Repeating Patterns

What you will learn

• Find the part of a pattern that repeats.
• Say what comes next in a repeating pattern, and what fits in a gap.
• Work out which shape sits in a given place, such as the $7$th.

Before you start: you should be able to count to $20$ and name a few shapes. Shapes All Around is a good start.

The idea

A pattern is something that follows a rule. Stripes on a scarf, the beat in a song, the tiles on a floor: each one repeats in a way you can predict. A repeating pattern has a short part that is used again and again. Once you spot that part, you know the whole pattern.

Finding the repeating part

Read the pattern from the left and look for where it starts to repeat.

The shortest piece that repeats is called the core of the pattern. Here the core is circle, square. We say it is an AB pattern, because it has two different things in turn.

Patterns with a longer core

Some patterns have three or more things in the core.

To find the core, look for the first shape to come back, then check that the same run of shapes comes again.

What comes next?

Once you know the core, carry on from where the last core ended. Find the place in the core where the last shape sits and say the next shape in the core.

The gap is a square, because after a circle in the core comes a square.

A shape far along the pattern

You do not need to draw out a hundred shapes to know the $100$th. Count in cores. If the core has $3$ shapes, then the $3$rd, $6$th, $9$th and so on all match the last shape of the core. The shapes after those are the first shape again.

For the pattern circle, square, triangle (core of $3$): the $7$th shape is one more than $6$, which is a multiple of $3$, so the $7$th is a circle. This links to counting in equal steps in Skip Counting in 2s, 5s and 10s.

Worked examples

Example 1: Name the core

Find the core of: square, triangle, triangle, square, triangle, triangle, square.

1. Look for the first shape to come back: square comes again at the $4$th place.
2. The shapes before it are square, triangle, triangle.
3. Check that the same three shapes repeat from the $4$th place: square, triangle, triangle. They do.

Answer: The core is square, triangle, triangle.

Example 2: Fill the gap

Find the missing number in $3, 4, 5, 3, 4, ?, 3, 4, 5$.

1. The core is $3, 4, 5$.
2. The gap is in the $6$th place, which is the last place of the second core.
3. The last place of the core is $5$.

Answer: $5$.

Example 3: The 10th shape

A pattern goes circle, square, circle, square, and so on. What is the $10$th shape?

1. The core has $2$ shapes: circle, square.
2. Even places ($2$nd, $4$th, $6$th, $8$th, $10$th) are squares, because they match the second shape of the core.
3. $10$ is even.

Answer: A square.

The core is the shortest repeating part. circle, square, circle, square is two cores, not one. Always look for the smallest piece that makes the whole pattern when it is repeated.

Why it matters

Patterns help us predict. Musicians count a repeating beat, builders lay repeating tiles and weather forecasters look for cycles. In maths, spotting a rule is the first step towards number sequences and algebra, where a letter stands for any number in the rule.