Patterns and rules: Explanation

Every pattern follows a set of instructions that tell us how the pattern works. This pattern, 2, 4, 6, 8, starts at 2 and goes up in twos. This is the pattern’s rule. Sometimes, we only have a rule, and we need to figure out the pattern. Let’s try it.

Take the rule “starts at 5 and goes up by 3s”. We know the pattern starts at 5. Then what? “Goes up by 3”. 5 add 3 is… 8, so the next number is 8. 8 add 3 is… 11.

That’s our next number. 11 add 3… that’s 14. 5, 8, 11, 14, and so on. Aha! We have generated a pattern from the rule!

Look at the next rule and see if you can find the matching pattern. The pattern that matches this rule is 25, 22, 19, 16, 13. The patterns we’ve looked at so far use adding or subtracting. Patterns can also have rules that use other operations, like multiplying and dividing. Try to generate a pattern from the next two rules.

The pattern that matches this rule is seven, 14, 28, 56, 112. The pattern that matches this rule is 64, 32, 16, 8. We can notice something about these patterns we’re making. The ones that use addition or multiplication go up. The ones that use subtraction or division go down.

This is a helpful trick to know when you need to match a pattern to a rule. For example, say you need to find which of the following patterns “starts at 24 and divides by 2”. Right away, we can rule out the patterns that go up. The only pattern that can be dividing is the first one, since it is going down. Our trick works the other way around, too – when we have a pattern and need to choose the rule it follows.

Take this pattern: 3, 6, 12, 24 Let’s choose which of these rules it follows: “starts at 3 and goes down by 3”; “starts at 3 and goes up by 3”; “starts at 3 and doubles”; “starts at 3 and halves”. The first rule uses subtracting, and the last rule uses dividing. Since the pattern is going up, we can rule those two out. Now we’re left with two options. It’s time to test them!

Let’s try, “Starts at 3 and goes up by 3”. First, let’s see what we need to add to get from the first number to the second. To get from 3 to 6, we need to add… 3. Then let’s see if this works to get to the next number. 6 add 3 is… 9, not 12.

Hmm, adding won’t work. So let’s try multiplying, instead. What do we multiply 3 by to get 6? We multiple by 2 – or double. What do we get if we double 6?

12! This rule is working! Let’s check it one more time. What’s double 12? 24!

Perfect. Now we can match the pattern to its rule starts at 3 and doubles. Let’s recap. To match a pattern to its rule, we can: First, check if the pattern is going up or down. A pattern that is going up is adding or multiplying.

A pattern that is going down is subtracting or dividing. Second, test an operation. See what you need to add or subtract to get from the first number to the second, then do the same again. If this gives you the next number in the pattern, you’ve found the rule! If it doesn’t, you can… Third, test a different operation.

Find out what you need to multiply or divide by to get from the first number to the second, then do the same again. If this gives you the next number in the pattern, you’ve found the rule! OK, over to you! Look at the next pattern and see if you can match it to the correct rule. This pattern’s rule is: starts at three and triples.

Now you know how to generate a pattern from a rule and match a pattern to a rule using any operation.