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.
There are six 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 altogether?
- Invite students to talk about sports they play after school or at the weekend and where they play these.
- Read the problem, talk about a tournament, 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.
- Ask students to estimate how many games will be played if each team plays each other once. List the estimations.
- Ask 6 volunteers stand and represent the 6 netball teams. They can be numbered 1, 2, 3, 4, 5, 6 or given school names of sports’ teams. Have the students work out how many games team 1 (student 1) plays. This could be 'shown' by 1 shaking hands with each student representing the other teams who then sit down once they have been ‘played’. When 1 has shaken hands with each other student, ask thow 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 see what is occurring.
- Have the class continue the investigation in their own way and discuss possible strategies they might use e.g., act it out, drawing, making lists etc.
- Have students share with the class their findings and show 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.
If another team joined the tournament, how many matches would be played?
What is the pattern for the number of games and can you find a rule?
Other Contexts for the Problem
Any sports’ team tournament.
Hand shaking at a party
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.,
1 plays 2, 1 plays 3, 1 plays 4, 1 plays 5, 1 plays 6
2 plays 3, 2 plays 4, 2 plays 5, 2 plays 6
3 plays 4, 3 plays 5, 3 plays 6
4 plays 5, 4 plays 6
5 plays 6
Another way to look at this is to see that the each team has to play five others. Since there are 6 teams, at first glance it looks as if there are
6 x 5 = 30 games. But we know that there are only 15 games, so where did we go wrong? What happened is that we counted each game twice. For instance when 1 played 2, we counted that game once as one of the five that 1 played and once as one of the games that 2 played. Since we counted each game twice the final answer should be 30 ÷2 = 15. You might be able to see now that for 10 teams there would be (10 x 9)/ 2 = 45 games. You might now see how to develop a formula for any number of teams.