# Lesson 8

Applications of the Pythagorean Theorem

## 8.1: Closest Estimate: Square Roots (5 minutes)

### Warm-up

The purpose of this warm-up is for students to reason about square roots by estimating the value of each expression. The values given as choices are close in range to encourage students to use the square roots they know to help them estimate ones they do not. These understandings will be helpful for students in upcoming activities where they will be applying the Pythagorean Theorem.

While four problems are given, it may not be possible to share every strategy. Consider gathering only two or three different strategies per problem.

### Launch

Display one problem at a time. Give students 30 seconds of quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all previous problems displayed throughout the talk. Follow with a whole-class discussion.

### Student Facing

Which estimate is closest to the actual value of the expression? Explain your reasoning.

1. $$\sqrt{24}$$

• 4

• 4.5

• 5

2. $$\sqrt7$$

• 2

• 2.5

• 3

3. $$\sqrt{42}$$

• 6

• 6.5

• 7

4. $$\sqrt{10}$$ + $$\sqrt{97}$$

• 13

• 13.25

• 13.5

### Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. After each student shares, ask if their chosen estimate is more or less than the actual value of each expression.

Speaking: MLR8 Discussion Supports.: Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . . ." Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)

## 8.2: Cutting Corners (15 minutes)

### Activity

The purpose of this activity is for students to use the Pythagorean Theorem to reason about distances and speeds to figure out who will win a race. Students must translate between the context and the geometric representation of the context and back (MP2). Identify students whose work is clearly labeled and organized to share during the whole-class discussion.

### Launch

Arrange students in groups of 2. Provide students with access to calculators. Ask students to read and answer the first problem and then give a signal when they have done so. Next, poll the class to see who students think will win the race and post the results of the poll for all to see. Select 2–3 students to share why they chose who they chose. The conversation should illuminate a few key points: Mai travels farther than Tyler, but she is also going faster, so it is not immediately clear who will win.

Give 2 minutes of quiet work time for the second problem, followed by partner and then whole-class discussions.

Representation: Access for Perception. Read the situation aloud. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Language
Reading: MLR6 Three Reads. Use this routine to support reading comprehension of this problem without solving it for students. In the first read, students read the problem with the goal of comprehending the situation (e.g., Mai and Tyler are racing from one corner of the rectangular field to the opposite corner.). In the second read, ask students to identify important quantities that can be counted or measured (e.g., the length and width of the field; Mai’s speed on her bike; Tyler’s running speed). After the third read, reveal the question: “Who do you think will win? By how much?” Ask students to brainstorm possible solution strategies to answer the question (e.g., Use the Pythagorean Theorem to calculate the length of the diagonal across the field; Use Mai’s and Tyler’s speeds to calculate how long it will take them to reach the opposite corner.). This will help students concentrate on making sense of the situation before rushing to a solution or method.
Design Principle(s): Support sense-making

### Student Facing

Mai and Tyler were standing at one corner of a large rectangular field and decided to race to the opposite corner. Since Mai had a bike and Tyler did not, they thought it would be a fairer race if Mai rode along the sidewalk that surrounds the field while Tyler ran the shorter distance directly across the field. The field is 100 meters long and 80 meters wide. Tyler can run at around 5 meters per second, and Mai can ride her bike at around 7.5 meters per second.

1. Before making any calculations, who do you think will win? By how much? Explain your thinking.

2. Who wins? Show your reasoning.

### Student Facing

#### Are you ready for more?

A calculator may be necessary to answer the following questions. Round answers to the nearest hundredth.

1. If you could give the loser of the race a head start, how much time would they need in order for both people to arrive at the same time?

2. If you could make the winner go slower, how slow would they need to go in order for both people to arrive at the same time?

### Anticipated Misconceptions

Some students may rush through reading the problem and mix up who is traveling which path at what speed. Encourage these students to label the relevant distances and information on the diagram.

### Activity Synthesis

The purpose of this discussion is for students to share how they organized their work for the problem and see how accurate students’ predictions at the start of the activity were. Select 1–2 previously identified students to share their work. Directly draw attention to how careful labeling of provided figures and organization of calculations when problems have multiple steps helps to both solve problems and identify errors. For example, in problems where multiple calculations are completed using a calculator, it is easy to copy an incorrect number, such as writing 14,600 instead of 16,400 for the sum $$100^2+80^2$$.

## 8.3: Internal Dimensions (15 minutes)

### Activity

The purpose of this task is for students to use the Pythagorean Theorem to calculate which rectangular prism has the longer diagonal length. To complete the activity, students will need to picture or sketch the right triangles necessary to calculate the diagonal length.

Identify groups using well-organized strategies to calculate the diagonals. For example, for Prism K some groups may draw in the diagonal on the bottom face of the cube and label it with an unknown variable, such as $$s$$, to set up the calculation for the diagonal as $$\sqrt{6^2+s^2}$$. Students would then know that to finish this calculation, they would need to find the value of $$s$$, which is can be done using the edge lengths of the face of the prism $$s$$ is drawn on.

The idea that the Pythagorean Theorem might be applied several times in order to find a missing length is taken even further in the extension, where students can apply it over a dozen times, hopefully noticing some repeated reasoning along the way, in order to find the last length.

### Launch

Arrange students in groups of 2. Provide access to calculators. Ask students to read and consider the first problem and then give a signal when they have done so. If possible, use an actual rectangular prism, such as a small box, to help students understand what length the diagonal shows in the image. Once students understand what the first problem is asking, poll the class to see which prism they think has the longer diagonal and post the results of the poll for all to see. Select 2–3 students to share why they chose the prism they did. The conversation should illuminate a few key points: Prism K has one edge with length 6 units, which is longer than any of the edges of Prism L, but it also has a side of length 4 units, which is shorter than any of the edges of Prism L. Prism K also has a smaller volume than Prism L.

Give 1 minute of quiet think time for students to brainstorm how they will calculate the lengths of each diagonal. Ask partners to discuss their strategies before starting their calculations. Follow with a whole-class discussion.

Representation: Internalize Comprehension. Provide access to 3-D models of rectangular prisms (for example, a tissue box) for students to view or manipulate. Use color coding to highlight and make connections between corresponding parts.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Conversing, Writing: MLR5 Co-Craft Questions. Before revealing the questions in this activity, display only the image of the rectangular prisms. Invite students to work with a partner to write mathematical questions that could be asked about the prisms. Select 1–2 groups to share their questions with the class. Listen for and amplify questions about the lengths of the diagonals. For example, “What is the length of the diagonal of Prism $$K$$?” and “Which rectangular prism has the longer diagonal length?” If no student asks these questions, ask students to adapt a question to align with the learning goals of this lesson. Then reveal and ask students to work on the actual questions of the task. This routine will help develop students’ meta-awareness of language as they generate questions about calculating the diagonal length of a rectangular prism.
Design Principle(s): Maximize meta-awareness

### Student Facing

Here are two rectangular prisms:

1. Which figure do you think has the longer diagonal? Note that the figures are not drawn to scale.
2. Calculate the lengths of both diagonals. Which one is actually longer?

### Activity Synthesis

Ask groups to share how they calculated the diagonal of one of the rectangular prisms. Display the figures in the activity for all to see and label the them while groups share.

If time allows and no groups pointed out how the diagonal length of the rectangular prisms are the square root of the sum of the squares of the three edge lengths, use Prism K and the calculations students shared to do so. For example, to calculate the diagonal of Prism K, students would have to calculate $$\sqrt{6^2+\left(\sqrt{41}\right)^2}$$, but $$\sqrt{41}$$ is from $$\sqrt{5^2+4^2}$$ which means the diagonal length is really $$\sqrt{6^2+5^2+4^2}$$ since $$\sqrt{6^2+\left(\sqrt{41}\right)^2}=\sqrt{6^2+\left(\sqrt{5^2+4^2}\right)^2}=\sqrt{6^2+5^2+4^2}$$. So, for a rectangular prism with sides $$d$$, $$e$$, and $$f$$, the length of the diagonal of the prism is just $$\sqrt{d^2+e^2+f^2}$$.

## Lesson Synthesis

### Lesson Synthesis

Tell students that there are many situations in the world where we can use the Pythagorean Theorem to solve problems. Ask, “What situations that you can think of involve right triangles?” Give brief quiet think time, then invite students to share their ideas. For example, thick wires (called guy-wires) are used to keep telephone poles upright and their length depends on how high up the pole they attach and how far away from the pole they hook into the ground.

## 8.4: Cool-down - Jib Sail (5 minutes)

### Cool-Down

It’s assumed that the tower makes a right angle with the ground. Since this is a right triangle, the relationship between its sides is $$a^2+b^2=c^2$$, where $$c$$ represents the length of the hypotenuse and $$a$$ and $$b$$ represent the lengths of the other two sides. The hypotenuse is the side opposite the right angle. Making substitutions gives $$a^2+15^2=17^2$$. Solving this for $$a$$ gives $$a=8$$. So, the other end of the cable should connect to the ground 8 feet away from the bottom of the tower.