Patterns in Mathematics

Why This Matters

Look around you. Patterns are everywhere.

The days of the week repeat. Monday, Tuesday, Wednesday, and round again. The seasons come back every year. The black and white keys on a harmonium repeat. The tiles on your floor follow a fixed design.

A pattern is something that repeats in a fixed way, or follows a rule you can spot. Once you see the rule, you can guess what comes next.

Maths is, in a big way, the hunt for patterns. And it does one more thing. It asks why the pattern happens. Not just “what comes next”, but “why does it always work?”

That is the fun part. In this chapter you will not just find patterns. You will understand them. You will see them with pictures made of dots. By the end, you will look at plain numbers like 1, 4, 9, 16 and see the hidden shape inside them.

The Big Idea

A list of numbers that follows a rule is called a number sequence. Many of these sequences have a hidden picture inside them. When you draw the numbers as dots, the pattern jumps out at you. And once you can see the pattern, you can also explain why it happens. That is the real magic of maths: not just spotting the pattern, but knowing the reason behind it.

Let’s Break It Down

What is a number sequence?

A sequence is just a list of numbers written in order, one after another, that follows a rule.

Here is a simple one:

2, 4, 6, 8, 10, …

The rule is easy. Each number is 2 more than the one before. The three dots ”…” at the end mean the list keeps going forever. The next numbers would be 12, 14, 16.

Here is another:

1, 2, 4, 8, 16, …

This time the rule is double the last number. 1 doubles to 2, 2 doubles to 4, and so on. The next number is 32.

So a number sequence is a list with a rule. Find the rule, and you can keep the list going.

Mathematicians have given special names to some sequences they love. Figure 1.1 below collects the main ones we will meet, with their names.

Tap to zoom

Three of these have wonderful hidden pictures: triangular numbers, square numbers, and cubes. Let us meet them next.

Triangular numbers — numbers shaped like a triangle

Take this sequence: 1, 3, 6, 10, 15, …

At first it looks random. But it is not. These are the triangular numbers. They are called that for a beautiful reason. If you have that many dots, you can arrange them into a neat triangle.

Figure 1.2 below shows how. Watch what happens as the triangle grows.

Tap to zoom

Do you see the rule? Each new triangle gets a fresh row at the bottom. The bottom row always has one more dot than the row above it.

• Start with 1 dot.
• Add a row of 2 → now 3 dots.
• Add a row of 3 → now 6 dots.
• Add a row of 4 → now 10 dots.

So the jumps are +2, +3, +4, +5, and so on. The jumps keep growing by one each time.

Let us use this rule to go a bit further:

Quick check to make sure the idea stuck:

Square numbers — numbers shaped like a square

Now take this sequence: 1, 4, 9, 16, 25, …

These are the square numbers, often just called squares. The name is a clue again. That many dots can be set out as a perfect square grid — the same number of dots across as down.

Before we draw them, let us refresh one small idea you will need.

Now look at Figure 1.3. See how each square number fills a perfect square box of dots.

Tap to zoom

So a square number is simply a number multiplied by itself:

1 = 1 × 1 = 1²

4 = 2 × 2 = 2²

9 = 3 × 3 = 3²

16 = 4 × 4 = 4²

That is why “squared” and “square number” share a name. Both come from the square shape made by the dots.

Cube numbers — numbers shaped like a cube

There is one more in this family: 1, 8, 27, 64, 125, …

These are the cube numbers, or cubes. A cube is a solid box-shape, like a dice or a sugar cube. It has the same length, width, and height.

A cube number is a number multiplied by itself three times:

1 = 1 × 1 × 1 = 1³

8 = 2 × 2 × 2 = 2³

27 = 3 × 3 × 3 = 3³

64 = 4 × 4 × 4 = 4³

The small raised 3 (as in 2³) means “use this number three times in a multiply”. You can imagine 8 small cubes stacked into a bigger 2-by-2-by-2 cube, and 27 small cubes into a 3-by-3-by-3 cube. The shape gives the name.

How a pattern continues

The whole skill here is the same every time. Find the rule, then use the rule to go further. You do not need to see all the numbers. You just need the rule.

Let us practise on a sequence where the rule is “double it”:

Adding odd numbers makes square numbers (and why!)

Here is one of the most beautiful patterns in all of maths. It looks almost like a magic trick. Watch what happens when you add up the odd numbers one by one.

1 = 1

1 + 3 = 4

1 + 3 + 5 = 9

1 + 3 + 5 + 7 = 16

1 + 3 + 5 + 7 + 9 = 25

Look at the answers on the right: 1, 4, 9, 16, 25. Those are exactly the square numbers! Every time you add the next odd number, you land on the next square.

This is amazing. But maths does not stop at “wow”. Maths asks: why does this always happen? Why should adding odd numbers give squares, forever?

Here is the lovely reason, and it is all in the picture. Think of building a square out of dots, but build it in L-shaped layers. Each L is just an odd number of dots. Figure 1.4 below shows it.

Tap to zoom

Do you see it? Each odd number — 1, 3, 5, 7 — is one L-shaped layer of dots. When you wrap a new L around a square, you always get the next bigger square. The L exactly fills the missing two sides plus the corner.

So:

1 + 3 + 5 + 7 = 16, and 16 = 4 × 4 = 4²

And because you can wrap an L around a square of any size, this trick never breaks. It works for 10 odd numbers, for 100 odd numbers, forever. Add the first 10 odd numbers and you get 100 (which is 10 × 10). That is the power of a picture: it shows you why, not just that.

Let us use this shortcut to answer a question fast:

Patterns in shapes

Patterns are not only in numbers. They live in shapes too. And often, a shape pattern is secretly tied to a number pattern.

Look at this family of shapes: a triangle, a square, a pentagon, a hexagon. They all have straight, equal sides. Now do one simple thing — count the sides of each. Figure 1.5 shows it.

Tap to zoom

The side-counts go 3, 4, 5, 6, …. That is just the counting numbers, starting from 3! So a pattern of shapes hides a pattern of numbers. This link between shapes and numbers is one of the prettiest ideas in maths, and you will meet it again and again.

Common Mistakes

These are the slips students make most often. Read them once and you will dodge them.

Quick Check

Try these. Each one checks an idea from the chapter.

Practice Problems

Try each one yourself first. Only then tap to see the full solution.

Easy

Medium

Challenge

Summary

• A pattern is something that follows a rule, so you can tell what comes next.
• A number sequence is a list of numbers in order that follows a rule, such as 2, 4, 6, 8, …
• Triangular numbers (1, 3, 6, 10, …) make a triangle of dots; each new one adds a longer bottom row.
• Square numbers (1, 4, 9, 16, …) make a square grid of dots; each is a number times itself, like 9 = 3 × 3.
• Cube numbers (1, 8, 27, 64, …) are a number times itself three times, like 8 = 2 × 2 × 2.
• Adding the odd numbers (1 + 3 + 5 + 7 …) always gives square numbers, because each odd number is an L-shaped layer that keeps the dots a perfect square.
• Shapes have patterns too, and they are often linked to number patterns — like the sides of a triangle, square, pentagon, hexagon counting up 3, 4, 5, 6.
• Maths is not just about finding a pattern. It is about understanding why the pattern is true.

What’s Next

Now that you can see patterns in numbers and shapes, it is time to look more closely at the shapes themselves. The next chapter is Lines and Angles. There you will meet the simple building blocks of every shape — points, lines, and the corners (angles) where lines meet. That is exactly what makes a triangle different from a square. See you there!