# Lesson 4: Building Cube Towers and Their Roots

Students will solve problems involving perfect cubes and cube roots of small perfect cubes.

In this lesson, you are a contractor building towers that are perfect cubes and during the building of the towers in the shape of cubes you will discover the cube roots of perfect cubes.

Discuss square roots. Explain to students that √9 is 3 because 3 x 3 or is 9. Students should understand that 3 is the square root of 9 because 3 multiplied by itself is 9.

Next discuss cube roots. Explain that ^{3}√8 is 2 because 2 x 2 x 2 or 2^{3} is 8.

**Lesson Objective:** Lesson is aligned to the Common Core State Standards for Mathematics – 8.EE.2 Expressions and Equations – Use square root and cube root symbols to represent solutions to equations of the form x^{2} = p and x^{3 }= 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.

**Materials Required**: cubes

## Lesson Procedure

In this lesson, you are a contractor building patios with cubes. The numbers of cubes used for each tower are perfect cubes. You discover the lengths of the sides of the cubes are the cube roots of the perfect cubes.

**Activity 1**

- With the cubes, build the smallest cube tower. The length of each of the sides of the tower should measure 1 unit.
- To find the volume of the tower, count the number of small cubes used to build the tower.
- How many cubes did you need to build this tower?

- Answer: 1

**Activity 2**

- With the cubes, build the next smallest cube tower. The length of each of the sides of the tower should measure 2 units.
- To find the volume of the tower, count the number of small cubes used to build the tower.
- How many cubes did you need to build this tower?

- Answer: 8

**Activity 3**

- With the cubes, build the next smallest cube tower. The length of each of the sides of the tower should measure 3 units.
- To find the volume of the tower, count the number of small cubes used to build the tower.
- How many tiles did you need to build this tower?

- Answer: 27

Discuss with students that the volume of a tower that is in the shape of a perfect cube is found by multiplying the lengths of the sides by itself (*s* x *s* x *s*) or cubing the side (*s*^{3}). Discuss with students when writing the volume in exponential form (*s*^{3}), the base number is the length of the side and the exponent is 3.

Discuss with students that a cube root is the inverse operation of cubing a number. You may relate this to subtraction is the inverse operation of addition and division is the inverse operation of multiplication.

Have your class complete **this table.**