Cube Numbers

Task 208 ... Years 6 - 10

Summary

Starting with a Size 2 cube students begin exploring how one size of cube can be made from a previous one by slicing away layers. Size 2 is straight forward. Size 3 becomes more interesting but the materials supplied allow students to touch and see the problem. Sizes 4 and 5 are more challenging but now sufficient materials are not supplied, so spatial perception skills become the basis of a mind experiment. The big challenge encourages students to take their visualisation further to explain how to calculate the difference between two cube numbers that are one size apart.

This task is related to Task 160, Painted Cubes. The From The Classroom section of Painted Cubes explores the difference between any two cube numbers, no matter how far apart their sizes.

 

Materials

Content

  • 2D representation of 3D objects
  • algebra, concept of a variable / function
  • algebra, cubic
  • algebra, factorisation
  • algebra, generalisation in words & symbols
  • equations, creating
  • graphical representation
  • numbers, cube
  • patterns, number
  • recording mathematics
  • spatial perception, 2D or 3D
Cube Numbers

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card. 		   

Cube numbers come from making cubes; just as square numbers come from making squares and triangle numbers come from making triangles. The smallest whole number cube is Size 1 (Colour A in this task). Students are asked to make Size 2 using Colours A and B. No other instruction is given because it doesn't matter how it is done, Colour A will be in a corner and the Size 2 will always be able to be transformed to match this diagram.

The next sequence shows how the Size 1 cube can be found using these steps:

  • Slice off the top layer, which counts 22 blocks.
  • Slice off the back wall, which counts 1 x 2 blocks.
  • Slice off the side wall (single cube), which counts 1 block to reveal the Size 1 cube.

                 

We started with a Size 2 cube and finished with a Size 1. What was sliced away must be the difference between them.

Another way to look at the diagrams is that if you start with the Size 2 and pull out the black Size 1, then the remain cubes represent the larger one subtract the smaller one, and their number can be counted by the slicing process.

Note: There are other ways of starting the slicing, for example, from the back wall, but they are equivalent to the above.

The 3 --> 2 case parallels the 2 --> 1 case. Begin with:

Cut the slices in the same way as above:

  • Slice off the top layer, which counts 32 blocks.
  • Slice off the back wall, which counts 2 x 3 blocks.
  • Slice off the side wall (square), which counts 22 blocks to reveal the Size 2 cube.
                 

  Finding more blocks, or carrying out a mind experiment leads to the generalisation that:
If two cubes are one size apart, the difference between them is:
  • a square measured by the size of the bigger cube plus
  • a rectangle measured by the product of the two size numbers plus
  • a square measured by the size of the smaller cube.
So if the two cubes are Size a and Size b and a - b = 1, then:
a3 - b3 = a2 + ab + b2

Extensions

  1. If we make ordered pairs like this (cube size, number of blocks) and plot the pairs, what do you expect to see?
  2. What happens if the cube sizes are more than one apart?
    This question leads to exploring the more general case of the difference between any two cubes and eventually makes sense of the rule:
    a3 - b3 = (a - b) (a2 + ab + b2)

    This exploration is what Shpetim carried out and recorded as a consequence of his exploration of Task 160, Painted Cubes. His journal is recorded in the From The Classroom section.

  3. What happens if we explore the sum of two cubes?
  4. What happens if we let a measure the side of the smaller cube and b measure the extra length in each direction to make the bigger cube? Then the bigger cube would be (a + b)3.
    If we take a cube apart using these measurements, what other expression can we make that is equal to (a + b)3?
    Maths With Attitude, Pattern & Algebra, Years 9 & 10 includes an investigation guide that explores this challenge.

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works. 		   

Twenty-seven cubes, which is what is needed to start this investigation, is almost the same as the number of students in most classrooms. Therefore it could be that this lesson follows one on Task 31, Cube Nets, which leads into choosing one of the net shapes to make a cube from cardboard. If you arrange for cardboard in the same colours and quantities as listed above in Materials, your students will make a class set which can be used as a floorboard model* to introduce this activity. It would also be best to have enough small cubes for pairs to have 27 in the colours above, but if you don't have that equipment, this DIY approach can still be used.

The structure of the task card and the iceberg notes above will guide the lesson.

[Floorboard Model*
We demonstrate lots of things on blackboards, whiteboards and interactive whiteboards but they are all two dimensional. The floor boards are often a great space to share mathematics as a community in 3D space.]

At this stage, Cube Numbers does not have a matching lesson on Maths300.

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner. 		   

The Cube Numbers task is an integral part of:

  • MWA Pattern & Algebra Years 9 & 10

Green Line
Follow this link to Task Centre Home page.