# Soft Drink Crates

### Task 211 ... Years 2 - 10

#### Summary

The story shell is a worker packing cans into a standard 6 x 4 crate. The worker notices that when the crate has a given number of cans in it, they are arranged so that the rows and columns are all even numbers. Given 10 cans as a starting point, the initial investigation revolves around the question:
• How might the cans be arranged?
The investigation is continued using questions such as:
• What happens if there is a different number of cans?
• Are there any even numbers of cans for which the 'even both ways' arrangement can't be done?
This cameo has a From The Classroom section which shows a journal entry from a primary student in a small rural school and a photo story of a whole class investigation of the task in a city school in England.

#### Materials

• 25 'pretend' soft drink cans

#### Content

• spatial perception
• odd & even numbers
• problem solving strategies - especially 'break the problem into smaller parts'

#### Iceberg

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

There are many solutions to the first problem, as can be seen in the exciting From The Classroom section below. Some are reflections and rotations of others. There are also several ways to find the solution and the tendency of teachers and students alike is to use guess, check and improve as their first approach. However, this problem and all the other variations in the task will better resolve with the more ordered approach of breaking the problem into smaller parts.

1. Make the columns correct...

2. Keep cans in the same column but alter their row...
3. Until the rows are also correct...
Similar reasoning leads to a solution for 18 cans. Again, there will be variations.
This is a task that encourages high level thinking without a correspondingly high demand for mathematical skills. The emphasis is on the concept of proof and the application of strategies to achieve proof. That makes it accessible to all of our students and as a result we may find out things about their mathematical reasoning abilities that we would never discover through a diet of number or symbol driven text work.

The last part of the card encourages trying all even numbers of cans. Clearly, 2 cans won't work and 24 cans must work. There is one other number of cans that doesn't work. Encourage your students to find it. 6, 10 and 14 cans are tricky, but will be found using the strategy above. Also, the strategy of 'have I solved a similar problem' is sometimes useful. For example if a block of 4, in 2 rows of 2, is added to the top right corner of the solution to 10 cans above, we will have a solution for 14 cans.

#### 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 task works beautifully as a whole class investigation if you have Poly Plug. The yellow/blue plugs even look like cans. If you don't have this resource you will have to find something appropriate - counters will do - and prepare a grid. There are two ways to use the plugs. Either two red boards can be masked to show the crate:

and then the yellow/blue plugs are used as cans:
or, the plugs can be used standing on this printable 'crate' (screw caps from bottled water will also work). Students can also record solutions on this Recording Sheet.

The initial investigation is as above, but we can also ask:

• Are there any even numbers of cans that can be arranged as 'odd both ways'?
• Are there any odd numbers of cans that can be arranged as 'even both ways'?
• Are there any odd numbers of cans that can be arranged as 'odd both ways'?
(Poly Plug can easily be changed into several different crate shapes.)

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 36, Soft Drink Crates, which includes an Investigation Guide.

Visit Soft Drink Crates 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 Soft Drink Crates task is an integral part of:

• MWA Number & Computation Years 3 & 4
• MWA Space & Logic Years 9 & 10

The Soft Drink Crates lesson is an integral part of:

• MWA Space & Logic Years 9 & 10

## From The Classroom

#### Winkie Primary School South Australia

Nic Dale
This journal entry from Brodie and Tyson shows the quality of work that can be expected from primary students

... and that these two need to do a bit more exploring into which numbers of cans can and can't be placed in even rows and columns.

You can read here about the influence of our 6 day professional development course Engineering 'aha' Moments in Number on Nic's teaching practice.

#### St. Edmund's Junior School England

Matthew Reames
Year 5

Matthew's photos tell their own story.

Words aren't really necessary to describe the involvement and enthusiasm of the children
...or who owns the problem.