Cubic Capacity

In this activity, the students explore cubic capacity using powers of 3. They will need to model cubic shapes and also (for question 3) visualise hollow cubic shapes, with the cubes that do not form part of the faces removed from the inside. Cubic capacity is significant because it forms the basic structure for measuring the volume of a shape as well as being a special type of number that has a form that is a consistent shape.
2³ is read as “two cubed” or “two to the power of three”, meaning the same as 2 x 2 x 2 in extended notation.
In question 1, you could use group discussion as an alternative to pairs. If necessary, prompt the students to recall how they work out the area of a square and to think out how the number of cubes in the whole shape i can be generated from their workings for the area of a square. You could use an imprinted decimetre cube as a model to demonstrate or verify answers or to review the dimensions of the length, width, and
height. Multilink cubes can be used to represent smaller cubes.
Rather than the students counting individual cubes, encourage them to think of each model as one layer times the number of layers. For example, 4 x 2 = 8 is 2 x 2 x 2 (2³).
Question 2 introduces a more formal recording process. This can be summarised in a brief table with the exponents fully expanded.

As an extension, you could challenge the students to find the number that, when cubed, has a value of 343 or to find other similar numbers. For example, “What is the largest number that, when cubed, could be shown on your calculator display?”
In question 3, the students may first try building a model to solve the simplest case or they could visualise and then check their thinking by building a model. For the bigger sized cubes, organising data systematically in a table may enable the students to predict the dimensions of the models and to see a pattern without
building them. They could observe that the hollow space from any solid cube will be 2³ less than the original cube. For example:
A solid shape of 5³ = 125 cubes. To find the hollow space in a 5³ cube:
(5 – 2)³ = 3³
= 27
Cubes in walls: 125 – 27 = 98 cubes

Some students may need to visualise the overlap between the adjacent faces of each cube pictorially before they calculate the inner space.

Answers to Activity

1. a. i. 8 (because 2 x 2 x 2 = 23, which is 8)
ii. 27 (because 3 x 3 x 3 = 33, which is 27)
iii. 64 (because 4 x 4 x 4 = 43, which is 64)
iv. 125 (because 5 x 5 x 5 = 53, which is 125)
b. Answers will vary.
2. a. 53 = 125
b. 33 = 27
c. 103 = 1 000
d. 43 = 64
e. 83 = 512
3. ii. 27 – 1 = 26 cubes
iii. 64 – 8 = 56 cubes
iv. 125 – 27 = 98 cubes
4. 2³ gives the volume of a cube that has edges of two cubes long.
2³ = 2 x 2 x 2
= 8 cubic units