11 plus Sequences & Patterns questions

What a Sequences & Patterns question looks like

Sequences & Patterns is the 11 plus maths topic that gives a run of numbers, or of shapes, following one rule, and asks for something that rule tells you. The numbers in a run are called its terms. What makes it its own topic, rather than a repeat of the number series questions in verbal reasoning, is where the gap sits: those almost always show a run and ask for the next one along, while here the missing term can be in the middle, at the very start, or the 8th term of a sequence never written out, and some questions ask for the rule itself instead of a number.

Most questions use three-digit numbers and steps such as 8, 12, 13 or 22, so the thinking is quick but the mental arithmetic has to be right. Some come dressed as a short story, or set the run out in a table or along a number line. A few swap numbers for a repeating line of shapes, a row of squares built from matchsticks, or dots making bigger squares. It is a sensible place to begin eleven plus algebra work.

Example one

What is the missing term in this sequence? 500, ___, 482, 473, 464

The gap is in second place, so the numbers either side of it are two steps apart, not one. Leave that pair alone and use two numbers sitting next to each other that you can both see. From 482 to 473 is 9, and 473 to 464 is 9 as well, so the sequence goes down by 9 each time. Work one step on from 500: 500 take away 9 is 491. Check it forwards, 491 take away 9 is 482, the number that comes next.

The answer is 491.

The tempting wrong answer is 490. These numbers sit near round hundreds, so a child in a hurry reads the step as 10 and gets something tidy. It fails the check: with a step of 10 the term after 490 would be 480, and the sequence says 482.

Example two

Zara writes a sequence. The rule is: multiply the position number by 14 and add 6. What is the 8th term in her sequence?

There is no run of numbers here, only the rule, and it works on the position of the term rather than on the term before it. The 8th term means position 8. Multiply 8 by 14, which is 112, then add 6.

The answer is 118.

The tempting wrong answer is 112, what a child gets by doing the multiplying and forgetting the second half of the instruction. When a rule comes in two parts, read it back once the answer is down and check both parts happened.

Example three

Ivy builds a row of squares out of matchsticks. Her first pattern is one square made from 4 matchsticks. Her second is two squares side by side sharing a middle matchstick, made from 7. Her third is three squares in a row, made from 10. How many does she need for the 8th pattern?

Work out what one extra square costs. The first pattern to the second adds 3 matchsticks, and so does the second to the third, because every new square borrows a side from the one before it. The first pattern uses 4, and reaching the 8th means adding 7 more squares, so the total is 4 add 7 lots of 3.

The answer is 25 matchsticks.

The tempting wrong answer is 28, from 8 lots of 3 add 4. It uses the right step, but pays for eight extra squares when only seven are added to the one you began with. Count the steps between patterns, not the patterns themselves.

How to answer it

1. Write the gap between each pair of terms sitting next to each other. Next to each other only: a pair with a blank between them covers two steps, and treating it as one is the commonest way to go wrong here.
2. If every gap is the same, the rule is to add or take away that amount. If they differ, try the three next most likely rules: a gap that grows by the same amount each time, a rule alternating between two steps such as add 7 then take away 3, and a rule with a multiply in it such as times 3 then take away 4.
3. Read the question again and say which term it wants. The next one, the one in the gap, the one the sequence started from and the 8th one are four different jobs.
4. Apply the rule in the right direction. Forwards, do what it says. Back to an earlier term, undo it: undo add 15 by taking away 15.
5. Put the answer back into the run and test it against the term on each side. That check catches nearly every slip.

Common mistakes

• Using the gap across the blank as if it were one step. In 500, ___, 482 the difference of 18 covers two steps, so the step is 9.
• Stopping one term short. Starting at 162 and adding 12 each time, the 5th term is 210, not 198, because there are four steps from the 1st term to the 5th. Count positions on your fingers as you write the terms out.
• Doing only the first half of a two-part rule. Multiply by 14 and add 6, or times 3 then take away 4: several wrong options come from stopping after the multiplying.
• Adding when the question wants you to work backwards. If the 3rd term is 375 and the sequence goes up by 25, the 1st term is 325, not 425.
• Assuming every sequence changes by the same amount. Square numbers go 1, 4, 9, 16, 25 with gaps of 3, 5, 7 and 9, and a growing triangle of tiles goes 1, 3, 6, 10, 15. Two unequal gaps in a row mean stop looking for a single step.

How to practise 11 plus Sequences & Patterns questions

Little and often beats a long session here, because the topic is two habits: spotting a rule, and being accurate with three-digit addition and subtraction under time pressure. Ten minutes a few times a week is plenty, the pattern we set out in how much 11 plus practice a child needs each day. When your child gets one wrong, ask them to put their answer back into the sequence and read it out with the terms either side. They usually hear the mistake themselves. 11+ Daily has a Sequences & Patterns project alongside the rest of the algebra work, on laptop, tablet or phone, and the whole list is on our 11 plus maths practice page. Every question is original to 11+ Daily, nothing is a past paper and nothing is licensed from another publisher, and the bank is re-checked in rounds.

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