# Arsenal collection: Half Time

What could the half time scores have been in these Arsenal matches?

## Problem

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When Arsenal played Aston Villa in the 2010/11 Premier League, the final score was $4 - 2$.

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What could the half time score have been?

Can you find all the possible half time scores?

How will you make sure you don't miss any out?

Later in the season Arsenal played Tottenham Hotspur. The final score was a draw, $3 - 3$.

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Can you find all the possible half time scores for this match?

## Getting Started

What could the score have been if Arsenal hadn't scored any goals by half time?

What could the score have been if Arsenal had scored one goal by half time ...?

## Student Solutions

## Teachers' Resources

### Why do this problem?

This challenge invites pupils to develop a systematic way of working and offers the opportunity for discussion in pairs, small groups and the whole class.

### Possible approach

This problem could be introduced in the context of any football team or match. The added appeal of a sport they enjoy and team they support may help many pupils engage in the solving of the problem.

Invite pupils to guess what the half time score was to the match. Take a few suggestions and then ask them to try and find all the possibilities. Give time for pupils to work in pairs on the task and look out for those children who are beginning to work in a systematic way.

After a suitable length of time, draw the whole group together and ask them how they are making sure they don't miss out any possibilities. You may wish to ask certain pairs to share their ways of working with the whole group. It might be handy to suggest that each different possibility is written on a separate strip of paper as this might aid later discussions.

Give them longer to work on the problem then bring everyone together once more to discuss findings. You could ask each pair how many different possible scores they think there are - they are unlikely to agree! This is where having the scores written on strips is useful as you can stick them on the board, or ask members of the class to hold them, then invite everyone to sort them
or re-order them. In this way, a system is imposed on the scores and any missing ones can be identified quickly.

You can then challenge pairs to find the possibilities for the $3 - 3$ match, using a similar system. The experience of working on the $4 - 2$ result all together should give them more confidence to tackle the second match in their pairs.

### Key questions

How do you know that may have been a half time score?

How can you be sure that you have found ALL the possible half time scores?

Suppose the final score was a draw, what then?

### Possible extension

You could ask "If there are $24$ possible different half time scores what could the final score have been?".

### Possible support

Some pupils may prefer to start with games where there are fewer goals, for example $0 - 1$, $1 - 0$, $1 - 1$ etc so that there are fewer possible half time scores.