Last week, mathematicians in Toroa started on a geometrical addition challenge:

Imagine you have an unlimited supply of interlocking cubes (all the same size) in different colours. Now imagine starting with one yellow cube. This is covered all over with a single layer of red cubes:

This is then covered with a layer of blue cubes.

How many red cubes have you used?

How many blue cubes have you used?

Find an easy way of working this out.

Now imagine adding a layer of green cubes.

How many green cubes are needed?

Have you found a quick method for working out the amounts that you would need for each layer?

Many teams used blocks to help understand the problem.

Diagrams were another effective strategy.

Being able to see the 'cube inside the cube' helped to focus on the outer layer of blocks.

Colour-coding to make our thinking clear was also helpful.

The problem became more difficult as we added more and more layers to our cube.

Several teams visualised the cube as being like a sandwich, with a whole top and bottom layer, plus hollow layers in the middle.

We explained our strategies to one another, convincing others that we had the correct answer.

By the end of the session, we all agreed that the blue layer of the cube would contain 98 blocks. Many of us started thinking about the next layers as well.

We used our past knowledge of geometry to illustrate our thinking on isometric dot paper. Wade wrote a particularly clear explanation of her mathematical thinking.

Mathematicians may choose their challenge for today:

Challenge A

Create a poster which clearly explains how to solve the cube problem. (How many cubes in the blue layer?) An effective poster will convince others that your solution is correct. It will include diagrams and thoughtfully written, step-by-step instructions.

Challenge B - Extra for experts!

Create a poster which clearly explains

*the rule*you would use to find how many blocks in the outer layer of

*any cube*(for example, how many blocks in the outer layer of a cube which is 35 blocks wide?).

Stop and think. Make a plan. You may use one or two pages of A3 paper. Your poster must be easy to read from a distance. Do a quick draft and use the entire page.

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