### Exploring Wild & Wonderful Number Patterns

EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.

### Sending Cards

This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?

### Dice and Spinner Numbers

If you had any number of ordinary dice, what are the possible ways of making their totals 6? What would the product of the dice be each time?

# Compare the Calculations

## Compare the Calculations

There are four multiplication calculations hidden below.

Your challenge is to put them in order, from easiest to hardest. Try to do this without actually calculating each answer if you can.

Click on 'Show' to see them.

70 x 40

70 x 57

70 x 21

70 x 100

How did you decide the order?

We would love to hear the reasons for your final order.

You might like to do exactly the same with the set of four division calculations hidden below.

350 $\div$ 7

350 $\div$ 1

350 $\div$ 25

350 $\div$ 3

Create a set of four multiplications or four divisions yourself, which you think could be put in order from easiest to hardest.
Give them to someone else to order.
Do they agree with your final order? Why or why not?

You may find it useful to print off this sheet, which contains the two sets of calculations. You could cut them up into two sets of four cards.

### Why do this problem?

This activity is designed to raise learners' awareness of different calculation methods and to help them recognise the value of choosing a method to suit a particular situation. If learners are encouraged to have a flexible approach to calculation, they are freed from feeling that they have to remember the 'right' method to use, and can therefore take greater ownership for their mathematics. This task focuses on multiplication and division, whereas Arranging Additions and Sorting Subtractions offers addition and subtraction examples.

Read more about the benefits of having a flexible approach to calculation in our Let's Get Flexible! article.

### Possible approach

Explain that you are going to show the class four calculations and, rather than being interested in the answers, you would like learners to order the calculations from easiest to hardest. Emphasise that you will be wanting them to be able to articulate how they decided on the order.

Reveal the four calculations (it doesn't matter whether you choose to use the set of multiplications or the set of divisions, or whether you use one before the other, or just one set). Give learners a few minutes to look at the whole set on their own to start with before asking them to work with a partner to agree an order. At this point, you may like to give out a set of the calculations, each calculation on a separate card, printed from this sheet (one set per pair). This will enable learners to physically move the calculations around as they discuss the ordering.

As the class works, listen out for pairs who are paying attention to the numbers involved and thinking carefully about how they would solve each one. You may like to stop everyone after five minutes or so to invite them to share some of their thoughts so far. How are they making decisions? Draw out the idea that just because the four calculations all involve the same operation, it doesn't mean we would do them all in the same way. We might be able to apply our knowledge of multiplication facts and/or place value; we might use compensation, or apply an algorithm for example.

Give everyone more time to come to a decision in their pairs before another whole class discussion. You might like to invite a few pairs to share their solution and reasoning, perhaps deliberately picking those who have not reached the same conclusion. It might be that you can reach a concensus on the method you would use to answer each calculation, in which case you could give each one a 'label' so that the whole class has a shared experience and you can refer back to these particular examples in the future.

As a follow-up activity, you could give each pair a piece of A4 paper and ask them to split it into four boxes (for example by folding). In one box, they could write one of the four calculations. In another box, they could work out the answer to that particular calculation (including a description of how they did this). In another, they could show how they would check their answer, using a different method. Finally, in the fourth box they could create a word problem that would be solved using that calculation. These would make a lovely classroom, or school corridor, display.

Of course you could do the same activity but with your own set of four calculations, to suit the experience and needs of your learners.

### Key questions

How would you do that calculation?
Why do you think that one is harder/easier than that one?

### Possible support

All children should have access to a range of materials to help them calculate, should they find it difficult not to actually work out the answers! This might include concrete objects as well as anything that facilitates jottings.

### Possible extension

Challenge learners to create their own set of four calculations, deliberately including a range of difficulty. Having decided on the order from easiest to hardest, they could swap sets with a partner.