## Netball tournament

This problem solving activity has a number (addition and subtraction) focus.

## About this resource

Specific learning outcomes:

To solve a problem by being systematic.

# Netball tournament

## Achievement objectives

NA2-1: Use simple additive strategies with whole numbers and fractions.

## Student activity

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?

## Description of mathematics

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 have become more and more important in statistics as well as in certain branches of mathematics. At the senior high school level, combinations are often needed in probability questions.

## Required materials

- six students (to represent the six teams and through systematic matching, eliminate all the possible combinations)

See **Materials that come with this resource** to download:

*Te whakataetae poitarawhiti CM te reo Māori*(.pdf)*Netball tournament CM English*(.pdf)

## Activity

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?

**1.**

Invite students to talk about sports they play after school or at the weekend and where they play them.

**2.**

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.

**3.**

Ask students to estimate how many games will be played if each team plays each other once. List the estimations.

**4.**

Ask 6 volunteers to stand and represent the 6 netball teams. They can be numbered 1, 2, 3, 4, 5, 6, or given the school names of the 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 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 sees what is occurring. Remind students to only record each match between two teams once (e.g., "team 1 plays team 6" would not need to be recorded if team "6 plays team 1" had already been recorded).

**5.**

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.

**6.**

Have students share with the class their findings and show the process they used to ensure that every team had played once. Discuss the different strategies used. Compare the 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 find a rule?

With 6 teams, 15 games would be played in total. Students could make lists, act them out, draw, or systematically match teams in columns. For clarity of work, 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 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.

### Solution to the extension

If another team joined the tournament, then an additional 6 matches would be played. The total number of matches would be 21.

The pattern of matches, when M = number of matches played and T = team number, can be recorded in the following way:

T | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

M | 6 | 5 | 4 | 3 | 2 | 1 | 0 |

Your students might notice that the M number is always 7 less than the T number (e.g., M = T - 7), or that the T numbers increase while the M numbers decrease. Displaying the numbers in a table should make it easier to notice these patterns.

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