# Surface Area with Tricubes

### Task 193 ... Years 4 - 10

#### Summary

A task designed to clarify the meanings of area, surface area and volume while at the same time exploring the spatial skill of representing 3D objects using isometric paper. The basic shape is one Tricube. Its volume is constant at three unit cubes. Its surface area is constant at 14 unit squares. But its base area will depend on how it is placed on the table. One possibility produces a remarkable base area. All challenges are solved using exactly four tricubes.

#### Content

• 2D representation of 3D objects
• measurement, area
• measurement, volume
• mental arithmetic
• position in space, 2D or 3D
• recording mathematics
• scale drawing
• 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.

If the students can interpret the first diagram on the card. They are on the way. To test their comprehension, and prepare them for the skill of isometric drawing, you might ask them to stand one Tricube on the table in a different way and draw it on isometric paper.

Once the doughnut in Question 1 is created there is an opportunity to ask the mathematician's question Can I check it another way?.

• What is the base area?...
• How do you know?...
• Can you check it another way?...
There are at least two ways to 'know' the base area:
• The base area is the same as the top area so I counted the top.
• When it's this way, the base of one Tricube is 3, so I multiplied by 4.
And the same mathematician's questions can be asked about the surface area.
• What is the surface area?...
• How do you know?...
• Can you check it another way?...
There are also at least two ways to 'know' the base area:
• Double the base area, then multiply one side area by 4, then do the same with the side areas in the whole, then add it all up.
• Four Tricubes have a total surface area of 56. Where the joins are two squars of surface are hidden, so take off 8.

Modelling these question often enough for the students to ask them for themselves leads to growing confidence through 'knowing' you are correct. (And much less of Is that right Miss?)

 Obviously the answers to all the challenges will be objects, which we have chosen not to draw or photograph ourselves. However we would be very happy to publish one photo or isometric drawing for each challenge. It's up to you to send them to us. The example here is from a student in Year 8 at Wade High School, Griffith. His structure represents [44, 0]. Encourage students to draw some of the solutions on isometric paper.

 If you have enough Tricubes, these creations from Luke and Beth, which they did at home, suggest further mathematical questions. You've gotta love the Tricube Christmas Tree.

#### 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.

This is a great task to turn into a whole class investigation, but you must have a class set of Tricubes. Basically the lesson follows the structure of the card, although you might explore 2-Tricube objects before tackling the Challenges. Doing so provides extended opportunity for discussion and development of concepts, developing awareness of counting strategies, refinement of isometric drawing skills and activation of the spatial perception required to tackle the Challenges.

Use the lesson to develop the 'check it another way' approach above, isometric drawing skills and measurement concepts. Then in the next lesson students will tackle the 'surface area' chapter exercises with vigour and comment on how easy they are.

These work samples are from Year 8, Wade High School, Griffith. Addtional ideas are:

• Ask the mathematician's question How many 2-Tricube objects are there?.
• Ask each pair to produce a different 4-Tricube object and work out its base and surface areas. Collect these measurements and present them in a future lesson as new challenges. Ownership is likely to generate further interest in the task, so this can be a useful revision lesson.
• Let's work together to make some letters of the alphabet. (See Luke above.) What are the base and surface areas of each?
• What's the largest Tricube Christmas Tree we can build with all the Tricubes in the room?
For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 165, Surface Area with Tricubes.

#### 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 Surface Area with Tricubes task is an integral part of:

• MWA Chance & Measurement Years 5 & 6
• MWA Chance & Measurement Years 7 & 8

The Surface Area with Tricubes lesson is not in any Maths With Attitude kit but it can be used to enhance either of the kits which include the task.