### Task 230 ... Years 2 - 7

#### Summary

Appealing to many children's experience of 'packing up neatly' the challenge is to find different ways of putting away the bears. The initial task is limited by having only six bears and three trays. The trays can take as many as six bears, but no tray has to be full. Simply and sweetly this task has children exploring sums of three numbers which add to six. But what happens if:
• We change the number of bears?
• We change the size of the trays?
• We change the number of the trays?

#### Materials

• Ten bears the same colour
• One board with 3 trays to fit 6 bears each

#### Content

• counting
• equations, creating
• equations, substitution & solution
• mental arithmetic
• patterns, number
• recording mathematics
• subitising

#### Iceberg

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

The examples shown for Question 1 are intended to answer questions about what is allowed before they are asked.

• The trays are 'forcing' the children to find sets of three numbers that sum to six.
• It doesn't matter where the bears are placed in the tray, it's only the number in any tray that matters.
• That number can be as high as ten, or as low as zero.
• Each time the six bears are packed a number expression is written to record the packing.

There are many answers to Questions 1 & 2. The double line on the card is a traditional task card signal to show your work so far to a teacher. It is important to check answers at this point to make sure the students are ready to tackle the challenges.

#### Challenges

1. The challenge How many different ways can you find to pack the bears? is open enough to offer success at different levels. Less experienced children can just keep on making until they can't make any more. More experienced children might start an ordered list.
• Let's have zero bears in one tray to start with. Now we have to find all the ways the six bears can be in the other two trays.

0 + 0 + 6 ... 0 + 1 + 5 ... 0 + 2 + 4 ... 0 + 3 + 3 ...
0 + 4 + 2 (?) ... 0 + 5 + 1 (?) ... 0 + 6 + 0 (?)

Immediately there is need for discussion of the word 'different'. For example do we think of 0 + 1 + 5 as the same as 0 + 5 + 1? The answer depends on whether we think the packing up should include 'order' or not. Possible arguments are:

1. 0 + 1 + 5 (for example) tells you top tray, middle tray and bottom tray (or left tray, middle tray, right tray) so it's meaning is different from 0 + 5 + 1.
2. Who cares which tray the bears finish up in as long as they are packed up? We only need to know that 0 + 1 + 5 means that one of the trays has 0, one has 1 and one has 5.
In a sense it doesn't matter which approach is taken, as long as the students agree. However in another sense it does matter because there will be many more arrangements (the first approach) than there are combinations (the second approach).

Once this discussion has been had children can continue the organised search - one bear in a tray to start, leaving five to share between the other two trays etc. Watch out for repeats if necessary.

2. The second challenge encourages repeating the problem to find triples which add to any other number up to 10. There are only 10 bears. However, because the trays are limited to six bears if the number chosen is greater that 6, there will be some theoretical answers which can't be made. For example, if you choose 7 bears with these trays, there cannot be a solution with 2 zeroes.

... But what happens if the tray size changes to always be the same as the number of bears?

3. The third challenge invites exploration of sets of four numbers that sum to six.

#### Extension

Referring to the original puzzle:
• 6 bears
• 3 trays
• no more than 6 bears in a tray
Prepare some working backwards questions. These might be pictorial...

or symbolic...

Find the number of missing bears

|______| + 2 + 1 = 6

#### 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.

Many schools have sets of 'counting bears' so all that is needed to run the investigation as a whole class experience is to design a sheet with trays the right size. If you don't have bears, then use blocks or other objects. If you have Poly Plug use 2 red boards to make the trays and pack up the bears by plugging yellow/blue plugs into the gaps. The other advantage of this material is the ease of changing the size and number of trays.

A lesson (for younger children) could begin by choosing two students to show how they 'play with my six pretend bears'. Then introduce the trays and the packing up. Finally show how to record the packing up. You might use poster paper and markers, or a record in the students' journals, or provide an erasable mini-whiteboard.

Then ask another pair of students if they would like to play with your bears. Repeat the process - which only takes one or two minutes.

• Is this the same packing up as the first team?
...No?
...So now we have found two ways to pack the bears into three trays. Today we are going to see how many different ways we can find to pack the bears.
As appropriate - which may be over several lessons - introduce the challenges and extensions above.

A variation on the development of this class lesson which uses 5 bears and 3 trays that will take up to 5 bears each can be found in the Poly Plug & Tasks link.

At this stage, Pack Up Your Bears does not have a matching lesson on Maths300. However Lesson 18, Addition Totals, involves finding all the pairs of numbers (2 trays) that sum to a given number and Lesson 36, Soft Drink Crates, involves packing soft drink cans into a crate using rules about odd and even numbers.

#### 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.

Pack Up Your Bears is not in any MWA kit. However it can be used to enrich the Number & Computation kit at Years 3/4.