Computing Volume Progression 1

Task

1. Amy wants to build a cube with $3$ $cm$ sides using $1$ $cm$ cubes. How many cubes does she need?

   

2. How many $1$ $cm$ cubes would she need to build a cube with $6$ $cm$ sides?

IM Commentary

This is the first in a series of four tasks that gradually build in complexity. The purpose of this series of tasks is to build in a natural way from accessible, concrete problems involving volume to a more abstract understanding of volume. The purpose of this first task is to see the relationship between the side-lengths of a cube and its volume.

Solution

1. Each single layer of cubes contains $3\times 3 = 9$ cubes. There are $3$ layers, so Amy needs $3 \times 9 = 27$ one cm cubes in all. Or, a shorter way: Amy needs $3 \times 3 \times 3 = 27$ one cm cubes.

2. Now each single layer of cubes contains $6\times 6 = 36$ cubes. There are $6$ layers, so Amy needs $6 \times 36 = 216$ one cm cubes in all. As before, we can simply compute $6\times6\times6$ to get the number of one cm cubes. Amy needs $216$ one cm cubes.