Number Sequences and Pattern Rules
By NeurofiED Editorial Team · Reviewed by NeurofiED Learning Science Team
NeurofiED resources are written for UK 11+ families and reviewed against the platform's retrieval practice, spacing, interleaving and feedback principles.
Help a Year 5 child identify, describe and test additive and multiplicative term-to-term rules, then continue sequences and find missing terms.
A number-sequence rule tells you how to move from one term to the next. For an additive sequence, add or subtract the same amount each time; for a multiplicative sequence, multiply or divide by the same amount each time. Describe the change precisely, apply it once per step, and test it across every pair of neighbouring terms before continuing the sequence.
For example, `17, 24, 31, 38, ...` has the rule add 7, so the next two terms are `45` and `52`. By contrast, `3, 6, 12, 24, ...` has the rule multiply by 2, so the next two terms are `48` and `96`. Looking only at whether numbers rise or fall is not enough: the operation and the amount both matter.
England’s non-statutory Year 5 curriculum guidance says pupils should recognise and describe linear number sequences, including sequences involving fractions and decimals, and find the term-to-term rule in words.[2] This article concentrates on whole-number additive and multiplicative rules so that the reasoning is visible. It does not teach equation solving or turn the task into times-table practice.
Contents
What a sequence rule describes
A term is one number in a sequence. A term-to-term rule describes the operation that takes one term to the next.
In a simple additive sequence, the difference between neighbouring terms stays constant:
`11, 16, 21, 26, ...`
Each move adds `5`. The rule is not merely “go up” or “add”; it is add 5 each time.
In a simple multiplicative sequence, the ratio between neighbouring non-zero terms stays constant:
`4, 12, 36, 108, ...`
Each move multiplies by `3`. The differences are `8`, `24` and `72`, so this is not a constant-difference sequence. The rule is multiply by 3 each time.
A child need not use the words difference and ratio immediately. “What happened to get from this number to the next?” is a good starting question. Precision can follow: name the operation, name the amount, then check every move.
A reliable find-test-continue routine
Use four steps:
A number line can make equal additive jumps visible, while grouped counters or a simple branching sketch can show repeated multiplication. The Education Endowment Foundation says representations such as number lines should be used purposefully, with a clear rationale for the mathematical idea being taught, and should help reveal structure rather than become an end in themselves.[3] At home, use a representation only while it clarifies the repeated change; remove it when your child can explain and apply the rule independently.
Worked example: an increasing additive sequence
Continue:
`17, 24, 31, 38, __, __`
Step 1: compare neighbouring terms.
The same difference appears each time, so the rule is add 7.
Step 2: continue carefully.
The completed sequence is:
`17, 24, 31, 38, 45, 52`
Step 3: check. The consecutive differences are all `7`: `24 − 17`, `31 − 24`, `38 − 31`, `45 − 38` and `52 − 45` each equal `7`.
A common error is to add `7` twice to the last printed term, writing `45` and then `45` again or jumping straight to `52`. Encourage your child to point to the current term before making each move.
Worked example: a decreasing additive sequence
Find the missing term and continue:
`66, 58, __, 42, 34, __`
The sequence decreases. Test the first change:
`58 − 66 = −8`
So a possible rule is subtract 8. Apply it across the gap:
That reaches the printed term `42`, so the missing number is `50`. Continue checking:
The completed sequence is:
`66, 58, 50, 42, 34, 26`
The printed `42` is useful evidence: it tests two linked moves around the missing term. If a guessed number fits on only one side, it has not solved the gap.
Worked example: a multiplicative sequence
Continue:
`3, 6, 12, 24, __, __`
The increases are not equal:
So “add the same number” fails. Now test multiplication:
The rule is multiply by 2. Continue:
The completed sequence is:
`3, 6, 12, 24, 48, 96`
Check backwards with the inverse operation: `96 ÷ 2 = 48`, `48 ÷ 2 = 24`, and so on. This is sequence reasoning, not a request to recite a multiplication table: the important work is identifying one transformation and showing that it explains every transition.
A decreasing multiplicative sequence works in the same way. In `160, 80, 40, 20, ...`, each term is divided by `2`, so the next term is `10`. Saying “subtract 80” describes only the first move; it immediately fails because `80 − 80` would be `0`, not `40`.
How to reject a rule that only partly works
Consider:
`5, 10, 15, 20, 25`
“Multiply by 2” works from `5` to `10`, but fails on the next move because `10 × 2 = 20`, not `15`. “Add 5” works across every gap, so it is the valid simple term-to-term rule.
Now consider:
`2, 6, 18, 54`
“Add 4” works once, from `2` to `6`, but `6 + 4 = 10`, not `18`. “Multiply by 3” works throughout:
This habit—propose, test, reject or retain—is more dependable than choosing a rule because it looks plausible.
Some sequences use alternating or changing rules, but do not assume complexity too soon. First test whether one constant additive or multiplicative rule explains every gap. If it does not, describe exactly what changes rather than forcing a near match.
Missing terms and working backwards
A missing term may be easier to find by reversing the rule.
Suppose the rule is multiply by 4:
`__, 12, 48, 192`
To move backwards, use the inverse operation:
`12 ÷ 4 = 3`
So the missing first term is `3`. Check forwards:
`3 × 4 = 12`, `12 × 4 = 48`, and `48 × 4 = 192`.
For an additive example with the rule subtract 9:
`__, 41, 32, 23`
Moving forwards subtracts `9`; moving backwards adds `9`:
`41 + 9 = 50`
The missing term is `50`, and the forward check is `50 − 9 = 41`.
Keep the distinction clear: reversing a sequence reverses the operation. The inverse of adding `6` is subtracting `6`; the inverse of multiplying by `5` is dividing by `5`.
Common mistakes and useful adjustments
Naming a direction instead of a rule
“Getting bigger” does not say how. Ask: Which operation? By how much? Every time? Accept “add 7 each time” or “multiply by 3 each time”.
Checking only the first pair
Cover the answer choices, have your child draw a small arrow between every pair, and write the proposed operation above each arrow. Any failed arrow rejects the rule.
Confusing constant difference with constant factor
For `2, 4, 8, 16`, subtraction gives changing differences: `2`, `4`, `8`. Division gives a constant factor: each term is `2` times the previous term. If mental comparison feels crowded, make two short rows labelled “difference” and “factor”.
Applying the rule to the original term repeatedly
To extend `10, 16, 22, ...`, the next terms are `28` and `34`. The second new term comes from `28 + 6`, not from calculating `22 + 6` again. Ask your child to underline each new current term before continuing.
Treating a calculation slip as a rule error
Separate the decisions. First ask whether the stated rule fits the printed terms. Then check the arithmetic used to continue it. A correct rule with one addition slip needs a calculation check, not a completely new strategy.
Moving to harder numbers too quickly
If the rule is unclear, reduce the arithmetic load without changing the reasoning. Try `4, 7, 10, 13` before `147, 150, 153, 156`. Once the child can state and test “add 3”, return to the original scale.
A parent action sequence
Try this short progression with paper and a pencil:
Listen for explanation before increasing difficulty. If your child can produce answers but cannot justify the rule, ask for the checks between terms. If the explanation is secure but arithmetic is slow, allow written working. If both are secure, place the missing term at the beginning or middle so that inverse reasoning is needed.
Short FAQ
Is every rising sequence additive?
No. `3, 6, 12, 24` rises by multiplying by `2`; its differences do not stay constant. Test both the operation and its amount across every transition.
Can a sequence go down?
Yes. An additive sequence might subtract the same amount, as in `66, 58, 50, 42`. A multiplicative sequence might divide by the same amount, as in `160, 80, 40, 20`.
How many terms should my child check?
Every displayed transition. A rule that explains only the first pair may fail immediately afterwards.
Should we start with a formula for the nth term?
Not for this goal. The focus here is a term-to-term rule stated in words, applied repeatedly and checked. Formulae and equation solving are separate work.
What should I do when my child is stuck?
Use smaller numbers, mark the gaps with arrows and compare one pair at a time. A number line can help for equal additive jumps; remove it once the structure is clear and your child can explain the rule unaided.
Next step
Choose one additive and one multiplicative example above. Ask your child to name the rule, prove it across every gap, continue two terms and then work one step backwards. When that explanation is secure, try a Maths lesson.
Sources
[2] Department for Education, National curriculum in England: mathematics programmes of study.
[3] Education Endowment Foundation, Improving Mathematics in Key Stages 2 and 3.
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