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KS3 Bitesize

Maths

Number patterns

A number pattern is a series of numbers that follow a rule.

Introduction

This Revision Bite covers:

Even and odd numbers

Even numbers

Any number that can be divided by 2 is called an even number.

Examples of even numbers are 2, 4, 6, 8, 10, 22, 144 and 2020.

If a number ends with an even digit, then it is even. For example 1 023 458 is even, because its final digit (8) is even.

Odd numbers

Any number that cannot be divided by 2 is called an odd number.

Examples of odd numbers are 1, 3, 5, 7, 9, 35, 177, 2435, etc.

If a number ends with an odd digit, then it is odd. For example 3 702 443 is odd, because its final digit (3) is odd.

Square numbers

Square numbers are formed by multiplying a number by itself.

The first four square numbers are
1 × 1 = 1
2 × 2 = 4
3 × 3 = 9
4 × 4 = 16

Each square number can be represented as a square of dots.

Square numbers represented by dots aligned as squares
Question

Write down the next six square numbers (from 5 onwards). Then check your answer.

Answer

5 × 5 = 25, 6 × 6 = 36, etc.

So the next square numbers are:
25, 36, 49, 64, 81 and 100

Cube numbers

Cube numbers are formed by multiplying a digit by itself three times.

For example 2 cubed is 2 × 2 × 2 = 8
3 cubed is 3 × 3 × 3 = 27
4 cubed is 4 × 4 × 4 = 64

Each cube number can be represented by a cube made up of unit cubes.

Cube numbers
Question

Use your calculator to find the next six cube numbers (from 5 onwards) - then check your answer.

Answer

125 = 5 × 5 × 5, 216 = 6 × 6 × 6, etc.

So the next six cube numbers are:

125, 216, 343, 512, 729 and 1000

Triangle numbers

Triangle numbers can be represented as a triangle of dots.

The triangle numbers are:

Triangle numbers represented by dots aligned to make equilateral triangles
Question

What is the next triangle number after 21?

Answer

The next row of the triangle will have 7 dots. So the next number is

1 + 2 + 3 + 4 + 5 + 6 + 7 = 28

28 dot equilateral triangle

Multiples

The multiples of a number are those numbers that it will divide into exactly.

For example, the multiples of 5 are 5, 10, 15, 20, 25, 30,....

The multiples of 7 are 7, 14, 21, 28, 35, 42,....

Remember: Multiples are like times tables.

1 × 5 = 5, 2 × 5 = 10, 3 × 5 = 15, 4 × 5 = 20.
Therefore, the multiples of 5 are 5, 10, 15, 20,....

Question

What are the first five multiples of 11?

Answer

11, 22, 33, 44, and 55

Remember that the first multiple is always the number itself.

Exploring number patterns

  • 2, 6, 10,... is a number pattern that follows the rule 'add 4'.

The next number is 10 + 4 = 14

  • 81, 27, 9, .... is a number pattern that follows the rule 'divide by 3'.

The next number is 9 ÷ 3 = 3

  • 5, 8, 14, ... is a number pattern that follows the rule 'subtract 1, multiply by 2'.

The next number is (14 -1) × 2 = 26

Each number in a number pattern is called a term. So in the number pattern 2, 6, 10 ... the first term is 2, the second term is 6 and the third term is 10.

Question

Write down the rule and the next two terms in the number pattern: 2, 4, 8,... Then check your answer.

Answer

The rule is 'multiply by 2' and the next 2 terms are 16 and 32.

Look at how patterns change from one term to another. See what rule gets you from the 1st term to the 2nd term, then check the same rule gets you from the 2nd term to the 3rd term.

If it doesn't, find a different rule to get to the 2nd number and then check that it gets you to the 3rd number.

Number patterns in diagrams

A number pattern in a diagram often requires counting shapes to find the rule. Again, look at how the pattern grows from one term to the next.

Example:

square number patterns
  • Pattern 1 has 0 orange tiles and 3 blue tiles, so 3 tiles altogether
  • Pattern 2 has 1 orange tiles and 5 blue tiles, so 6 tiles altogether
  • Pattern 3 has 2 orange tiles and 7 blue tiles, so 9 tiles altogether
Question

Q1. Based on the number patterns above, draw pattern 4.

Q2. Look again at the number patterns above. Write the rule and the fourth terms for:
a) orange tiles
b) blue tiles
c) all tiles

Answer

A1.

square numbers pattern

Pattern 4 has 3 orange tiles and 9 blue tiles, so 12 tiles altogether

A2.
a) The rule for the number pattern of orange tiles is 'add 1' and the 4th term is 3.
b) The rule for the number pattern of blue tiles is 'add 2' and the 4th term is 9.
c) The rule for the number pattern of all tiles is 'add 3' and the 4th term is 12.

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