Write the gaps between the terms before you look at the numbers themselves. Almost every sequence in this test gives itself up at that point.
Maths carries 25% of the marks. Almost all of it arrives as worded problems rather than bare calculations, so the reading is part of the maths.
Differences first, always
- Write the difference between each pair of terms, above the gaps.
- Are they all the same? Then it is a constant step. Done.
- If not, look at the differences themselves. Are they going up by a fixed amount?
- If still not, try multiplying. Divide each term by the one before it.
- Check your rule against every term, not just the first two.
That last step is where marks are saved. A rule that explains the first two terms and fails on the third is not the rule.
The patterns worth knowing by sight
| Sequence | Rule | Next |
|---|---|---|
| 3, 7, 11, 15 | Add 4 | 19 |
| 2, 6, 18, 54 | Multiply by 3 | 162 |
| 1, 4, 9, 16 | Square numbers | 25 |
| 1, 3, 6, 10 | Triangular: add 2, then 3, then 4 | 15 |
| 1, 1, 2, 3, 5, 8 | Add the two before it | 13 |
Square numbers and triangular numbers are worth learning outright. A child who recognises 1, 4, 9, 16, 25 instantly saves twenty seconds every time one appears, and they appear often.
When the gaps keep changing
Take 2, 5, 10, 17. The differences are 3, 5 and 7, which are not constant, so there is no single step to apply.
But look at the differences themselves: 3, 5, 7 go up by 2 each time. So the next difference is 9, and the next term is 17 plus 9, which is 26.
This is called looking at the second differences, and it is the single most useful trick in the topic. When the first differences do not settle, take the differences of the differences.
And they will be worded
You will rarely see a bare list. You will see a pattern of tiles, a growing arrangement of matchsticks, or a savings account, and the sequence has to be extracted from the sentences first.
So write the first four terms out as numbers before doing anything else. Converting the words into a row of figures is the step that makes the rest routine.
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Convert the words into numbers before you think
Every year a large number of students find worded problems very difficult, and the majority of the 11+ maths is based on worded problems. Sequences are a clear case. The pattern is rarely handed to you as a list of numbers; it is buried in a description of tiles or matchsticks.
So the first move is mechanical: get the first four terms written out as plain figures in the booklet. Once they are numbers in a row, the sequence is a routine problem. While they are still sentences, it is a reading problem pretending to be a maths one.
That is why I am so insistent about comprehension in mathematics. The child has to understand what is being asked first, and only then apply what they know. The speed of answering is what causes a lot of children to make mistakes here, because they start calculating before they have finished reading.
Mushtaque Ahmed, Founder of GrammarPath and Bright Mentors
Frequently asked questions
How do you solve 11+ number sequence questions?
Write the difference between each pair of terms first. If the differences are all the same it is a constant step; if not, look at whether the differences themselves follow a pattern; if still not, try dividing each term by the one before to test for multiplication.
What is a second difference?
The difference between the differences. In 2, 5, 10, 17 the gaps are 3, 5 and 7, which rise by 2 each time, so the next gap is 9 and the next term is 26. It is the most useful trick in the topic and solves sequences that look impossible.
Which number patterns should my child memorise?
Square numbers 1, 4, 9, 16, 25 and triangular numbers 1, 3, 6, 10, 15 at minimum. Recognising them instantly saves around twenty seconds each time, and they appear regularly.
What if my rule only works for the first two terms?
Then it is not the rule. Always check a candidate rule against every term in the sequence, not just the pair you worked it out from. This is one of the commonest ways children lose a question they understood.
Are 11+ sequence questions given as plain lists?
Rarely. They usually arrive as worded problems: a pattern of tiles, a growing matchstick arrangement, or money saved each week. Write the first four terms out as numbers before doing anything else.