Fishtank Learning

8th Grade Math
Unit 7: Pythagorean Theorem and Volume
Lesson 6

Objective

Understand the Pythagorean Theorem as a relationship between the side lengths in a right triangle.

Warm Up

Use the following set of images to answer the questions below.

Fishtank Learning curriculum image

a. What do you notice? What do you wonder?

b. Do you think the right angle in these triangles are important?

Anchor Problems - Problem 1

Use the square grid cut outs to find examples of right triangles. Use the right angle guide on the second page of the cut outs to aid in your precision. Here is an example:

Fishtank Learning curriculum image

a. For each right triangle you can create, record the side lengths and areas of the squares that created it.

b. Look at your data. What pattern(s) do you observe?

Purpose

This problem engages students in an exploration of right triangles in a hands-on way using square cut outs. They use the sides of the squares to create closed right triangles, finding sets of three that work, and exploring patterns in the areas of the squares. Allow for some slightly off measurements, since students are using paper manipulatives, and there will be time for precision in later problems when students use the algebraic formula. At the end of the investigation, the Pythagorean Theorem is introduced and connected to their exploration.

Setup

Have students form partner pairs or small groups.
Print out one copy of Square Grid Cut Outs for each pair or small group of students. Print single-sided and use card stock if it is available.
Hand out a pair of scissors to each student and have them cut out the squares from the grid. Students should save the bottom half of the second page to use the right angle as a guide.

Facilitation Guidance

Launch the task by having students become familiar with the square cut outs and adding labels that will help them work efficiently in their exploration time, such as the side length and area of each square.
Have a student in each group recreate the example shown in the problem with their own cut outs to ensure they understand how to work with the squares.
- How did the right angle guide help you line up your square pieces?
- What are the side lengths of this triangle?
- Can you create an example that does not make a right triangle?
Give groups 1-2 minutes to discuss how they will organize and record their work and findings. Have groups share with the class so that different options are available.
Supporting Access for Students

If students need executive functioning or organization support, you can suggest any of the following:
- In labeling the squares, have students record:
  - Side length: $$5u$$
  - Area: $$5^2$$ or $$25u^2$$
- Organize the squares on someone’s desk from smallest to largest
- Create a table to keep track of their findings. Possible headers to include:
  
  Side length 1 Area of square 1 Side length 2 Area of square 2 Side length 3 Area of square 3 Notes
Give students time to work in their groups to find right triangles. Circulate as students work to check for an acceptable level of precision/accuracy and to ask guiding questions:
- What is your approach to trying out different side lengths?
- What are you looking for to see if you made a right triangle?
- Can you show me one that you found?
- Can you show me one that was not a right triangle?
- How are you organizing your findings? Will this support you in finding patterns in Part (b)?
Once it looks like students have found several right triangles, have them shift to Part (b). As you walk around, listen to the observations students make. Quietly write them down verbatim somewhere everyone can see.
Supporting Access for Students

Additional supports to help students find patterns in their data:
- Have students look at one single set of squares that made a triangle. You can also nudge them to look at the areas of the squares and see if they observe anything.
- Have students look at some of the smaller numbers in their data as that may be easier to observe the pattern in the square areas.
- Have students compare and contrast a set of squares that made a triangle with a set of squares that did not make a triangle. Using smaller numbers can also help here.
- Have students reorganize their data so the first two sides they look at are the smaller sides, and the third side is the largest side.
- Have students recreate their data with just the areas, organized from smallest to largest across each row.
Debrief the problem.
- First, gather a list of the right triangles student groups found. Record these on the board for everyone to see.
- Then, open the discussion of Part (b). Show students the collection of their thoughts that you recorded earlier. Ask if students see any trends, patterns, what questions arise, and what they agree or disagree with. Generally, guide students to the conclusion that there is a relationship between the square areas of the side lengths of a right triangle. When you add the area of two of the squares, you get the area of the third.
- Conclude the problem by introducing the Pythagorean Theorem, $$a^2+b^2=c^2$$, where $$a$$ and $$b$$ are the two shorter sides of the triangle and $$c$$ is the longest side. (Students will learn the terminology legs and hypotenuse in the next Anchor Problem.)
  - Make the explicit connection between the area of the squares as numbers and represented as square numbers (i.e. the square with side length $$3$$ has an area of $$9$$ which is $$3^2$$).
  - Model this equation using a set of squares and connect it back to the data students collected and the pattern they observed.

Anchor Problems - Problem 2

Use the Pythagorean Theorem to show the relationship between the sides of the right triangles shown below. In each triangle, which measures represent the legs and which measures represent the hypotenuse?

Two right triangles with side lengths labeled, one with hypotenuse √40 and legs 6 and 2, the other with hypotenuse √12 and legs √3 and 3.

Purpose

The purpose of this Anchor Problem is to reinforce the relationship students discovered in Anchor Problem 1 and to give them practice applying the theorem, including instances with square root values for side lengths. Students are also introduced to new terminology used with right triangles: legs and hypotenuse. They informally observe that the hypotenuse is the longest side of a right triangle.

Facilitation Guidance

Launch the problem by having students look closely at the two triangles in the problem.
- Which angle in each triangle is the right angle?
- What tool(s) can you use to determine this if you can’t tell by inspection?
- Label the right angle with a right angle marker.
Introduce the terms legs and hypotenuse as vocabulary used to describe specific side lengths in right triangles. The hypotenuse is the longest side length of the right triangle and sits across from the right angle. The two legs meet at a right angle. Write down the Pythagorean Theorem and connect the variable $$c$$ to the hypotenuse and the variables $$a$$ and $$b$$ to the legs, with either variable referring to either leg.
Model using the Pythagorean Theorem with the triangle on the left, substituting in the values for the legs and the hypotenuse, and evaluating to demonstrate the equivalence of each side of the equation. If needed, review the fact that $$\left ({\sqrt{40}} \right )^2=40$$ because $${\sqrt{40} }$$ is the solution to the equation $$s^2=40$$.
Give students time to work on the second triangle independently. As students work, circulate and ask guiding questions:
- Which side lengths are the legs?
- Which is the hypotenuse?
- Which length is the longest in the triangle?
- Do you think the hypotenuse is always longer than either leg on a right triangle?
- What does your final statement mean? What does it tell you about the triangle?
Supporting Access for Students

If students need more practice applying the Pythagorean Theorem with rational numbers, give them triangles with whole number side lengths to practice using the theorem apart from square root work (i.e. Pythagorean triples such as 6, 8, 10 or 5, 12, 13).
Debrief the problem by reviewing responses and connecting the work to Anchor Problem 1.
- If these were triangles that you made like the ones you made in Anchor Problem 1, what would the squares look like?
- How does the algebraic form of the Pythagorean Theorem support the conclusion you came to in Anchor Problem 1?
- Were there any triangles in Anchor Problem 1 that were really close to being right triangles, but didn’t fit the pattern? How could we use the formula to see what’s going on with these triangles? What do you think we’ll see?

Problem Set

View Problem Set

Target Task - Problem 1

Lee tried to use the Pythagorean Theorem on the triangle shown below and found that the relationship did not hold true.

Triangle with sides labeled 7, 5, and √80, showing Pythagorean Theorem application.

Explain why the Pythagorean Theorem did not show a true relationship in Lee’s triangle.

Target Task - Problem 2

Which three measures could be the side lengths of a right triangle? Explain or show your reasoning.

$$5$$ $$2$$ $$7$$ $$\sqrt{24}$$