1. What makes a pattern?

Some patterns are easy to spot. Others make you stop and look twice.

In this lesson, your job is to be a pattern detective. Try to work out the rule before anyone tells you the mathematical name for it.

What comes next?

Try it

Look at each row. What should come next?

●■●■?
Reveal answer
Purple circle. The rule repeats: circle, square, circle, square…
▲▲●▲▲●?
Reveal answer
Blue triangle. The piece ▲ ▲ ● keeps repeating.
★★★★★★?
Reveal answer
Four stars. This pattern is not repeating the same block — it is growing by one star each time.

Important question: Don’t just say what comes next. Can you explain why?

Find the mistake

One object in this pattern is wrong. Can you find it?

●■▲●■◆●■▲
Reveal answer
The pink diamond is the mistake. The rule is purple circle, orange square, blue triangle, repeated again and again. So the sixth object should be a blue triangle.

A pattern inside a pattern

Now things get more interesting. Sometimes two different rules are happening at once.

Look only at the shapes.

●■●■●■

Shape rule: circle, square, circle, square…

Now ignore the shapes. Look only at the colours.

●●●●●●

Colour rule: purple, orange, orange, purple, orange, orange…

Pattern detective

Put the two rules back together:

●■●■●■???

Can you predict the next three objects?

Reveal answer
The next three are purple circle, orange square, orange circle. The shape rule repeats every 2 objects, while the colour rule repeats every 3. Together, the whole pattern repeats every 6 objects.

So what is a mathematical pattern?

Definition
A mathematical pattern follows a rule

If we can describe the rule, we can work out what belongs in the pattern — and often predict what should come next.

Some patterns repeat. Some grow. Some have more than one rule at the same time.

Make your own repeating pattern

If the activity does not load, open the pattern builder in a new tab.

The smallest repeating piece

Look at this pattern:

●■●■●■

You could say that all six objects repeat. But you don’t need all six to make the pattern. The smallest piece is just ● ■.

New word
Repeating unit

The smallest piece that repeats to make a repeating pattern is called its repeating unit.

Practice puzzles

1. What is the repeating unit?

◆●●◆●●
Reveal answer
The repeating unit is pink diamond, blue circle, blue circle.

2. Repair the pattern

▲●▲●●●

One object is wrong. Which one should you change?

Reveal answer
The fifth object should be a purple triangle. The rule alternates purple triangle, orange circle.

3. Two rules

The shapes alternate circle, square. The colours repeat pink, blue, blue. Starting with a pink circle, draw the first 6 objects. (You might want to use the interactive tool above)

Reveal answer
● ■ ● ■ ● ■

Go further

Can the same beginning have more than one possible rule?

Suppose a pattern starts:

1, 2, 3, …

One rule could be “add 1 each time”, so the next number would be 4. But can you invent a different rule that also starts 1, 2, 3 and then does something else?

One possible idea
You could say: “Start with 1, 2, 3, then repeat 1, 2, 3.” That would give 1, 2, 3, 1, 2, 3… So seeing only the beginning does not always tell us the rule for certain.

You’ve now met repeating patterns, growing patterns and patterns with two rules at once. Next we’ll ask a very different pattern question: can every shape cover a floor without gaps or overlaps?

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