Curriculum  /  Math  /  8th Grade  /  Unit 7: Pythagorean Theorem and Volume  /  Lesson 13

Pythagorean Theorem and Volume

Lesson 13

Math

Unit 7

8th Grade

Lesson 13 of 16

Objective

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

Common Core Standards

Core Standards

  • 8.EE.A.2 — Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
  • 8.NS.A.2 — Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

Foundational Standards

  • 6.EE.B.5
  • 7.NS.A.3

Criteria for Success

  1. Understand the cube root of a number, $$p$$, is written as $$\sqrt[3]{p}$$ and represents the number that when cubed is equal to $$p$$.
  2. Understand the side length of a cube with volume $$p$$ is equal to $$\sqrt[3]{p}$$.
  3. Evaluate cube roots of perfect cubes.
  4. Solve equations in the form $$x^2=p$$ and $$x^3=p$$.
  5. Understand that there is no solution to the equation $$x^2=-p$$ because there is no real number that when squared is a negative value. 

Tips for Teachers

  • This lesson is a parallel lesson to Lesson 1, where students investigated and learned about square roots. This lesson is placed at this point in the unit so it immediately precedes students' study of volume. 
  • Similar to Lesson 1, as students are introduced to cube roots, they must make sense of the value of a cube root, as well as its relationship and connection to cubic volume (MP.2). 
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Anchor Problems

25-30 minutes

Problem 1

Two cubes are shown below with their volume in cubic units. 

a.   What is the side length of each cube?

b.   Describe the relationship between the side length of a cube and its volume.

c.   Another cube has a volume of $${{{64}}}$$ cubic units. Write an equation to show the relationship between the side length, $$s$$, of this cube and the volume.

d.   This equation can be solved by asking, “What number, when cubed, equals $${{{64}}}$$?” Another way of saying this is, “What is the cube root of $${{{64}}}$$?” Use this information to write two ways to represent the solution to the equation.

Guiding Questions

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Student Response

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Problem 2

a.   Determine if there is a solution to each equation below. If yes, then give the exact solution. If no, then explain why there is no solution. 

  1. $${x^3=-27}$$
  2. $${x^2=-9}$$

b.   Evaluate the square and cube roots below, if possible. If not possible, explain why not.

  1.  $${-\sqrt{64}}$$
  2. $${\sqrt{-64}}$$
  3. $${-\sqrt[3]{64}}$$
  4. $${\sqrt[3]{-64}}$$

Guiding Questions

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Student Response

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Problem 3

Compare each pair of values with $${<}$$, $${>}$$, or $$=$$. 

a.   $$\sqrt{200}$$ and $$\sqrt[3]{200}$$

b.   $$\sqrt{64}$$ and $$\sqrt[3]{125}$$

c.   $$\sqrt{16}$$ and $$\sqrt[3]{64}$$

d.   $$\sqrt{8}$$ and $$\sqrt[3]{50}$$

Guiding Questions

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Student Response

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Problem Set

15-20 minutes

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Target Task

5-10 minutes

Problem 1

Solve the equations. Give an exact answer.

a.   $${x^2=20}$$

b.   $${x^3=1000}$$

c.   $${x^3=81}$$

Student Response

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Problem 2

What is a solution to $${x^3=-1}$$? Select all that apply.

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Student Response

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Additional Practice

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.

  • Challenge: The volume of a cube is 125 cubic inches. What is the length of the inside diagonal line from the bottom right, back corner to the top left, front corner?

Next

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

Lesson 14
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Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Irrational Numbers and Square Roots

Topic B: Understanding and Applying the Pythagorean Theorem

Topic C: Volume and Cube Roots

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