Problem Teachers' Notes Hint Solution

# Playing Connect Three

### Why do this problem?

This problem offers students a chance to analyse a game which involves adding and subtracting positive and negative numbers, and requires them to work out the probability of the different possible outcomes. Teachers could explain that by knowing about the mathematics behind a game we can sometimes determine a winning strategy (or more realistically, a strategy that improves our chances of winning).
Teachers may find it interesting to read the NCETM Mathemapedia Entry:
Games as challenges to stimulate curiosity and support learning

### Possible approach

The game could be introduced using an interactive whiteboard by asking for two volunteers to play, each linked to one of the coloured counters, or the class could be divided into two teams.
After the introduction students could play in pairs at individual computers, or this board could be printed for students to play away from the computers. This spreadsheet can be used to simulate throwing the two dice.
To switch attention from consolidation of number skills to the mathematics behind the game, this needs to be followed by a whole class discussion that focuses on emerging strategies, observations, explanations and justifications.
Students can then go back to working in pairs to establish the numbers of ways of achieving the different totals. At the end of the lesson a plenary discussion can offer students a chance to present their findings. The discussion can compare the merits of the different approaches used (eg listing possibilities vs sample space diagrams).
Following on from this problem, students could take a look at:
Weights
Consecutive Negative Numbers

### Key questions

Are there some numbers that we should be aiming for? Why?
Are certain numbers easier to 'cover' than others? Why?
"Have you got all the solutions?" "How do you know?"

### Possible extension

If appropriate, teachers could set up an open ended activity for the rest of the week in which students investigate changes they could make to the game.
For example:
* Different shaped boards
* Boards where some numbers appear more than once
* Different dice (e.g. dodecahedral) or different numbers on the dice
* Allowing multiplication and division and changing the board accordingly

### Possible support

Students who struggle with adding and subtracting negative numbers can play First Connect Three , a simpler version of the game that just requires dice with positive numbers.
Teachers may like to take a look at the article on Adding and Subtracting Negative Numbers