Cube NumbersTask 208 ... Years 6  10SummaryStarting 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

IcebergA 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:
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:

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:So if the two cubes are Size a and Size b and a  b = 1, then: a^{3}  b^{3} = a^{2} + ab + b^{2} Extensions

Whole Class InvestigationTasks 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. 
Twentyseven 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* 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 handson learning kits available from Years 310 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:
