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Why play this game?
is a motivating context in which learners can apply simple addition and subtraction. However, the real challenge here is to find a winning strategy that always works and this involves working systematically, conjecturing, refining ideas, generalising and using knowledge of factors and multiples.
All the notes that follow assume that the game's default setting is a target of $23$ using the numbers $1$ to $4$.
Introduce the game to the class by inviting a volunteer to play against the computer. Do this a couple of times, giving them the option of going first or second each time (you can use the "Change settings" button to do this).
Ask the students to play the game in pairs, either at computers or on paper. Challenge them to find a strategy for beating the computer. As they play, circulate around the classroom and ask them what they think is important so far. Some might suggest that in order to win, they must be on $18$. Others may have thought further back and have ideas about how they can make sure they get to $18$,
and therefore $23$.
After a suitable length of time bring the whole class together and invite one pair to demonstrate their strategy, explaining their decisions as they go along. Use other ideas to refine the strategy.
Demonstrate how you can vary the game by choosing different targets and different ranges of numbers. Ask the students to play the game in pairs, either at computers or on paper, using settings of their own choice. Challenge them to find a winning strategy that will ensure they will always win, whatever the setting.
How can I work out the 'stepping stones' that I must 'hit' on my way to the target?
Is there an efficient way of finding the first 'stepping stone'?
When is it better to go first and when is it better to let the computer go first?
If the computer says $1$, I say...?
If the computer says $2$, I say...?
If the computer says $3$, I say...?
Two more demanding games, requiring similar strategic thinking, are
You could demonstrate the game a few more times at the start. Alter the settings on the game to have a lower target and a shorter range of numbers (for example a target of $10$ using the numbers $1$ and $2$). As you play, note down the running totals to refer back to later.
Handouts for teachers are available here (word document
, pdf document
), with the problem on one side and the notes on the other.