2. Using games to explore rectangular numbers

From square numbers we move to rectangular numbers. The only requirement now is that there must be at least 2 rows and 2 columns. Each row must have the same number of crosses, e.g.

NumberRectangular Patterns
62x3, 3x2

This allows for ‘rectangular’ numbers. Would you expect there to be more or fewer ‘rectangular’ numbers than ‘square numbers’, and can you explain why?

Looking at the numbers that form rectangular shapes is one way for pupils to explore multiplication (or division) through seeing and doing, as well as carrying out mental and written sums.

Trying out the investigations yourself to see where pupils may find difficulties and planning how you can help those who do will help you be more effective in supporting their learning.

Case Study 2: Playing a game to find factors

Mrs Ali planned to ask her pupils to find different rectangular numbers by using multiplication facts.

She decided to have a class competition. She divided the class into two teams and asked each team to choose a scorekeeper. The game was that she would write a number on the board and the first pupil who gave her two correct factors for that number scored a point for their team. Mrs Ali explained that there would be more than one correct answer – sometimes many. She then showed the class an example by writing 6 and saying that she would have given a point to any one who said ‘2 times 3’ or ‘3 times 2’ or ‘1 times 6’ or ‘6 times 1’. The class enjoyed the game and became quite excited. Mrs Ali was very happy, as she had planned ahead that this game would help her pupils with their next activity.

In later tasks, she often played this game with her pupils when she had five minutes left at the end of the day.

Activity 2: Multiplication using counters

You will need 20 counters, bottle tops, beans or stones, for each group of four/five pupils.

  • Begin by dividing the class into their groups and handing out the counters.
  • Copy or draw the table in Resource 2: Table of multiplications on the board for each pupil to copy to record their findings.
  • Ask the groups to take 6 counters and arrange them in rows of equal numbers, exploring all the possible arrangements they can make.
  • After five minutes, let the groups share the possible arrangements they found for the number 6. Check that at least one group included an arrangement with only one row. Ask them to fill in their table for 6 as shown in Resource 2.
  • Next, let them try number 12 but, before they make the arrangements, they must predict the number of possible arrangements, and then check if their predictions are true.
  • Repeat with all the numbers on the table.

Resource 5 has more examples of maths games that you can use with your pupils

3. Looking for patterns