# Tricubes

### Task 69 ... Years 2 - 10

#### Summary

How can such an apparently simple arrangement of 4 identical objects develop into such a challenge? We are even shown a step by step breakdown of where each piece is to be placed. This brilliantly designed puzzle (attributed to Geoff Giles) leads us into understanding that representing three dimensional objects in two dimensions by using isometric drawing must result in loss of some information.

Tricubes also appears on the Picture Puzzles Shape & Space A menu where the problem is presented using one screen, two learners, concrete materials and a challenge. You will find it as a challenge within the Tricube Building B Picture Puzzle on that menu.

#### Content

• 3D spatial perception
• isometric drawing
• problem solving

#### Iceberg

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

There is no advantage in supplying the answer to this task. You and your students can work it out. However, appropriate hints might be:

• The construction is viewed in the same orientation each time. Nothing is rotated from the first position.
• Shading has been used to draw attention to what is happening to the 'platform'.
It might take some discussion before students are convinced, but the drawings are correct. Isometric drawing only has to show edges of objects, not the 'cracks' in surfaces where two objects join to make a common plane.

The task illustrates that when three dimensions are 'filtered' to two dimensions, something must be lost. Isometric drawing is one type of 3D to 2D filter and it can confuse us in several ways:

1. There may be some part of an object unable to be shown because it is hidden.
2. Two points (or lines) may look coincident when they are not.
3. Objects apparently closer to the viewer may appear shorter than objects of the same height further back.
It is the third of these which is the root of the challenge in this task.

Note that the task does ask that each step is recorded. On the final drawing students could be asked to indicate, perhaps by colour, how the four pieces fit.

There are many objects which can be made from four Tricubes and these can provide further challenge. Ask students to create their own four drawings to challenge a classmate. If these are pasted onto card and laminated you can easily build a class set of such puzzles.

Note: This investigation has been included in Maths At Home. In this form it has fresh context and purpose and, in some cases, additional resources. Maths At Home activity plans encourage independent investigation through guided 'homework', or, for the teacher, can be an outline of a class investigation.
• Visit the Home Page for more Background.
• For this specific activity click the Learners link and on that page use Ctrl F (Cmd F on Mac) to search the task name.

#### 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.

If you want the whole class to work on this investigation simultaneously, you will need a class set of Tricubes. These can be made in art or woodwork class as a cross-faculty project, or by community groups such as a Mens' Shed (see Tasks, Kits & Resources). The Maths At Home reference above includes simple instructions for making Tricubes from square section timber. Plastic cubes which join in three dimensions - most schools have sets of these - can work but they are awkward because there is always one lug which gets in the way.

If you only have Tricubes in the this task, you can use the task in a work station, probably with other spatial tasks, and rotate students through the station over a period of time. Keep a class scrapbook at the work station in which each group records its new puzzle as suggested above.

At this stage, Tricubes 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.

Tricubes is not in any Maths With Attitude kit, however, it is included as a key component of the Mixed Media Unit titled Points of View: Representing 3D Objects in 2D.