Problem
There are six A Grade netball teams in the local inter-school netball tournament. During the tournament, they all need to play each other once. How many matches will be played once they have done this?

What is this problem about?
This problem requires students to make lists, be systematic, maintain a running addition total, eliminate possibilities and explain their results. (Some students may also identify the sequential pattern and devise a rule.)
The rule that comes from this problem (see Solution) is an example of a combination problem. Combination problems are one of several classes of counting problems that become more and more important in statistics as well as in certain branches of mathematics. At senior high school level, combinations are often needed in probability questions.

Achievement Objectives
Number, Level 2
- make sensible estimates and check the reasonableness of answers Mathematical Processes
- effectively plan mathematical exploration
- devise and use problem-solving strategies to explore situations mathematically
- record, in an organised way, and talk about the results of mathematical exploration

Resources required
- Paper for making lists and matching teams or drawing
- 6 students could represent the 6 teams and through systematic matching, eliminate all the possible combinations
- Copymaster of the problem (English)

Specific learning outcomes
The children will be able to:
- solve a problem by being systematic
- compare an estimate with the result

Teaching Sequence

1. This problem would mesh well with a real sports’ tournament event that the school is involved in. The teams could be netball, rugby, soccer, cricket or whatever. In reading the problem to the class the teacher could highlight the actual event and ask why it would be helpful to know how many games will be played e. g., this helps to determine how many courts are required, how tired the players might get, how long the tournament will last etc.
2. Ask students to estimate how many games will be played if each team plays each other once. List the estimations.
3. Ask for 6 volunteers to come and stand at the front to represent the 6 netball teams. They can be numbered 1, 2, 3, 4, 5, 6 or given school names of sports’ teams. Ask the students to work out how many games would team 1 (student 1) play. Students may need guidance in matching team 1 (student 1) with each of the other teams. To show that team 1 has played another team she or he could shake hands with each student representing the other teams and they could sit on the floor once they have been ‘played’. When student 1 (team 1) has skaken hands with each other student ask the class how many ‘games’ have been ‘played’. Then ask student 1 (team 1) to join the rest of the class. Ask the remaining volunteers to stand and repeat the process with student 2 (team 2) until the class seem to follow what is occurring.
4. Ask the class to continue the investigation in their own way and discuss possible strategies they might use e.g., either continuing to act it out in a simulated fashion (one student representing one team), drawing, making lists etc
5. Ask students to share with the class their findings showing the process they used to ensure that every team had been played once. Discuss the different strategies used. Compare with initial estimation and evaluate the reason for the differences between the estimations and the result.

Extension
If another team joined the tournament, how many matches would be played?
What is the pattern for the number of games and can you devise a rule?

Other Contexts for the Problem
Any sports’ team tournament.
Hand shaking at a party

Solution
With 6 teams, 15 games would be played in total. Students could make lists, act it out, draw or systematically match teams in columns. For clarity of working it may help students to progressively list all the combinations starting with team 1 e.g.,