Why do this problem?
is a great way for students to take responsibility for their own learning. They can avoid negative numbers if they are not confident or they can push themselves to calculate negative answers. In analysing the game more fully, rather than just playing it, the idea is that learners can develop a system for
finding all the possible ways of making each number on the grid, so they can justify which are the easiest to get.
This printable worksheet may be useful: First Connect Three.
You could introduce the game by playing against the class, or by splitting the class into two teams to play against each other, or with the class playing against the computer. Students can play against each other in pairs to get more of an idea of the game. You can print off this board if the students are not playing at a computer.
After a suitable length of time, ask the suggested questions in 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 a chance to present findings and you can draw attention to those methods which were particularly efficient. This would then lead to a discussion about how their findings might affect the way they play the game to win.
Are there some numbers that we should be aiming for? Why?
Which numbers on the grid are the easiest to get? Why?
Which numbers are most difficult to get? Why?
Further challenges could be provided by asking what would happen if:
- there was a differently shaped board
- numbers appeared more than once on the board and you could place more than one counter in a turn
- you could use dodecahedral dice or, for example, $1-12$ spinners
- you wanted to design a board for a game where you allowed multiplication and division
For students who are able to add and subtract both positive and negative numbers, the game Connect Three
and the problem Playing Connect Three
are suitable extensions.
If some pupils are struggling, you could adapt the board so that it only contains the numbers $1-12$.