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6-8 Accelerated 7 Unit 8 Lesson 12

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IM K-5 Math
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Lesson 12

12.1 Warm-up 12.2 Activity 12.3 Activity 12.4 Activity Student Lesson Summary Glossary

Unit 8, Lesson 12

Edge Lengths, Volumes, and Cube Roots

Accelerated 7

12.1

Warm-up

Let , , , , , and be positive numbers.

Given these equations, arrange , , , , , and from least to greatest. Explain your reasoning.

12.2

Activity

Your teacher will give you a set of cards. Each card has a number line with a plotted point, an equation, or a square or cube root value.

For each card with a letter and square or cube root value, match it with the location on a number line where the value exists, and the equation that the value makes true. Record your matches and be prepared to explain your reasoning.

12.3

Activity

The value of a cube root of a number lies between two integers. Which are those consecutive whole numbers for the following? Be prepared to explain your reasoning.

12.4

Activity

The numbers , , and are positive, and:

A number line that shows the integers from negative 3 to 9

Plot , , and on the number line. Be prepared to share your reasoning with the class.
Plot on the number line.

Student Lesson Summary

For a square, its side length is the square root of its area. For example, this square has an area of 16 square units and a side length of 4 units.

Both of these equations are true:

A square with a side length of 4 units on a square grid.

For a cube, the edge length is the cube root of its volume. For example, this cube has a volume of 64 cubic units and an edge length of 4 units:

Both of these equations are true:

A solid cube composed of 64 unit cubes. Each edge length is 4 unit cubes.

is pronounced “the cube root of 64.”

Like square roots, most cube roots of whole numbers are irrational. The only time the cube root of a number is a rational number is when the number we are taking the cube root of is a perfect cube. For example, 8 is a perfect cube, and .

We can approximate the values of the cube root of a number by observing the integers around it and remembering the relationship between cubes and cube roots. For example, is between 2 and 3 since and , and 20 is between 8 and 27. Similarly, since 100 is between and , we know is between 4 and 5.

The cube root of a number is the number whose cube is . It is also the edge length of a cube with a volume of . The cube root of is written as .

The cube root of 64 is written as . Its value is 4 because is 64.

is also the edge length of a cube that has a volume of 64.