# Pattern Cube

### Task 209 ... Years 2 - 10

#### Summary

An easy to state 3D spatial problem, that is not necessarily easy to complete without applying strategies such as break the problem into smaller parts. There are several solutions but essentially only one process to find them and for generally, if made randomly, the solutions are transformations of each other.

#### Materials

• 27 wooden cubes, 9 each of red, yellow & black

#### Content

• 2D representation of 3D objects
• measurement, volume
• numbers, cube
• patterns, colour
• patterns, visual
• position in space, 2D or 3D
• recording mathematics
• spatial perception, 2D or 3D #### Iceberg

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

Pattern Cube grows out of the task Latin Squares. A Latin Square is a square made from the same number of colours as the number of units in the base and has no colour the same in any row or column. For example: 3x3 Latin Square

To create the Pattern Cube, begin with a Latin Square in the base. Notice that each row is formed from the previous one by a cyclic movement of colours. The bottom row (front left) is:

Red, Black, Yellow
The next row is formed as if we have taken the Red from the left end, put it on right end and 'pushed' the other two blocks 'up' to make:
Black, Yellow, Red
Repeating this process with the Black from the left end makes the top row:
Yellow, Red, Black
To make the cube, build a Latin Square upward from each row. The three Latin Square 'walls' you make will become a pattern cube. One solution is: 3x3x3 Pattern Cube

There are variations on this solution depending on how the cube is rotated, however, these are essentially the same solution. To highlight this, students could be asked to sit opposite each other with a completed cube between them. They then draw and shade the cube on the isometric paper. They are looking at the same solution, but their diagrams will appear to be different.

So, are there other unique solutions for Pattern Cube? Perhaps the students could begin to answer this, as a mathematician might, by breaking the problem into smaller parts and asking if there are other unique Latin Squares that could be used in the base.

• Investigate the colours that make diagonals on each face.
• Investigate the colour of the hidden cube in the middle of the 3x3 cube.
• Start with a 4x4 Latin Square and try to make a 4x4 pattern cube from it.

#### 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.

To convert this task to a whole class investigation you only need plenty of blocks or linking cubes. The iceberg above suggests the direction of the investigation, which is not only to create a Pattern Cube, but to investigate the number of unique pattern cubes that can be made.

At this stage, Pattern Cube 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 Pattern Cube task is an integral part of:

• MWA Space & Logic Years 5 & 6
• MWA Space & Logic Years 9 & 10
This task is also included in the Task Centre Kit for Aboriginal Students under the title Aboriginal Cube. 