# Growing Tricubes

### Task 234 ... Years 4 - 12

#### Summary

Tricubes are an incredibly useful teaching aid because they support learning in several content areas. This task focuses on pattern and algebra. The key to this area is discovering that 'four Tricubes make a Tricube'. Then the questions which power the investigation of the popular Task 166, Sphinx, can be brought into three dimensions. Growing Tricubes is also a partner of its two dimensional version, Task 238, Growing Trisquares. In all three of these tasks, the new, scaled up, shape or object provides a 'template' for constructing the next size, and the next, and the next... The visual pattern can also be represented as a number pattern and this leads to algebra and functions, graphing, equation work and scale factors.

#### Content

• 2D representation of 3D objects
• algebra, concept of a variable / function
• algebra, cubic
• algebra, generalisation in words & symbols
• algebra, linear
• equations, creating
• equations, substitution & solution
• graphical representation
• measurement, area
• measurement, length
• measurement, perimeter
• measurement, volume
• mental arithmetic
• numbers, cube
• numbers, square
• patterns, number
• patterns, visual
• powers of 2
• ratio & proportion
• scale drawing
• sequences & series
• 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.

This may be the students' first experience with Tricubes, so the early questions are to familiarise them with measurements related to them. The volume of a Tricube doesn't change. It is always 3 unit cubes. The surface area doesn't change either. It is always 14 unit squares. However the base area changes depending on how the object is placed on the table. If you want students to explore these measurements further then at some stage they should try Task 193, Surface Area with Tricubes. (It is possible to stand the Tricube so it has a surface area of zero. Try standing it to make an 'arch'.)

The challenge in this problem really begins with Questions 6 & 7. Putting four Tricubes together to make a 'tricube' is not necessarily straightforward for some students, especially after making their own shapes in Question 5, but they will get it eventually. However, what they make with four Tricubes is not really a Tricube. One dimension has not been doubled, so it is not proportional, or alternatively, not made from three cubes in the same sense as a Size 1 Tricube is.

The Size 2 Tricube actually looks like this: It shows that 8 Tricubes make a Tricube

But now we have a new Tricube. So what do we know? We know 8 Tricubes make a Tricube and that leads us to the next size, Size 4. The task doesn't supply enough Tricubes to make this - let's face it, 64 Tricubes wouldn't fit in the bag - but they do have enough to make one 'slice' and hence imagine the Size 4 object.

Imagining the next size can lead to significant brain-ache. It will be Size 8 and will have eight slices. The length and width (or, using the photo, length and height) will be doubled, so just one slice would need 64 Tricubes. Students might like to join a pages of square dot paper together and try to sketch out a bird's eye view of how they would fit together to make the slice. Or they might cut out Trisquares and build the same view in a similar way to the student work in Task 238, Growing Trisquares.

So clearly there is a visual pattern here. And wherever there is a visual pattern there will be a number pattern and vice versa. For each size of Tricube (placed as on the card) its base perimeter, base area, surface area, number of unit Tricubes, or volume in unit cubes can be counted. Choosing whichever of these is appropriate to you students will produce one of the columns in this table.

 Size Base Per. Base Area Surf. Area No. Tricubes Volume 1 8 3 14 1 3 2 16 12 56 8 24 4 32 48 224 64 192 8 ... ... ... ... ...

Younger students can find number patterns going down (although Surface Area is a bit tricky). Older students can find number pairs where the first number is S, for Size, and depending on which column it is paired with the result will be a linear, quadratic or cubic function. The table also begs the question: What happened to Size 3?.

It suggests that if Size 3 did exist, it would need 27 Tricubes to make it. So students again don't have enough Tricubes to make the object. Or do they... with a little imagination combined with some mathematics?

If the Size 3 can be made the 27 Tricubes might be equally distributed among the three cubic sections of it. That would mean nine Tricubes per section. Nine Tricubes have a volume of 27 unit cubes and 27 is a cube number. The students do have enough Tricubes to find out whether or not nine Tricubes can make a cube. And if they can, then it must be possible to make the Size 3 Tricube in at least one way. And there are other ways to make Size 3. For example:  Now there is reason to expand the table ... and for those with graphic calculators (or graph paper will do) to do a little interpolation. Extrapolation too, for the next questions are surely about Sizes 5, 6 & 7 which might exist between Sizes 4 and 8.

#### 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. There are 100 in the set so each student in a class of 25 can start with four. Packing your set in sandwich bags of four makes it easier to distribute and pack up. So, together the pair can make Size 2; Size 4 requires eight pairs to work together; and there will still be some Tricubes left for exploring further questions.

One of the learning features that makes this lesson such a success, is that every student gets to contribute their Tricubes to some part of the various solutions.

The card will guide you through the lesson. The 4 Tricubes make a Tricube ... hmm, no 8 Tricubes make a Tricube is obviously a critical step. It is also helpful to use this to generate the idea of building the next Size in slices. Apart from these points, choose your own adventure depending on whether you want spatial outcomes, measurement outcomes, pattern and algebra outcomes, or connected curriculum outcomes. Perhaps you can visit the task several times through the year.

These Year 11 students at Settlebeck High School, Cumbria, UK are very excited about their 'aha' moment of building the Size 4.  And these teacher trainees at Högskolan Malmö, Sweden appear very satisfied that they have completed a sequence. See Growing Tricubes in Cube Tube for videos showing the struggle and achievement of a similar class at this same university. Altogether they used 100 Tricubes, just the number in a class set. ... Hmm, wait a minute. Tricubes are built with cube numbers of unit Tricubes. So, adding up 4 cube numbers gave a square number. That's neat!

Hmm, I wonder if...?

At this stage Growing 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.

The Growing Tricubes task is an integral part of:

• MWA Chance & Measurement Years 9 & 10 