lesson 6 homework practice use the pythagorean theorem

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

Pythagorean Theorem and Volume

Lesson 6 of 16

Criteria for Success

Tips for teachers, anchor problems, problem set, target task, additional practice.

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

Common Core Standards

Core standards.

The core standards covered in this lesson

8.G.B.6 — Explain a proof of the Pythagorean Theorem and its converse.

The essential concepts students need to demonstrate or understand to achieve the lesson objective

• Understand the relationship between the legs and the hypotenuse of right triangles, named the Pythagorean Theorem :  $${a^2+b^2=c^2}$$ .
• Use the Pythagorean Theorem to verify the relationship between the legs and hypotenuse of right triangles.
• Understand that the hypotenuse of a right triangle is the longest side of the triangle located across from the right angle.

Suggestions for teachers to help them teach this lesson

In this lesson, students are introduced to the famous relationship that exists between the side lengths of right triangles. In the next lesson, students will investigate informal proofs of the Pythagorean Theorem, followed by lessons where students will apply the theorem in various problems.

Lesson Materials

• Graph Paper (1 sheet per student)

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Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Several right triangles are shown below (not drawn to scale). In a right triangle, the two side lengths that form the right angle are called legs, and the side length opposite the right angle is called the hypotenuse. 

Use the triangles to investigate the question: What relationship do you see between the measures of the legs and the measure of the hypotenuse?

Guiding Questions

Use the Pythagorean Theorem to show the relationship between the sides of the right triangles shown below.

A right triangle has side lengths $$6$$  units,  $$4$$ units, and  $${\sqrt{20}}$$ units. 

Which side length represents the hypotenuse?

A set of suggested resources or problem types that teachers can turn into a problem set

Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

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

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

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

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

Student Response

An example response to the Target Task at the level of detail expected of the students.

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

• Include problems where students use given side lengths or identify side lengths from a group for a right triangle and verify the relationship using the Pythagorean Theorem.
• Open Up Resources Grade 8 Unit 8 Practice Problems — Lesson 6 #2-3
• Open Middle Pythagorean Theorem 2 — Challenge
• Open Middle Pythagorean Theorem 1

Topic A: Irrational Numbers and Square Roots

Define, evaluate, and estimate square roots.

Understand that some numbers, including $${\sqrt{2}}$$ , are irrational. Approximate the value of irrational numbers.

8.NS.A.1 8.NS.A.2

Locate irrational values approximately on a number line. Compare values of irrational numbers.

Represent rational numbers as decimal expansions.

Represent decimal expansions as rational numbers in fraction form.

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Topic B: Understanding and Applying the Pythagorean Theorem

Understand a proof of the Pythagorean Theorem.

Use the converse of the Pythagorean Theorem to determine if a triangle is a right triangle.

Find missing side lengths involving right triangles and apply to area and perimeter problems.

Solve real-world and mathematical problems using the Pythagorean Theorem (Part I).

Solve real-world and mathematical problems using the Pythagorean Theorem (Part II).

Find the distance between points in the coordinate plane using the Pythagorean Theorem.

Topic C: Volume and Cube Roots

Define and evaluate cube roots. Solve equations in the form  $${x^2=p}$$  and $${x^3=p}$$ .

8.EE.A.2 8.NS.A.2

Solve real-world and mathematical problems involving the volume of cylinders and cones.

Solve real-world and mathematical problems involving the volume of spheres.

Solve real-world problems involving multiple three-dimensional shapes, in particular, cylinders, cones, and spheres.

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Unit 8: Pythagorean theorem and irrational numbers

Lesson 2: side lengths and areas.

• Square roots of perfect squares (Opens a modal)
• Square roots Get 5 of 7 questions to level up!
• Roots of decimals & fractions Get 3 of 4 questions to level up!

Lesson 3: Rational and irrational numbers

• Intro to rational & irrational numbers (Opens a modal)
• Classifying numbers: rational & irrational (Opens a modal)
• Classify numbers: rational & irrational Get 5 of 7 questions to level up!

Lesson 4: Square roots on the number line

• Approximating square roots to hundredths (Opens a modal)
• Comparing irrational numbers with a calculator Get 5 of 7 questions to level up!

Lesson 5: Reasoning about square roots

• Approximating square roots (Opens a modal)
• Approximating square roots walk through (Opens a modal)
• Approximating square roots Get 3 of 4 questions to level up!

Extra practice: Irrational numbers

• Classifying numbers (Opens a modal)
• Classifying numbers review (Opens a modal)
• Comparing irrational numbers Get 5 of 7 questions to level up!

Lesson 6: Finding side lengths of triangles

• Intro to the Pythagorean theorem (Opens a modal)
• Use area of squares to visualize Pythagorean theorem Get 3 of 4 questions to level up!

Lesson 7: A proof of the Pythagorean Theorem

• Bhaskara's proof of the Pythagorean theorem (Opens a modal)

Lesson 8: Finding unknown side lengths

• Pythagorean theorem example (Opens a modal)
• Pythagorean theorem intro problems (Opens a modal)
• Use Pythagorean theorem to find right triangle side lengths Get 5 of 7 questions to level up!

Lesson 9: The converse

• No videos or articles available in this lesson
• Right triangle side lengths Get 3 of 4 questions to level up!

Lesson 10: Applications of the Pythagorean Theorem

• Pythagorean theorem word problem: carpet (Opens a modal)
• Pythagorean theorem word problem: fishing boat (Opens a modal)
• Pythagorean theorem word problems Get 3 of 4 questions to level up!

Extra practice: Pythagorean theorem

• Pythagorean theorem proof using similarity (Opens a modal)
• Another Pythagorean theorem proof (Opens a modal)
• Pythagorean theorem in 3D Get 3 of 4 questions to level up!

Lesson 11: Finding distance in the coordinate plane

• Use Pythagorean theorem to find perimeter Get 3 of 4 questions to level up!
• Distance between two points Get 5 of 7 questions to level up!

Lesson 12: Edge lengths and volumes

• Dimensions of a cube from its volume (Opens a modal)
• Cube roots Get 5 of 7 questions to level up!

Extra practice: Square roots and cube roots

• Equations with square roots & cube roots Get 3 of 4 questions to level up!
• Equations with square roots: decimals & fractions Get 3 of 4 questions to level up!
• Square and cube challenge Get 3 of 4 questions to level up!

Lesson 14: Decimal representations of rational numbers

• Converting a fraction to a repeating decimal (Opens a modal)
• Writing fractions as repeating decimals review (Opens a modal)
• Writing fractions as repeating decimals Get 5 of 7 questions to level up!

Lesson 15: Infinite decimal expansions

• Converting repeating decimals to fractions (part 1 of 2) (Opens a modal)
• Converting repeating decimals to fractions (part 2 of 2) (Opens a modal)
• Writing repeating decimals as fractions review (Opens a modal)
• Converting repeating decimals to fractions Get 5 of 7 questions to level up!
• Converting multi-digit repeating decimals to fractions Get 3 of 4 questions to level up!

• → Resources
• → 8th Grade

The Pythagorean Theorem Lesson Plan

Get the lesson materials.

The Pythagorean Theorem Guided Notes with Doodles Legs Hypotenuse Right Triangle

Ever wondered how to teach the Pythagorean Theorem in an engaging way to your 8th grade students?

In this lesson plan, students will learn about the Pythagorean Theorem proofs, legs and hypotenuse, right triangles, and their real-life applications. Through artistic, interactive guided notes, check for understanding, a color by code activity, and a real-life application example, students will gain a comprehensive understanding of the Pythagorean Theorem.

The lesson culminates with an exploration of baseball diamond highlighting the importance of understanding the Pythagorean Theorem in real-world scenarios.

• Type : Lesson Plans
• Grade : 8th Grade
• Standard : CCSS 8.G.B.7

Learning Objectives

After this lesson, students will be able to:

Identify the hypotenuse and legs of a right triangle

Use the Pythagorean Theorem to find the length of a missing side in a right triangle

Apply the Pythagorean Theorem to solve real-life problems

Prerequisites

Before this lesson, students should be familiar with:

Basic understanding of triangles (including how to identify right triangles) and their properties, such as side lengths and angle measures

Basic algebra skills, including solving for unknown variables and simplifying equations

Familiarity with squaring and square roots

Colored pencils or markers

Pythagorean Theorem

Guided Notes

Key Vocabulary

Right triangle

Square Roots

Introduction

As a hook, ask students why the Pythagorean Theorem is important in real life, such as finding the dimensions of a baseball diamond. Use the guided notes to introduce the Pythagorean Theorem.

Walk through the key points of the topic of the guided notes to teach. Refer to the FAQs for a walk-through on this, as well as ideas on how to respond to common student questions.

For Page 1 of the guided notes, introduce the Pythagorean Theorem with the worked example "what do you notice."

For Page 2 of the guided notes, explain how to know which side is a, b, or c, and check for understanding. Based on student responses, reteach concepts that students need extra help with.

Have students practice using the Pythagorean Theorem with the Maze activity. Walk around to answer student questions.

Fast finishers can dive into the Doodle Math activity for extra practice. You can assign it as homework for the remainder of the class.

Real-Life Application

Bring the class back together, and introduce the concept of using the Pythagorean Theorem to determine the dimensions of a baseball diamond. Explain that the Pythagorean Theorem can be used to find the distance between any two points, even if they are not directly next to each other. In the case of a baseball diamond, the distance from home plate to first base, home plate to second base, and so on can be found by using the Pythagorean Theorem. Refer to the FAQ for more ideas on how to teach it!

Hands-on project

A fun, no-prep way for students to practice is my Pythagorean Spiral project . Students will use a square to draw successive triangles and create a spiral. They will then measure the spiral and solve for the length of the hypotenuse for every triangle in their Pythagorean spiral. At the end of the activity, students can turn their Pythagorean spiral into an art project.

Additional self-checking digital practice

If you’re looking for digital practice for the Pythagorean theorem, try my Pixel Art activities in Google Sheets. Every answer is automatically checked, and correct answers unlock parts of a mystery picture. It’s incredibly fun, and a powerful tool for differentiation.

There’s two version depending on the focus with your students:

Finding the hypotenuse and leg

. It’s available with fall , winter , Valentine’s Day , and St. Patrick’s Day themed images.

Challenge: Finding the distance between two points

. It’s available with Valentine’s Day and St. Patrick’s Day

themed images.

What is the Pythagorean theorem? Open

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In equation form, it can be written as a^2 + b^2 = c^2, where c is the length of the hypotenuse and a and b are the lengths of the other two sides.

How do you use the Pythagorean theorem to find the length of a missing side in a right triangle? Open

To use the Pythagorean theorem to find the length of a missing side in a right triangle, follow these steps:

Identify the two sides of the right triangle that are known.

Assign one of the sides as "a" and the other as "b."

Identify the hypotenuse of the right triangle

, which is the side opposite the right angle. Assign the hypotenuse as "c."

Plug the known values into the Pythagorean theorem

, which states that a^2 + b^2 = c^2. Solve for the unknown side by isolating it on one side of the equation and taking the square root of both sides.

For example, if the length of side a is 3 and the length of side b is 4, and you want to find the length of the hypotenuse (c), you would use the equation 3^2 + 4^2 = c^2. Simplifying, you get 9 + 16 = c^2, or 25 = c^2. Taking the square root of both sides gives you c = 5. Therefore, the length of the hypotenuse (c) is 5.

What is a hypotenuse? Open

A hypotenuse is the longest side of a right triangle and is opposite the right angle. It is the side that connects the two legs of the triangle.

What are the legs of a right triangle? Open

The legs of a right triangle are the two sides that form the right angle. They are usually labeled as 'a' and 'b'.

How do you identify a right triangle? Open

To identify a right triangle, you need to look for a triangle with one angle that measures 90 degrees, also known as a right angle. This means that one side of the triangle will be perpendicular to another side, forming the right angle.

What are some real-life applications of the Pythagorean theorem? Open

The Pythagorean theorem has a wide range of applications in real life. Some examples include:

Architecture:

Architects use the Pythagorean theorem to make sure that building corners are square, and to calculate the length of sloping roofs.

Navigation:

The Pythagorean theorem is used in navigation to determine distances between two points. This is particularly useful in aviation and marine navigation.

In sports, the Pythagorean theorem can be used to calculate the distance between bases on a baseball diamond, or to determine the length of a shot in basketball.

Surveyors use the Pythagorean theorem to measure distances between points on the earth's surface, and to calculate elevations.

In the classroom, students can apply the Pythagorean theorem to real-world problems, such as finding the distance between two points on a map or the height of a building. One particularly fun application is to use the Pythagorean theorem to find the dimensions of a baseball diamond, which involves using the theorem to calculate the distance between bases.

How do you find the dimensions of a baseball diamond using the Pythagorean theorem? Open

To find the dimensions of a baseball diamond using the Pythagorean theorem, you need to calculate the distance between the bases. Here's how:

Identify the two bases you want to calculate the distance between.

Draw a right triangle connecting the two bases and home plate, with the two bases forming the legs and the distance between them forming the hypotenuse.

Measure the distance between the bases and record it as one leg of the triangle.

Measure the distance from the starting base to home plate and record it as the other leg of the triangle.

Use the Pythagorean theorem (a^2 + b^2 = c^2) to solve for the hypotenuse, which represents the distance between the two bases.

Want more ideas and freebies?

Get my free resource library with digital & print activities—plus tips over email.

Chapter 3, Lesson 6: Using the Pythagorean Theorem

• Extra Examples
• Group Activity Cards
• Personal Tutor
• Self-Check Quizzes

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Chapter 3, Lesson 6: Using the Pythagorean Theorem

• Extra Examples
• Group Activity Cards
• Personal Tutor
• Self-Check Quizzes

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