Describe & continue number patterns
Theory
A number pattern follows a rule that turns each term into the next. The rule may add or subtract the same number each time, or multiply or divide by the same number. This Year 6 maths page shows how to find and describe the rule, continue a pattern, fill gaps, jump to a far-off term, and test whether a number will ever appear.
Each number in a pattern is a term. The rule tells you how to get from one term to the next.
If the gaps between terms are all the same, the rule adds (gaps going up) or subtracts (gaps going down) that number.
If the gaps keep growing (or shrinking), check whether each term is the one before multiplied or divided by the same number, such as \(5, 15, 45, \ldots\) (multiply by \(3\)).
To find a far-off term of an adding pattern, add the step the right number of times: the \(10\)th term is the first term plus \(9\) steps.
A number will appear only if it fits the pattern: in \(1, 6, 11, 16, \ldots\) every term ends in \(1\) or \(6\).
Finding the rule:
| Pattern | Gaps | Rule |
|---|---|---|
| \(11, 20, 29, 38\) | \(+9\) each time | add \(9\) |
| \(84, 77, 70, 63\) | \(-7\) each time | subtract \(7\) |
| \(5, 15, 45, 135\) | growing | multiply by \(3\) |
| \(400, 200, 100, 50\) | shrinking | divide by \(2\) |
Finding the rule
- Find the gaps between neighbouring terms.
- Same gap? The rule is add or subtract that gap.
- Growing or shrinking gaps? Divide a term by the one before; if the answer is always the same, the rule multiplies (or divides) by it.
Finding a missing term
- Use two neighbouring terms to find the rule.
- Apply it to fill the gap, then check the next term.
| \(38 + 9\) | \(=\) | \(47\) |
| \(47 + 9\) | \(=\) | \(56\) |
Rule: add \(9\). Next terms \(47\) and \(56\).
| \(63 - 7\) | \(=\) | \(56\) |
| \(56 - 7\) | \(=\) | \(49\) |
Rule: subtract \(7\). Next terms \(56\) and \(49\).
| \(135 \times 3\) | \(=\) | \(405\) |
Rule: multiply by \(3\). Next term \(405\).
| \(50 \div 2\) | \(=\) | \(25\) |
Rule: halve (divide by \(2\)). Next term \(25\).
| \(21 - 13\) | \(=\) | \(8\) |
| \(21 + 8\) | \(=\) | \(29\) |
\(29\). Check: \(29 + 8 = 37\).
From the 1st term to the 10th term there are \(9\) steps of \(9\).
| \(11 + 9 \times 9\) | \(=\) | \(92\) |
\(92\).
No. Adding \(5\) each time from \(1\) makes every term end in \(1\) or \(6\), and \(64\) ends in \(4\).
(ii) A tank holds \(300\) L and \(15\) L is used each day. How much is left after \(6\) days?
| \(20 + 5 \times 7\) | \(=\) | \(55\) |
| \(300 - 6 \times 15\) | \(=\) | \(210\) |
(i) \(\$55\) (\(5\) steps from week 1 to week 6)
(ii) \(210\) L
Common pitfalls
Frequently asked questions
How do you find the rule of a number pattern?
Look at the gaps. Equal gaps mean add or subtract; growing gaps often mean multiply.
What is the next term in 11, 20, 29, 38?
\(47\) (add \(9\)).
How do you find a missing term?
Find the rule from two neighbouring terms, then use it to fill the gap.
How do you find the 10th term without listing them all?
Add the step \(9\) times to the first term.
How can you tell a number will never appear?
Check it against the pattern, for example its last digit or whether it is odd or even.