Problem
Ms Mataira comes across a bubble gum machine when she is out shopping with her twins. Of course, the twins each want a bubblegum. What’s more they want one of the same colour. Ms Mataira can see that there are only blue and yellow bubblegum balls in the machine. The bubblegum balls cost 50c each. How much money will Ms Mataira have to spend to make sure that she gets two bubble gum balls of the same colour?

What is the problem about?
This problem requires the child to identify the possible outcomes of an event (in this case selecting bubblegum balls from a machine). It also involves a slight twist as the child realises that after three draws Ms Mataira will have at least two of one of the colours. Questions B and C encourage the children to look for patterns in their answers to the questions. The important issue is not the specific formula involved, but the process of trying different examples, organising information, looking for patterns, expressing patterns symbolically and explaining patterns.

The goal of the extension to the problem is to to find a formula so that, if someone tells you the number of children and the number of colours, the formula will tell them the maximum number of balls they need to buy.

Achievement Objectives
Statistics (Level 4)
- find all possible outcomes for a sequence of events, using tree diagrams

Mathematical Processes
- interpret information and results in context (logic, be systematic)

Resources
bag of bubblegum balls to introduce the problem.
Blackline master of the problem (English)
Blackline master of the problem (Maaori)

Specific learning outcomes
The children will be able to:
- identify the possible outcomes of an event
- follow a logical argument

1. Use the bag of bubblegum balls to introduce problem A.
2. Let the children work on the problem in pairs.
3. Share solutions to A.
4. Let the children work on parts B and C.
   What is the most that Mr Smith could spend without getting three the same? (2 of each colour )
   Share your thinking about this problem with me?
   Could you make up your own problem?
   Do you think that there are relationships between each of these questions? How could you find out?
5. Encourage the children to create and solve examples of their own. As they do this ask them to look for a way to organise the information so that they can look for patterns in the answers.
   Do you have any general ideas about what is happening in this problem?
   How did you arrive at this idea?
6. Share solutions to the problems and the general ideas or conjectures that the children have written.

Extension to the problem
If someone tells you the number of bubblegum colours and the number of children, find a rule that tells you the maximum number of bubblegum balls you need to buy if all the children are to get the same colour.

Other contexts for the problem
Drawing pairs of socks from a number of single socks. (You have 4 pairs of socks in your drawer that are unmatched. How many do you need to draw out to make sure that you have a pair?)

Solution to the problem
A 3 ($1.50)
B 4 ($2.00)
C 7 ($3.50)
If the children form a table for the twin problem they will probably see that the number of balls needed is one more than the number of bubblegum colours.

In the case of the 3 children it is helpful to think about "what is the most that Mr Smith could spend without getting 3 the same?". You can then see that the worst-case scenario is getting exactly 2 of each colour. Once this has happened the next gum ball must give him 3 of one of the colours.

Therefore for say, 5 children and 3 colours the worst-case scenario involves 4 of each of the 3 colours (4 x 3) + 1 or (5-1)x 3 + 1.