- In the mall there is a bubble gum machine. Ms Mataira's twins each want a bubblegum ball of the same colour. There are only blue and yellow balls in the machine and each ball costs 50c. How much money must Ms Mataira spend to be sure to get two bubble gum balls of the same colour?

- The next day Ms Mataira sees a different bubblegum machine. This one has three colours – blue, yellow and red. What is the most Ms Mataira might have to spend?

- Mr Smith passes the 3-colour machine with his three children each of whom wants the same colour ball. How much will he have to spend?

This problem requires the students to identify the possible outcomes of an event (buying bubblegum balls from a machine.)

Students should be encouraged to try different examples, to organise information, look for patterns, express patterns using symbols and explain patterns.

Students are challenged to find a generalized rule that will tell them the maximum number of balls they need to buy for n children and n colours.

Bag of bubblegum balls to introduce the problem.

### Problem

- In the mall there is a bubble gum machine. Ms Mataira's twins each want a bubblegum ball of the same colour. There are only blue and yellow balls in the machine and each ball costs 50c. How much money must Ms Mataira spend to be sure to get two bubble gum balls of the same colour?
- The next day Ms Mataira sees a different bubblegum machine. This one has three colours – blue, yellow and red. What is the most Ms Mataira might have to spend?
- Mr Smith passes the 3-colour machine with his three children each of whom wants the same colour ball. How much will he have to spend?

### Teaching sequence

- Use the bag of bubblegum balls to introduce problem A.
- Have students work on the problem.
- Share solutions.
- Pose parts B and C and ask:
*What is the most that Mr Smith could spend without getting three the same?**(2 of each colour).*

Share your thinking about this problem.

Could you make up your own problem?

Do you think that there are relationships between each of these questions? How could you find out? - Encourage the students 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? - Pose and share solutions to the Extension problem.

#### Extension to the problem

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

#### Other contexts for the problem

Drawing pairs of socks from a number of single socks

### Solution

A. 3 ($1.50) After three draws Ms Mataira will have at least two of one of the colours.

B. 4 ($2.00)

C. 7 ($3.50)

If the students 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.

Number of colours | Number of bubblegum balls needed |

2 | 3 |

3 | 4 |

4 | 5 |

5 | 6 |

c | c+1 |

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.

#### Solution to the extension:

For **c** colours of gum and **n** children the formula is:

c(n-1) + 1