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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
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
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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Proofs of the Pythagorean Theorem Teacher Guide Alpha Version January 2012
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:
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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
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(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?
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• How could Penelope or Nadia improve her solution? 147
Proofs of the Pythagorean Theorem Teacher Guide Alpha Version January 2012
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