Algebra

Patterns & Sequences

3rd Class · 4th Class · 5th Class · 6th Class

✓By the end of this lesson students will be able to identify and describe simple number patterns.
✓By the end of this lesson students will be able to continue a given number pattern by applying its rule.
✓By the end of this lesson students will be able to create a table of values to represent a number sequence.
✓By the end of this lesson students will be able to find the rule for a simple number sequence.

Key concepts

What is a Pattern?

A pattern is a set of numbers, shapes, or objects that are arranged according to a rule. In maths, we often look at number patterns where numbers follow a predictable order.

What is a Sequence?

A sequence is a list of numbers that follow a specific rule or pattern. Each number in the sequence is called a 'term'. For example, in the sequence 2, 4, 6, 8, ... the first term is 2, the second term is 4, and so on.

The Rule of a Pattern

The rule tells us how to get from one term to the next in a sequence. For example, if a pattern is 3, 6, 9, 12, ... the rule is 'add 3' to the previous number. Sometimes the rule might be 'multiply by 2' or 'subtract 5'.

Tables of Values

A table of values is a way to organise the numbers in a sequence. It usually has two rows or columns: one for the position of the term (e.g., 1st, 2nd, 3rd) and one for the value of the term itself. This helps us see the pattern clearly and find the rule.

Key facts to remember

1A pattern is an arrangement of numbers or objects that follows a rule.
2A sequence is a list of numbers in a specific order, following a rule.
3Each number in a sequence is called a 'term'.
4The 'rule' tells you how to get from one term to the next, or how a term relates to its position.
5Common rules involve adding, subtracting, multiplying, or dividing by a constant number.
6Tables of values help to organise sequences by showing the position of a term and its value.
7To find the rule, look for the difference or ratio between consecutive terms.

Worked examples

Example 1

Look at the pattern: 5, 10, 15, 20, ... (a) Describe the rule for this pattern. (b) Write down the next three numbers in the pattern.

I(a) To find the rule, look at how the numbers change from one to the next:

II 10 - 5 = 5

III 15 - 10 = 5

IV 20 - 15 = 5

V Each number is 5 more than the previous number.

VI(b) To find the next three numbers, continue adding 5:

VII 20 + 5 = 25

VIII 25 + 5 = 30

9 30 + 5 = 35

Answer

(a) The rule is 'add 5' to the previous number. (b) The next three numbers are 25, 30, 35.

Always check the difference or relationship between at least two pairs of consecutive numbers to be sure of the rule.

Example 2

Complete the table of values for the sequence where the rule is 'multiply the position number by 4'.

IFor Position 1: 1 x 4 = 4

IIFor Position 2: 2 x 4 = 8

IIIFor Position 3: 3 x 4 = 12

IVFor Position 4: 4 x 4 = 16

VFor Position 5: 5 x 4 = 20

Answer

The completed table is: | Position | Term | |----------|------| | 1 | 4 | | 2 | 8 | | 3 | 12 | | 4 | 16 | | 5 | 20 |

Sometimes the rule relates the term directly to its position, not just to the previous term.

Example 3

A sequence starts with 20. The rule is 'subtract 3 from the previous number'. (a) Write down the first five terms of the sequence. (b) Create a table of values for these five terms.

I(a) Start with 20 and subtract 3 repeatedly:

II 1st term: 20

III 2nd term: 20 - 3 = 17

IV 3rd term: 17 - 3 = 14

V 4th term: 14 - 3 = 11

VI 5th term: 11 - 3 = 8

VII(b) Organise these terms into a table:

VIII Position 1: Term 20

9 Position 2: Term 17

10 Position 3: Term 14

11 Position 4: Term 11

12 Position 5: Term 8

Answer

(a) The first five terms are: 20, 17, 14, 11, 8. (b) The table of values is: | Position | Term | |----------|------| | 1 | 20 | | 2 | 17 | | 3 | 14 | | 4 | 11 | | 5 | 8 |

Be careful with subtraction, especially when numbers get smaller. Double-check your calculations.

Common mistakes

✗Not checking the rule across several terms; sometimes the pattern changes.
✗Making calculation errors when adding, subtracting, multiplying, or dividing to find the next terms.
✗Confusing the position number with the term's value, especially when creating tables.
✗Assuming the rule is always 'add' or 'subtract' when it could be 'multiply' or 'divide'.

Exam tips

★Read the question carefully to understand if you need to describe the rule, continue the pattern, or both.
★Show your working for finding the rule and for calculating the next terms.
★Use a pencil and paper to write out the terms and differences clearly.
★If asked to create a table, make sure it is neat and clearly labelled with 'Position' and 'Term'.

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