Draw a number line on a piece of paper, marked from $0$ to $20$, like this:
(You could print off this sheet which has several number lines drawn on it.)
This challenge is a game for two players. The first player chooses two numbers in this grid and either times or divides them.
He or she then marks the answer to the calculation on the number line.The second player then chooses two numbers and either $\times$ or $\div$, and marks that number in a different colour on the number line.
If the answer is too big or too small to be marked on the number line, the player misses a go.
The winner is the player to get four marks in a row with none of their opponent's marks in between.
What good ways do you have of winning the game?
Does it matter if you go first or second?
[This game is adapted from a SMILE Centre card.]
Why do this problem?
gives children the opportunity to estimate answers to calculations in a motivating context, and gives plenty of practice in multiplication and division. Playing strategically involves higher-order thinking and the need to think ahead.
You could play the game on the board against another adult (or a child who has been told the rules), but without saying anything about the rules to the whole group. You could write down each calculation as you go along. Invite the class to watch the game in action and after a few moves, ask them to suggest what they think the rules might be. How might the game be won?
Having discussed the possibilities as a whole class, explain the rules as described in the problem itself and set the children off playing in pairs. (This sheet has several numberlines to print).
As they play, you should find that if each child wants to win, they will automatically scrutinise the answers of their opponent carefully and this means that the pupils will be required to explain and justify their thinking, and to check their own calculations thoroughly. There is also scope here for the children themselves to make decisions about the rules of the game. For example, will
they allow a player to multiply or divide a number by itself, or must the two numbers chosen be different? Will they notice that the zero can't be used?
During the plenary, encourage learners to explain any strategies that they developed. You may also wish to draw attention to some particularly good examples of justification and explanation which you heard as the children played the game, or examples of insightful comments in general.
How are you deciding which number to aim for next?
Can you find a winning strategy?
Children can be encouraged to tweak the game and to try out their new version. For example, they might change the number line, the grid of numbers, the operations, the number of numbers needed to win ...
You might wish to have calculators available and/or adapt the grid/number line to suit the children with whom you are working. Alternatively, you could tweak the game so that children throw two dice and have to use any operation with the two numbers thrown to create a new number to be marked on the number line. The same rules then apply, that the the winner is the first to create
four (or three) numbers in a row without the other person squeezing one in between. (With thanks to Jenni Back for this idea.)