11 plus Function Machines & Inverse questions

What a Function Machines & Inverse question looks like

A function machine is a chain of boxes, each one doing a single thing to a number. A number goes in at one end, the first box multiplies it by 4, the second box subtracts 3, and the answer comes out at the other end. In the 11 plus they turn up as small drawn machines with arrows between the boxes, as a sentence about someone thinking of a number, or as a short table of the numbers that went in and came out.

These questions run in both directions. Some give the number going in and ask what comes out, some give the number coming out and ask what went in, and a few give neither and ask what the boxes must be doing. Its nearest neighbour, Inverse / Think of a Number, only ever runs backwards from a result to the starting number, while this one also sends numbers forwards through the boxes and asks a child to work out a machine’s hidden rule from a table of inputs and outputs.

Machines here have one, two or three boxes, a few join two of them so the answer from the first becomes the number going into the second, and the hardest swap the starting number for n and ask for the answer as an expression: add 4 then multiply by 3 gives 3n + 12.

Example one: forwards through two boxes

A machine multiplies by 4 then subtracts 3. The input is 5. What is the output? The options are 2, 17, 20, 8 and 23.

Send the 5 into the first box: 5 × 4 = 20. That 20 is the number that now goes into the second box: 20 - 3 = 17.

The answer is 17.

20 is what you hand in if you do the first box and forget there is a second one. 8 is what you get by doing the boxes the other way round, (5 - 3) × 4, and it tempts a child because taking 3 from 5 feels like the easier place to start.

Example two: backwards to the number that went in

A machine multiplies by 5 then subtracts 7. Which input gives an output of 43? The options are 9, 72, 12, 10 and 50.

Going backwards means starting at the last box and undoing it. The last box subtracted 7, so put the 7 back: 43 + 7 = 50. The first box multiplied by 5, so divide by 5: 50 ÷ 5 = 10.

The answer is 10. Check it forwards: 10 × 5 = 50, and 50 - 7 = 43.

50 is where a child stops after undoing one box and forgetting the other. 12 comes from taking the 7 away instead of putting it back, 43 - 7 = 36, then dividing by 3. Undoing means the opposite: if the machine subtracted, you add.

Example three: working out what the boxes do

Other questions give numbers that have already been through the machine and ask for the rule. Here it multiplies by one number then adds another.

What comes out when 10 goes in? The options are 51, 50, 60, 53 and 41.

Compare the first two rows. The numbers going in went up by 3, from 2 to 5, and the numbers coming out went up by 15, from 11 to 26. Each 1 going in is worth 5 coming out, so the machine multiplies by 5. Now use one row for the rest: 2 × 5 = 10, but 11 came out, so it adds 1 after multiplying. Test that on the third row: 8 × 5 = 40, and 40 + 1 = 41, which matches.

So 10 × 5 = 50, and 50 + 1 = 51. The answer is 51.

50 is the tempting one: a child who has spotted the multiply by 5 can hand it in before adding the 1. 41 is there for anyone who reads the wrong row and copies an answer already printed.

How to answer it

1. Decide which way the question runs. Is the missing number the one going in or the one coming out? Everything else follows from that.
2. Going forwards, do one box at a time in the order they are written, and write down the number that comes out of each box before starting the next. Never do the easier sum first.
3. Going backwards, start at the last box and undo it, then the one before it, and so on. Adding undoes subtracting, subtracting undoes adding, dividing undoes multiplying, multiplying undoes dividing.
4. If the boxes are hidden, use two rows to find the number the machine multiplies by, one row to find what is added or taken away, then test the rule on a row you have not used.
5. Check by running your answer forwards through the machine. It should land exactly on the number the question gave, and a mismatch is nearly always a reversed step rather than a sum.

Common mistakes

• Doing the boxes in the wrong order. With multiply by 4 then subtract 3 on an input of 5, (5 - 3) × 4 gives 8 instead of 17. Children reach for whichever sum looks friendlier, and it is rarely the first box.
• Stopping halfway. The halfway number is on the option list every time, because it is the most common wrong answer.
• Repeating instead of undoing. Going backwards through a subtract 7 box and taking 7 away again, rather than adding it.
• Undoing from the wrong end. If a machine adds 8 then multiplies by 3, the way back is divide by 3 first and subtract 8 second. Starting with the 8 feels orderly and gives the wrong number.
• Finding half the rule and stopping. In a table question, spotting the multiply by 5 and handing in 50 rather than 51. The added number is small and easy to forget.

How to practise 11 plus Function Machines & Inverse questions

This is a format where a child improves quickly: there is one method, and most errors are about sequence rather than sums. The habit worth building is the check: after every answer, run it forwards and watch it land on the number the question gave. Asking your child to say each box aloud as they go, “times four, now take three”, stops most order mistakes on its own. Short and regular beats long and occasional here, which is the pattern we set out in how much 11 plus practice a child needs each day. 11+ Daily has a Function Machines & Inverse project alongside the rest of the eleven plus maths topics, on laptop, tablet or phone, and the full list is on our 11 plus maths practice page. Everything in it 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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