whole class discussion to consider the different approaches used in the sample work. Focus the discussion on parts of the task students found difficult. Ask the students to compare the different solution methods.

Which approach did you like best? Why? Which approach did you find most difficult to understand? Why?

To support the discussion, you may want to use Slides P-3, P-4, P-5, and P-6 of the projector resource.

Review solutions to The Difference of Two Squares (10 minutes) Give out the sheet How Did You Work? and ask students to complete this questionnaire.

Some teachers set this task as homework.

The questionnaire should help students review their progress.

If you have time, ask your students to read through their original solution and using what they have learned, attempt the task again. In this case, give each student a blank copy of the assessment The Difference of Two Squares.


SOLUTIONS Assessment task: The Difference of Two Squares

There are many ways students could answer these questions. Below are a few possible examples. Proofs of conjectures usually will involve manipulating quadratic functions that may be beyond many students at this level. However, systematic investigations that develop patterns and that show an exhaustive consideration of cases may be possible.

1. Michael is not correct. Students may have shown their answers in a table, for example:

Difference of 1 Difference of 2 Difference of 3 Answer

12 − 02 1

Cannot be expressed as the difference of two squares. 2

22 − 12 3

22 − 02 4

32 − 22 5


42 − 32 7

32 − 12 8

52 − 42 32 − 02 9


62 − 52 11

42 − 22 12

72 − 62 13


82 − 72 42 − 12 15

52 − 32 16

From this table, it may be conjectured that the only integers that cannot be expressed as the difference of two squares are those that are of the form 2(2n-1).

To explain why certain integers cannot be expressed as the difference of two squares, students may try to argue systematically, for example:

For the integer 2, Try 22 − 02 = 4, or 22 − 12 = 3. Neither of these expressions make 2.

Try 32 − 02 = 9, or 32 − 12 = 8, or 32 − 22 =5. None of these expressions make 2.

As the first number in the expression gets greater then the answer will also get greater. All answers will be greater than 2.


2. From the table students may notice that all odd numbers can be expressed as the difference of two square numbers. The two integers in these expressions are always consecutive. Numbers that are multiples of 4 can also be expressed as the difference of two squares. The two integers in these expressions differ by 2.

There are many ways students could explain the situation. For example, students may use a two-way table for x2 − y2:

x

y

0 1 2 3 4 5 6 Diagonal patterns Horizontal patterns

0 0 1 4 9 16 25 36 x2

1 0 3 8 15 24 35 Multiples of 5, x2 − (x − 5)2 = 10x − 25 x2 − 12

2 0 5 12 21 32 Multiples of 8, x2 − (x − 4)2 = 8x − 16 x2 − 22

3 0 7 16 27 Go up in 6's, x2 − (x − 3)2 = 6x − 9 x2 − 32

4 0 9 20 Multiples of 4, x2 − (x − 2)2 = 4x − 4 x2 − 42

5 0 11 Odd numbers, x2 − (x − 1)2 = 2x − 1 x2 − 52

6 0

The table shows how some integers can be expressed as the difference of two squares. The algebraic expressions in the table all assume the patterns continue.

Diagonal patterns:

Students may notice a pattern in the structure of these expressions: if the difference between the two numbers is n, then the expression can be written in the form 2nx − n2. This argument is not a proof as it assumes the pattern continues.

Alternatively, students may use area diagrams:

Area = (x + 1)2 − x2 = 2x +1

This shows all odd numbers can be expressed as the difference of two squares.

Area = (x + 2)2 − x2 = 4x + 4

This shows all multiples of 4 can be expressed as the difference of two squares.

Area = (x + 3)2 − x2 = 6x +9

This shows all numbers of this form expressed as the difference of two squares.

Students may then assume that when the difference between the two numbers is n the integer answer is 2nx + n2.

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The Difference of Two Squares Michael writes:

1. Is Michael correct? Explain your answer.

2. What do you notice about the integers that can be expressed as the difference of two square integers? Explain what you notice.


Sample Student Response: Aakar

Explain Aakar’s table.

Can you see any more diagonal number patterns in the table? Write each pattern in terms of x.

How can Aakar use the number patterns to explain why some integers cannot be expressed as the difference of two squares?


Sample Student Response: Tanya

What Math did Tanya do well?

What assumption did Tanya make?

What do the expressions 2n + 1 tell you about the integer answers?

What further work can Tanya do?


Sample Student Response: Heng

What Math did Heng do well?

What further work can Heng do?







How Did You Work? Mark the boxes and complete the sentences that apply to your work.

1 Our group work better than my own work

Our group solution was better because

2 Our solution is similar to one of the sample responses OR Our solution is different from all the sample responses

Our solution is similar to

Add name of the student

Our solution is different from all the sample responses because:

I prefer our solution / the student’s solution because

4 I made some assumptions

My assumptions were

3 What advice would you give a student new to this task about potential pitfalls


Planning a Joint Method

P-1

1  Take turns to explain your method and how you think your work could be improved.

2  Listen carefully to each other. - Ask questions if you don’t understand.

3  Once everyone in the group has explained their method, plan a joint method that is better than each of your separate ideas.

4  Make sure that everyone in the group can explain the reasons for your chosen method.

5  Write a brief outline of your method on one side of your sheet of paper.


Evaluating Student Sample Responses

1.  Take it in turns to work through a students’ solution. -  Write your answers on your mini-whiteboards.

3.  Explain your answer to the rest of the group.

4.  Listen carefully to explanations. –  Ask questions if you don't understand.

5.  Once everyone is satisfied with the explanations, write the answers

below the students’ solution. -  Make sure the student who writes the answers is not the

student who explained them.

P-2


Sample Student Response: Aakar

P-3


Sample Student Response: Tanya

P-4


Sample Student Response: Heng

P-5



P-6






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The Difference of Two Squares - svmimac.orgsvmimac.org/images/L72_The_Difference_of_Two_squares_draft... · The Difference of Two Squares ... if students attempt to make generalizations