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# Factors and Multiples Game

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This game can replace standard practice exercises on finding factors and multiples. In order to play strategically, learners think about numbers in terms of their factors, utilising primes and squares to develop winning moves. The switch from a competitive to a collaborative game gives an opportunity for learners to work together and try to find the longest chain, a problem that they could keep coming back to over and over again!

The game also offers the chance to focus on any of the five key ingredients that characterise successful mathematicians. The collaborative version lends itself particularly to fostering a positive attitude to mathematics as learners' resilience may be tested!

*The following printable worksheets may be useful: Factors and Multiples Game, Factors and Multiples Puzzle, resources page for 1-100 square grids.*

*This activity featured in an NRICH video for students in June 2020 and an NRICH student webinar in November 2020. It also featured in an NRICH webinar for teachers in June 2020.*

**Dozens may be a good problem for students to tackle, before embarking on this challenge.**

Play the game as a class, on the board, to introduce the rules, perhaps dedicating the last twenty minutes of each lesson for a week, to playing in pairs. When pupils have finished a game, they could play the next game against someone they've not yet played. At the end of each game, ask pairs to analyse why the last few moves led to its end - working out better moves that could have been
made.

To start with you could choose not to mention the initial rule that restricts the starting number to a positive even number that is less than 50. When pupils discover that the first player can win after just three numbers have been crossed, discuss the need to restrict the initial number to an even number smaller than 50.

As learners are playing the game, listen out for those who are considering the probable next few moves when placing a counter/crossing out a number. Game strategies form a natural context for developing deductive logic. You may want to invite a pair of pupils to play against another pair. This gives a 'reason to reason' as each player will need to justify their choice of next move in order for their partner to agree it is the best way to proceed.

The collaborative challenge could run for an extended period: the longest sequence can be displayed on a wall or noticeboard and pupils can be challenged to improve on it. Any improved sequences can be added (perhaps once they have been checked by someone else!). You could then set aside some time in a future lesson to discuss the longest sequence. Does the class think that it is possible to create a longer one? Why or why not?

Do you have any winning strategies?

Are there any numbers you should try to avoid?

Use a smaller number board, eg 1-50 (or 1-49 in a square). Here is a large 1-50 grid and here is a sheet of smaller grids which could be given to pupils. This makes the mental calculations much easier, without watering down the mathematics. The lesson could focus on accuracy of calculation - with teacher interventions to get pupils sharing their mental strategies.

Handouts for teachers are available here (Word document, pdf), with the problem on one side and the notes on the other.

For the collaborative version of the game, ask pupils to explain why their choice of numbers is good.

Xavi's T-shirt is a suitable follow-up problem

Xavi's T-shirt is a suitable follow-up problem