Topics · Primary Maths · Number Patterns
Number patterns: finding the rule
A pattern is a rule applied over and over. Find the rule and every question about the pattern answers itself — including the ones about terms far beyond the end of the list.
- 282 exam papers analysed
- appears in 61% of P3 Maths papers
- ~3 min read
01 Look at the gaps, not the numbers
5, 8, 11, 14, … — the numbers themselves say little, but the gaps between them say everything: 3, 3, 3. The rule is add 3.
So the first move on every pattern question is to write the differences underneath. A constant difference means adding or subtracting the same amount each time, which is most primary patterns.
If the differences are not constant, look at what you would multiply by instead: 2, 6, 18, 54 has differences 4, 12, 36 — not constant — but each term is three times the last.
02 Working out a far-off term
Continuing a pattern one step at a time is fine for the next two or three terms and hopeless for the 50th. For those, count the jumps rather than the terms.
- Find the rule from the differences. For 5, 8, 11, 14 it is add 3.
- Count the jumps, not the terms. Getting from the 1st term to the 10th takes 9 jumps, not 10. This off-by-one is the commonest error on the topic.
- Multiply and add. The 10th term is 5 + 9 × 3 = 32.
- Check with a small case. The 2nd term should be 5 + 1 × 3 = 8, which matches the list — so the method is right.
03 Patterns made of shapes
Figure patterns are the same idea with something to count. Count the sticks, dots or squares in each figure, write the numbers down, and it becomes an ordinary number pattern.
Then look for what stays and what is added: a row of squares made of matchsticks starts with 4 and adds 3 for each new square, because each new square borrows one side from its neighbour.
That structure is usually visible in the picture, and seeing it is quicker than counting a big figure stick by stick.
04 How to say them
These are words most children have never said out loud, and a word you cannot say is a word you avoid using. Capitals mark the syllable you lean on. Tap the speaker to hear it.
- patternsayPAT-uhnmore than onepatterns
A sequence made by applying the same rule over and over.
- termmore than oneterms
One number in a pattern. The 1st term, the 2nd term, and so on.
- rulemore than onerules
What is done to get from one term to the next, such as add 3 or multiply by 2.
- differencesayDIF-uh-runssmore than onedifferences
The gap between two terms next to each other. Constant differences mean an adding rule.
- sequencesaySEE-kwuhnssmore than onesequences
Another word for a number pattern.
05 What each level is tested on
The same topic, asked differently as it goes up. The percentage is the share of papers at that level in which it appears.
- P2
Counting on and back in steps, and completing simple number patterns.
Practise P2 Number Patterns — 10 questions →48% of P2 papers
- P3
Finding the rule of a pattern and continuing it.
Practise P3 Number Patterns — 10 questions →61% of P3 papers
- P4
Patterns made of shapes, and finding a term further along.
Practise P4 Number Patterns — 10 questions →59% of P4 papers
- P5
Patterns inside word problems, and describing a rule in words.
Practise P5 Number Patterns — 10 questions →28% of P5 papers
- P6
Patterns leading to a general rule, and the first algebraic expressions for them.
Practise P6 Number Patterns — 10 questions →22% of P6 papers
06 Mistakes worth knowing about
Each of these is wrong. Decide why before you open it.
“The 10th term is 5 + 10 × 3 = 35.”
It takes 9 jumps to get from the 1st term to the 10th, not 10. The answer is 5 + 9 × 3 = 32.
“The pattern goes up, so the rule is add.”
Not always. 2, 6, 18, 54 goes up by multiplying by 3. Check whether the differences are constant before assuming.
“I continued the pattern one step at a time to the 50th term.”
One slip anywhere ruins it, and it takes far longer. Find the rule and jump straight there.
“Each new square in the matchstick pattern adds 4 sticks.”
It adds 3 — the new square shares one side with the one before. Only the first square needs all 4.
“5, 8, 11 — the rule is 3.”
The rule is add 3. A rule is an instruction, and the word matters when the pattern turns out to be multiply by 3 instead.
Now try it
50 questions across 5 levels, marked as you go with the working shown.