# Can Stack

### Task 27 ... Years 5 - 10

#### Summary

The task makes a connection with part-time work in which many young people become involved. Younger children of course are aware of piles of cans stacked in the aisles of supermarkets. Cans are always stacked in a visual pattern. Wherever there is a visual pattern, there is always a number pattern to describe it, albeit non necessarily a simple one. The task asks the students to interpret and continue the visual pattern on the card and then use the data from the stack to predict for ever larger stacks.

#### Materials

• Around 22 pretend cans

#### Content

• spatial and numeric patterns
• number skills such as addition and multiplication
• generalisation
• symbolic algebraic representations
• substitution and solution of equations
• problem solving strategies

#### Iceberg

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

The card doesn't reveal how Harry was stacking the cans in the first place. Perhaps, since they were apparently unstable it was like this:

This is a visual pattern and the number pattern involved to find the total of cans needed to make L layers is:
1 + 2 + 3 + 4 + ... + (L - 1) + L
The challenge of summing this series may be sufficient for some students.

Digging deeper, the 'boss' in the story on the card actually leaves the worker with an unclear instruction about how to stack the cans in a more stable way. Did she mean:

Build two rows on the bottom first Harry. They can be as long as you like. Then you can stack your way. That will make the stack more stable.
For example:

or did she mean:
Choose the size of the bottom layer first. The number of rows should be one less than the number of columns. Then stand every can above on four other cans. Except the top one. You just put that there to finish off.
For example:

Either interpretation leads to a significant investigation and appropriate questions for either are:
• How many cans in the next layer? ...and the next ...and the next?
• In each case, how many cans altogether are needed to build the stack?
• If I tell you any number of layers, can you tell me how many cans in the bottom layer?
• If I tell you any number of layers, can you tell me how many cans altogether?
Of course, instead of these can stack designs, students might like to design their own can stacking pattern and try to predict the total of cans needed for, say, 10 layers.

#### 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.

Students do like stacking things, so a lesson based on this task could harness that predisposition, but you will need lots of pretend cans. The task uses empty film canisters and you may be able collect sufficient of these. An alternative might be off-cuts circular dowel; perhaps 20/25mm diameter dowel cut into 15mm lengths. If you have Poly Plug, the yellow/blue plugs are can like (and silent when the stacks tumble), but it is a good idea when using these to insist that the blue is always the bottom of the can, otherwise, the stacks can become visually distracting.

At this stage, Can Stack does not have a matching lesson on Maths300.

Visit Can Stack on Poly Plug & Tasks.

#### 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 Can Stack task is an integral part of:

• MWA Pattern & Algebra Years 5 & 6
• MWA Pattern & Algebra Years 9 & 10