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Play: What comes next?

Patterns and sequences · step 1 of 5

A question to start

Look at these shapes. Which one comes next?

?

You probably said "a circle" straight away. Your brain found the rule without being told: circle, square, triangle, and then start again. That rule is a pattern, and noticing patterns is one of the most important things in all of mathematics.

The idea

There are two kinds of pattern you will meet again and again.

1. Repeating patterns. A small group of things, called the unit, repeats over and over.

  • red, blue, red, blue, red, blue ... The unit is red, blue. It is 2 long.
  • circle, square, triangle, circle, square, triangle ... The unit is 3 long.
  • Days of the week: Monday, Tuesday, ..., Sunday, then Monday again. The unit is 7 long.

2. Growing patterns. Each picture or number is bigger than the one before, by a rule.

14916 ?

The squares of dots hold 1, 4, 9, 16 dots. The next square is 5 dots wide and 5 dots tall, so it holds 5 × 5 = 25 dots.

How to find a rule. Ask yourself three questions:

  1. Does something repeat? Find the unit and count how long it is.
  2. Does something grow? Look at how it changes from one step to the next.
  3. Can I say the rule in one sentence? If you can, you can continue the pattern as far as you like.

Worked example

The pattern: 3, 6, 9, 12, ... What comes next, and what is the 10th number?

  1. Does it repeat? No, the numbers keep getting bigger.
  2. How does it change? From 3 to 6 is +3. From 6 to 9 is +3. From 9 to 12 is +3. It goes up by 3 every time.
  3. The rule in one sentence: start at 3 and add 3 each time.

So the next number is 12 + 3 = 15. Carrying on: 18, 21, 24, 27, 30. The 10th number is 30.

Did you notice? These are the numbers in the 3 times table: 3 × 1, 3 × 2, 3 × 3, ... so the 10th one is 3 × 10 = 30. Seeing a pattern in two different ways is a sign that you really understand it.

Try it

  1. What comes next? 🔴 🔵 🔵 🔴 🔵 🔵 🔴 🔵 ...
  2. What are the next two numbers? 5, 10, 15, 20, ...
  3. What are the next two numbers? 1, 2, 4, 8, ...
  4. A pattern repeats red, yellow, green. What colour is the 10th one?
  5. Today is Monday. What day will it be in 15 days?
Answers
  1. 🔵. The unit is red, blue, blue, 3 long. After red, blue the unit still needs one more blue.
  2. 25, 30. Add 5 each time.
  3. 16, 32. Each number is double the one before. This is a different kind of growing: multiplying, not adding.
  4. Red. The unit is 3 long, so the 3rd, 6th and 9th are green and the 10th starts the unit again: red.
  5. Tuesday. 14 days is exactly 2 weeks, back to Monday; one more day is Tuesday.

Common mistake

Guessing the rule from only two numbers. If you only see 1, 2, ..., the next number could be 3 (add 1) or 4 (double). Always check your rule on every number you are given before you trust it.

Where it leads