Lesson 7

The purpose of this warm-up is to give students a chance to study a diagram that they will need to understand for an upcoming proof of the Pythagorean Theorem. The construction depends on the triangles being right triangles, so students get to contrast it with a similarly constructed figure with non-right triangles. In that case, the composite figure is not a square.

Arrange students in groups of 2. Display the diagram for all to see. Give students 1 minute of quiet work time to identify at least one thing they notice and at least one thing they wonder about the diagram. Ask students to give a signal when they have noticed or wondered about something. When the minute is up, give students 1 minute to discuss their observations and questions with their partner. Follow with a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After each response, ask the class whether they agree or disagree and to explain alternative ways of thinking, referring back to the images each time.

Tell students that when you take a square and put a congruent right triangle on each side as shown on the left, they form a larger square (they will be able to prove this in high school). But it doesn’t work if the triangles are not right triangles. We will use this construction in the next activity.

The purpose of this activity is for students to work through an area-based algebraic proof of the Pythagorean Theorem (MP1). One of the figures used in this particular proof, G, was first encountered by students at the start of the year during a unit on transformations and again in a recent lesson where they reasoned about finding the area of the triangles.

While there are many proofs of the Pythagorean Theorem similar to the one in this activity, they often rely on \((a+b)^2=a^2+2ab+b^2\), which is material beyond the scope of grade 8. For this proof, students reason about the areas of the two squares with the same dimensions. Each square is divided into smaller regions in different ways and it is by using the equality of the total area of each square that they uncover the Pythagorean Theorem. The extension uses this same division to solve a challenging area composition and decomposition problem.

Begin by explaining to students how the two figures are constructed. Each figure starts with a square with side length \(a+b\).

• Figure F partitions the square into two squares and two rectangles.
• Figure G takes a right triangle with legs \(a\) and \(b\) and puts one identical copy of it in each corner of the square. The copies touch each other because the short leg of one and the long leg of the one next to it add up to \(a+b\), so they fit exactly into a side. So they form a quadrilateral in the middle. We know the quadrilateral is a square because
  • The corners must be 90 degree angles:
    • The two acute angles in each triangle must sum to 90 degrees because the sum of the angles in a triangle is 180 degrees, and the third angle is 90 degrees.
    • The two smaller angles along with one of the corners of the quadrilateral form a straight angle with a measure of 180 degrees, that means that the angle at the corner must also be 90 degrees.
  • All four sides are the same length: they all correspond to a hypotenuse of one of the congruent right triangles.

Arrange students in groups of 2. Give 3 minutes of quiet work time for the first two problems. Ask partners to share their work and come to an agreement on the area of each figure and region before moving on to the last problem. Follow with a whole-class discussion.

Begin the discussion by selecting 2–3 groups to share their work and conclusion for the third question. Make sure the last group presenting concludes with \(a^2+b^2=c^2\) or something close enough that the class can get there with a little prompting. For example, if groups are stuck with the equation looking something like \(a^2+ab+b^2+ab=\frac12ab+\frac12ab+\frac12ab+\frac12ab+c^2\), encourage them to try and combine like terms and remove any quantities both sides have in common on each side.

After groups have shared, ask students how they see the regions in each figure matching the regions in the other figure. For example, since the two small squares in Figure F match the one large square in Figure G, how do the rectangles and triangles match? After some quiet think time, select 1–2 students to explain how they see it. (The area of the two rectangles is the same as the area of two of the triangles since, if I put two of the triangles together, I get a rectangle that is \(a\) wide and \(b\) long.) Show students an image with the diagonals added in, such as the one shown here, to help make the connection between the two figures clearer.

Note how these figures can be made for any right triangle with legs \(a\) and \(b\) and hypotenuse \(c\).

Before this lesson, students could only find the length of a segment between the intersection of grid lines in a square grid by computing the area of a related square. The Pythagorean Theorem makes it possible to find the length of any segment that is a side of a right triangle. 

Arrange students in groups of 2. Give students 3 minutes of quiet work time followed by partner and then whole-class discussions. 

Invite a few students to share their reasoning with the class for each unknown side length. As students share, record their steps for all to see, showing clearly the initial setup with \(a^2+b^2=c^2\).

In this activity, students explore a transformations-based proof of the Pythagorean Theorem, calling back to their work with transformations earlier in the year. Since this proof is not one students are expected to derive on their own, the focus of this activity is on understanding the steps and why they are possible from a transformations perspective.

Listen for students using precise language when describing their transformations.

The digital activity demonstrates the same proof in a slightly different way. Students have the opportunity to explore three different right triangles in the applets.

Select previously identified groups to share their explanations for each transformation and their conclusion to the last problem. If possible, have them show each transformation for all students to see.

An important takeaway for this activity is that this proof can be generalized to any right triangle. To help students see why, ask them to consider how the diagonal lines in Figure 1 were created. Give 1–2 minutes for partners to discuss and then select 2–3 groups to share their ideas. Hopefully, at least one group noticed that the diagonals create two congruent right triangles with sides of length \(a\), \(b\), and \(c\)—the same as the original right triangle. This means that this process could be duplicated to show that for any right triangle with legs \(a\) and \(b\) and hypotenuse \(c\), \(a^2+b^2=c^2\) is true.