Proofs of the Pythagorean Theorem -...

Lesson #30

© 2012 MARS University of Nottingham

Mathematics Assessment Project

Formative Assessment Lesson Materials

Proofs of the Pythagorean Theorem

MARS Shell Center University of Nottingham & UC Berkeley

Alpha Version January 2012 !!!!

Please Note:!!These materials are still at the “alpha” stage and are not expected to be perfect. The revision process concentrated on addressing any major issues that came to light during the first round of school trials of these early attempts to introduce this style of lesson to US classrooms. In many cases, there have been very substantial changes from the first drafts and new, untried, material has been added. We suggest that you check with the Nottingham team before releasing any of this material outside of the core project team. !!!!

If you encounter errors or other issues in this version, please send details to the MAP team c/o [email protected].

Proofs of the Pythagorean Theorem Teacher Guide Alpha Version January 2012

© 2012 MARS University of Nottingham 1

Proofs of the Pythagorean Theorem 1

Mathematical goals 2

This lesson unit is intended to help you assess how well students are able to produce and evaluate geometrical 3

proofs. In particular, this unit aims to identify and help students who have difficulties in: 4

• Interpreting diagrams. 5

• Identifying mathematical knowledge relevant to an argument. 6

• Linking visual and algebraic representations. 7

• Producing and evaluating mathematical arguments. 8

Common Core State Standards 9

This lesson relates to the following Standards for Mathematical Content in the Common Core State Standards 10

for Mathematics: 11

G-CO: Prove geometric theorems. 12

G-SRT: Prove theorems about triangles 13

This lesson also relates to the following Standards for Mathematical Practice in the CCSS: 14

3. Construct viable arguments and critique the reasoning of others. 15

7. Look for and make use of structure. 16

Introduction 17

This lesson unit is structured in the following way: 18

• Before the lesson, students attempt the assessment task individually. You then review students’ work 19

and formulate questions that will help them to improve. 20

• The lesson begins with a whole class discussion of a related diagram. Students then work collaboratively 21

in pairs or threes on the assessment task, to produce a better collective solution than those they produced 22

individually. Throughout their work they justify and explain their decisions to peers. 23

• In the same small groups, students critique examples of other students’ work. 24

• In a whole class discussion, students explain and evaluate the arguments they have seen and used. 25

• Finally, students work alone on a new task similar to the original assessment task. 26

Materials required 27

• Each individual student will need copies of the pre- and post-assessment tasks Proving the 28

Pythagorean theorem and Proving the Pythagorean theorem again and some squared paper to work 29

on. 30

• Each small group of students will need a large sheet of paper, copies of Sample Responses to Discuss 31

(1) and (2) and the sheet Comparing the methods. 32

• Provide copies of the extension activity, Proving the Pythagorean theorem using similar triangles, as 33

necessary. 34

• There are some projectable resources to help with whole-class discussion. 35

Time needed 36

Twenty minutes before the lesson, a one-hour lesson, and twenty minutes in a follow up lesson (or for 37

homework). All timings are approximate, exact timings will depend on the needs of your students. 38

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Before the lesson 41

Pre-Assessment Task: Proving the Pythagorean Theorem task (20 minutes) 42

Have the students do this task, in class or for homework, a day or more before the formative assessment lesson. This will give you an opportunity to assess the work and identify students who have difficulty. You will then be able to target your help more effectively in the follow-up lesson.

Give each student a copy of Proving the Pythagorean Theorem and a sheet of squared paper.

Introduce the task briefly, and help the class to understand the work they are being asked to do.

Spend twenty minutes working individually, answering these questions.

Write all your reasoning on the sheet, explaining what you are thinking.

It is important that, as far as possible, students answer the questions without assistance. 43

Students who sit together often produce similar answers, and then when they come to compare their work, they 44

have little to discuss. For this reason, we suggest that when students do the task individually, you ask them to 45

move to different seats. At the beginning of the formative assessment lesson, allow them to return to their usual 46

places. Experience has shown that this produces more profitable discussions. 47

Assessing students’ responses 48

Collect students’ responses to the task. Read through their scripts and make some notes on what their work 49

reveals about their current levels of understanding and their different approaches to producing a proof. 50

We strongly suggest that you do not score students’ work. Research shows that this is counterproductive, as it 51

encourages students to compare scores and distracts their attention from what they are to do to improve their 52

mathematics. 53

Instead, help students to make further progress by asking questions that focus attention on aspects of their work. 54

Some suggestions for these are given on the next page. These have been drawn from common difficulties 55

observed in trials of this unit. 56

We suggest that you write your own lists of questions, based on your own students’ work, using the ideas below. 57

You may choose to write questions on each student’s work, or, if you do not have time for this, just select a few 58

questions that apply to most students and write these on the board when the assessment task is revisited. 59

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Proofs of the Pythagorean Theorem Student Materials Alpha Version December 2011

© 2011 MARS University of Nottingham S-1

Proving the Pythagorean theorem

Marty is attempting to prove the Pythagorean theorem.

He draws the following sketch showing two congruent right triangles joined by a line:

1. Using a pencil and a ruler, re-draw Marty’s diagram so that it is more accurate. Label your diagram clearly.

2. Write down what you know about all the lengths, angles, shapes and areas on the diagram. Give reasons for your statements.

3. Using the diagram, construct a proof to show that in any right triangle with sides a, b, c and hypotenuse c, a2+b2=c2



64

Common issues: Suggested questions and prompts:

Inaccurate construction of diagram For example: The student has not made lengths with the same label equal.

• Look carefully at the diagram Marty drew. Which lengths are equal?

• Label the sides of the triangles in Marty’s diagram. Use this to help you draw accurately.

Insufficient mathematical knowledge elicited

For example: The student does not deduce that the angle between the sides marked c is right angled.

• What can you say about the angles in this diagram?

• What different geometrical figures can you see? • What do you know about the areas of these

figures?

Use of relevant mathematical structure

For example: The student does not realize that the side length of the large square is the sum of a and b.

• How does your area formula apply to the trapezoid in your diagram?

• How could you write the length of this side using algebra?

Incomplete solution

For example: The student has written some relevant theorems and noticed some relevant structure but has lost direction.

• What do you already know? • What do you want to find out? • Try working backwards: what will the end result

be?

Visual solution

For example: The student recognizes that the area of the square side c on the first diagram is equal to a2 + b2 on the second diagram, but doesn’t justify this.

• The rearrangement shows that c2 = a2+b2, but can you explain why this is true using words and algebra?

Empirical solution For example: The student has measured the sides of the triangle and used those measures in length/area calculations.

• Think of your triangle as representing any right triangle with sides a, b, c.

• How does your diagram help you show the Pythagorean Theorem is true for any right triangle?

Complete solution Provide copy of the extension task, Extension: Proving the Pythagorean theorem using similar triangles.

• Here is a new diagram. Use it to prove the Pythagorean Theorem.

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Suggested lesson outline 66

Whole-class introduction (10 minutes) 67

Remind students of the assessment task A Proof of the Pythagorean Theorem they completed prior to this lesson: 68

What is the Pythagorean Theorem? 69

Are there any other ways of stating the theorem? 70

Prompt students for both length and area expressions. 71

Check that students understand when the Pythagorean Theorem can be applied: 72

For what kind of triangles is Pythagorean Theorem true? 73

Is it true for any right triangle? Even one that looks like this? [Draw some extreme cases] 74

There are many different ways of proving the Theorem. 75

We are going to look at some this lesson. 76

Re-introduce Marty’s attempt to prove the theorem using projectable resource. Step through the construction of 77

Marty’s diagram, step by step (see below). Don’t try to complete the proof at this stage, just help students to 78

understand the construction of the diagram. Research shows that students benefit from reconstructing and 79

analyzing diagrams. At each stage, prompt students to use clear mathematical language to describe this 80

construction process, such as congruent, perpendicular, reflect in a horizontal/vertical line and rotate by 90° 81

(counter) clockwise. 82

Describe how this triangle has moved. 83

How do you know that the area in the square is a right angle? 84

If the sides of the triangle are a and b, what is the total length of this line? 85

How do you know that the orange area never changes? 86

Collaborative group work (15 minutes) 87

Organize students into groups of two or three. Distribute a large sheet of paper to each group. 88

I have read your solutions to the task I gave you last time and I have some questions about your work. 89

Use my questions to review your work and then work together to come up with a better solution. 90

Spend a few minutes on your own first, reading my questions and then share your ideas with the rest of 91

the group. 92

Explain that when proving, it is useful to gather knowledge of the mathematical structure: 93

When thinking about your joint solution, you may first want to discuss how the diagram is constructed. 94

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Think about what you already know about lengths, areas and angles and think about what you know of 95

this diagram in particular. 96

Think about how you can use the diagram to figure out the statement of the Pythagorean Theorem. 97

Make sure you explain everything really clearly on your poster paper. 98

You have two tasks during small group work: 99

• Note different student approaches to the task. Notice how students make a start on the task, if they get 100

stuck, and how they respond if they do come to a halt. Note which properties they identify, and how 101

they choose which information may be useful. Notice when students look for different ways of writing 102

the same information. Do they use visual/geometrical as well as algebraic language? Do they produce a 103

general (algebraic or visual) solution, or do they introduce measures and work empirically? 104

• Support student problem solving. Try not to make suggestions that move students towards a particular 105

approach to this task. Instead, ask questions to help students clarify their thinking. If several students in 106

the class are struggling with the same issue, write a relevant question on the board. You might also ask 107

a student who has performed well on a particular part of the task to help a struggling student. 108

The following questions and prompts have been found most helpful: 109

• Which side(s) of the triangle form(s) the side of the square? 110

• What is the length of this line segment? 111

• How could you calculate the area of this square? Is that the only way to figure out the area? 112

• How do you know this is always true? Maybe it only works for these numbers. 113

• You have assumed that…. Why do you believe that is true? 114

Collaborative analysis of Sample methods to discuss (20 minutes) 115

Give each small group of students a copy of the two sheets: Sample methods to discuss, a blank sheet of paper to 116

write their responses on, and a copy of the sheet: Comparing the methods. Draw students’ attention to the 117

questions on these sheets (projectable resource). 118

• Read through these proofs. None of them is perfect. 119

• Describe what each student has done on the blank sheet. 120

• Will their approach lead to a proof? Why? 121

• Explain how the work can be improved. 122

123

• When you have done this, evaluate the completed arguments. 124

• Which approach do you find most convincing? Why? 125

• Produce a complete correct solution using your preferred method. 126

I have some grid paper if you want to use it. 127

This analysis task will give students the opportunity to engage with and evaluate different types of argument 128

without providing a complete solution strategy. It also raises questions about what counts as a good 129

mathematical argument for discussion in the plenary. 130

We have included methods here that reflect some common issues in student reasoning. Penelope and Sophie 131

have only considered specific cases. In Penelope’s case, her reasoning can be generalized, using algebra, but 132

Sophie’s diagram will only work for isosceles right triangles, so her method cannot be generalized. Nadia’s 133

approach shows the beginnings of an algebraic method, but this is incomplete and contains an error. 134

During small group work, support student thinking as before. Also, listen to see what students find difficult. 135

Identify one or two aspects of these approaches to discuss in the plenary. Note similarities and differences 136

between the sample approaches and those the students took in small group work. If students require more work, 137

ask them to produce a proof using Penelope’s diagram and compare this with their improved version of Marty’s 138

(visual and algebraic) solution method. 139

140



Whole class discussion comparing different approaches (15 minutes) 141

Organize a whole class discussion to analyze the different methods of the Sample Responses to Discuss. These 142

are all given on the projector resources. The intention is that you focus on getting students to explain the 143

methods of working, and compare different styles of argument, rather than just checking numerical or algebraic 144

solutions. 145

Penelope has measured the diagram and found the area of the figure in two different ways. She finds that the whole trapezoid has an area that is roughly equal to the three triangles added up.

This does not amount to a proof of the Pythagorean theorem of course, because it only considers a special case. She has some material here for developing a proof, notably the (implied) equation:

!

12(a + b)2 =

12ab " 2 +

12c 2 .

If this were simplified, it could be made into a proof.

• Penelope assumes the shape is a trapezoid. How do you know she is correct?

• How can we develop her ideas into a proof?

Nadia’s method is rather like Marty’s, but she has not tried to transform the diagram, but has tried to find the area again in two different ways. Some students may notice that Nadia’s diagram is like two of Penelope’s placed on top of one another.

There is an algebraic error in the last line (a common one), but if this were corrected the proof could be completed. By equating

!

(a + b)2 = 2ab + c 2 • Nadia assumes the angle in the inner

quadrilateral is a right angle. Is she correct? How do you know?

• Your group used a similar solution method to Nadia. Can you explain the solution?

• How can we develop her ideas into a proof?

Sophie’s method does not generalize. It can be used only for isosceles right triangles, in which case a = b. Emphasize the need to check that diagrams work for all possible cases.

• Will Sophie’s method work for all right angled triangles?

• What would happen if you tried to draw Sophie’s diagram making the sides of the right triangle unequal?

146

• How could Penelope or Nadia improve her solution? 147



• Was it hard to understand these approaches? 148

Once students have explained both methods, ask them to compare them. 149

• Which did you think was the most convincing proof? Why? 150

Students work on a further approach (post assessment) (20 minutes) 151

Invite students to work individually on the sheet: Proving the Pythagorean theorem again. This may be used as a 152

post-assessment to see if students can apply what they have learned from the lesson. This may be completed in a 153

follow-up lesson or for homework. 154

Optional extension task 155

A further method for proving the Pythagorean theorem is begun on the extension sheet: Proving Pythagorean 156

theorem using similar triangles. This may be given to students that have been successful with the other 157

approaches shown in this lesson as a further challenge. 158

Solutions 159

Proofs of the Pythagorean Theorem 160

Students may choose visual, empirical or algebraic approaches with any of the diagrams. We provide sample 161

algebraic proofs in the Sample Responses to Discuss (below). 162

Analysis of Sample Responses to Discuss 163

Marty’s solution 164

Marty relies on visual transformation of the area. This visual transformation is a powerful mathematical tool, but 165

it leaves too much for the reader to do; it does not constitute a full proof. Marty provides no explanation of his 166

approach. He moves the four congruent triangles to form two rectangles of side lengths a and b, with two 167

squares of sides length a, b respectively making up the rest of the large square area. 168

Marty could strengthen his solution by showing the connection between the variables he uses to label the side 169

lengths and the areas of the constituent parts of the figures more explicitly. In particular, he needs to describe 170

how the algebra he uses links the lengths a, b, c to the transformed areas in his second diagram. He needs to 171

provide much more explanation of his work to make it clear for the reader. 172

For example, he could write: 173

“The side length of the large square is a+b. So the area of the large square is (a+b)2 = a2+b2+2ab. 174

Now I can find the area of the individual pieces of the large square. 175

The inner square has side length c and area c2. 176

Each of the right triangles has area

!

12 ab. Two of these triangles form a rectangle ab. There are four of them. So 177

this gives an area of 2ab from the triangles. 178

Adding together all the pieces gives the area of the large square. So the area of the large square is c2+2ab. 179

I now have two ways of writing the area of the large square. So c2+2ab= a2+b2+2ab. 180

Subtract 2ab from each side to see that c2= a2+b2.” 181

Penelope’s solution 182

Penelope gives a clear explanation of her approach at the beginning of her solution, that she is finding the area of 183

the trapezoid as a whole, and as the sum of three triangle areas. She provides more information for the reader 184

than the others, so her solution is a better explanation. However, she takes an empirical approach, using the 185

measures of the sides of the triangles to find that the area is approximately equal whether found as a trapezoid or 186



as the sum of the areas of three triangles. This is not a proof of the Pythagorean Theorem because it is not a 187

general argument. 188

Penelope’s solution can be improved using some of the formulas she identified. However, she first needs to 189

show that the quadrilateral is a trapezoid. The two (apparently) vertical sides are parallel, since they meet the line 190

segment ab at the same (right) angle. 191

Then area of the trapezoid =

!

12 h(a + b) = 1

2 (a + b)(a + b) = 12 (a + b)2 = 1

2 (a2 + 2ab + b2) 192

Penelope can also find the area of the trapezoid as the sum of three triangles. Area of triangle with side lengths a, 193

b =

!

12 ab. There are two of these, with total area

!

ab . 194

Penelope needs to establish that the angle between the segments of length c is right before using these to 195

calculate the area of the triangle with side lengths c. Since the triangle with sides a, b is right, the two unknown 196

angles sum to 90°. The angle in the triangle with side lengths c forms a straight line with these two angles. So 197

the missing angle in the triangle with side lengths c is 90°. 198

The area of triangle with perpendicular side lengths c is

!

12 c2 . So the total area of the trapezoid is

!

12 c2 + ab 199

Since both methods give the total area of the trapezoid,

!

12 c2 + ab =

!

12 (a2 + 2ab + b2 ) = 1

2 a2 + ab + 12 b2 200

Cancelling ab from both sides,

!

12 c2 = 1

2 a2 + 12 b2 , so

!

c2 = a2 + b2 as required. 201

Nadia’s solution 202

This shows just one half of Marty’s diagram and therefore does not show any transformations of the geometric 203

shapes. There is one mistake in assuming that

!

(a + b)2 = a2 + b2, but if this were corrected one could deduce 204

that:

!

(a + b)2 = 2ab + c 2, which leads to a solution. 205

Proving the Pythagorean Theorem again 206

The post-assessment task is another dissection proof. The sides a, b are perpendicular and so can be used to 207

calculate the area of each right triangle. The hypotenuse of the right triangle, c, forms the side of the large 208

square. 209

Area of large square = c2. 210

Area of large square = area of four triangles + area of small square. 211

Area of one triangle =

!

12

base " perp.height =12

ab. 212

Area four triangles =

!

4 "12

ab = 2ab . 213

Side length of inner square =

!

a " b . 214

Area of inner square =

!

a " b( )2 = a2 " 2ab + b2. 215

So area of the large square is

!

c 2 = 2ab + a2 " 2ab + b2 = a2 + b2 as required. 216

Proofs of the Pythagorean Theorem Student Materials Alpha Version January 2012


Proving the Pythagorean theorem

Marty is attempting to prove the Pythagorean theorem.

He draws the following sketch showing two congruent right triangles joined by a line:

1. Using a pencil and a ruler, re-draw Marty’s diagram so that it is more accurate. Label your diagram clearly.

2. Write down what you know about all the lengths, angles, shapes and areas on the diagram. Give reasons for your statements.

3. Using the diagram, construct a proof to show that in any right triangle with sides a, b, c and hypotenuse c, a2+b2=c2



Grid Paper



Sample methods to discuss (1)

Below are three attempts at a proof of the Pythagorean theorem.

1. Describe what each student has done.

2. Will this approach lead to a proof of the theorem?

3. Explain how the work can be improved.

Penelope’s method



Sample methods to discuss (2) Nadia’s method

Sophie’s method



Comparing the methods Compare the three solutions.

Whose solution method do you find most convincing? Why?

Produce

Produce a complete correct solution using your preferred method.

Your teacher has grid paper if you want to use it.

Produce

Produce



Proving the Pythagorean theorem again

Four congruent right triangles and a square may be rearranged to get from diagram 1 to diagram 2.

1. Using a pencil and a ruler, re-draw the diagrams so that they are more accurate. Label your diagrams clearly.

2. Write down what you know about all the lengths, angles, shapes and areas on the diagrams. Give reasons for your statements.

3. Using the diagrams, construct a proof to show that in any right triangle with sides a, b, c and hypotenuse c, a2+b2=c2



Extension: Proving the Pythagorean theorem using similar triangles.

Max has started to prove the Pythagorean theorem using similar triangles.

1. Explain why the three triangles are similar.

2. Try to complete Max’s proof.

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© 2012 MARS, University of Nottingham Alpha Version January 2012 Projector Resources:

Analyzing and comparing

1.  Describe what each student has done. 2.  Will the approach lead to a proof of the theorem? 3.  Explain how the work can be improved. Compare the three solutions. 1.  Whose solution method do you find most

convincing? Why? 2.  Produce a complete correct solution using your

preferred method.

14


!

Penelope’s method

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© 2012 MARS, University of Nottingham Alpha Version January 2012 Projector Resources: !

Nadia’s method

16


!

Sophie’s method

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Proving the Pythagorean theorem again

18

!


Extension: Using similar triangles.

19 !

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