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Word problems and multi-step logic

Choosing the operation, bar modelling and working through several steps.

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Word problems hide the maths inside a story. The calculations are ones you already know. The skill is working out which calculations to do, and in what order. In this lesson you will learn:

  • a simple plan for any word problem
  • how to choose the right operation: +, −, × or ÷
  • how to draw a bar model to see what is going on
  • how to break a problem into steps
  • how to work backwards to find a starting number

1. A plan for every word problem

Key idea Do not rush to calculate. Read, picture, plan, and only then work it out.
  1. 1Read it twice. Underline the numbers and the question at the end.
  2. 2Picture it. Draw a quick bar model or sketch.
  3. 3Plan it. What do you need to find first? Then what?
  4. 4Solve it, one step at a time. Write each step down.
  5. 5Check it. Did you answer the question asked? Is the answer sensible? Units: £, cm, pupils?

2. Choosing the operation

Key idea Ask yourself what is happening in the story: are things being put together, taken away, made into equal groups, or shared? Clue words help, but always read the whole sentence.
What is happening?OperationClue words
Putting together, finding a total+altogether, total, sum, more than
Taking away, finding a difference−left, fewer, difference, how many more
Equal groups, repeated adding×each, every, times as many, rows of
Sharing, or grouping equally÷share equally, each get, how many groups
Watch out: clue words can trick you “Mia has 7 more stickers than Leo. Mia has 20. How many does Leo have?” It says more, but Leo has fewer, so you take away: 20 − 7 = 13.

3. Bar models: draw the story

Key idea A bar model turns words into a picture. Draw a bar for each amount, and a ? for what you need to find. The picture shows you the calculation.

Part and whole

A school has 324 pupils. 158 are girls. How many are boys?

324 pupilsPupilsgirls 158boys ?

The whole is 324 and one part is 158, so the other part is 324 − 158 = 166 boys.

Comparing

Sam has 46 marbles. Jo has 19 fewer. How many does Jo have?

Sam46Jo?19 fewer

Jo’s bar is 19 shorter than Sam’s, so Jo has 46 − 19 = 27 marbles.

Equal parts

4 friends share £84 equally. How much does each get?

£84 shared by 4Each????

The bar is split into 4 equal parts, so each part is £84 ÷ 4 = £21.

Times as many

Ella has 3 times as many cards as Finn. Together they have 48. How many does Finn have?

Finn?Ella???Together: 48

There are 4 equal boxes altogether, worth 48. One box is 48 ÷ 4 = 12, so Finn has 12 (and Ella has 36).

4. Problems with several steps

Key idea Many 11+ problems need two or three calculations. Ask: what do I need to know first? Solve that, then use the answer in the next step.

Example

Tickets cost £7 for adults and £4 for children. 2 adults and 3 children go to the cinema. They pay with a £50 note. How much change do they get?

  1. First find the adults’ cost.2 × £7 = £14
  2. Then the children’s cost.3 × £4 = £12
  3. Add them to find the total.£14 + £12 = £26
  4. Take the total from £50 to find the change.£50 − £26 = £24
Answer£24 change
Watch out Answer the question that is asked. Here the question is the change (£24), not the total cost (£26).

5. Working backwards

Key idea When you know the end and need the start, work backwards. Undo each step in reverse order, using the inverse: + undoes −, and × undoes ÷.

Example: I think of a number. I multiply it by 3, then subtract 5. The answer is 31. What was my number?

?→× 3 →− 5→31
  • The last thing done was − 5. Undo it with + 5: 31 + 5 = 36.
  • Before that was × 3. Undo it with ÷ 3: 36 ÷ 3 = 12.
Answer12

Check forwards: 12 × 3 = 36, and 36 − 5 = 31.

The mistakes that cost marks

  1. Grabbing the numbers and adding them. Every number in the story gets added, whatever is happening. ask what is happening first: putting together, taking away, equal groups or sharing.
  2. Trusting a clue word without reading the sentence. “7 more than Leo” does not always mean add. read the whole sentence and decide who has more. A bar model makes it clear.
  3. Stopping one step too early. Finding the total cost (£26) when the question asks for the change. read the question again at the end. Did you answer exactly what it asks?
  4. Undoing the steps in the wrong order. In “× 3, then − 5”, dividing by 3 first. undo the last step first: + 5, then ÷ 3.
  5. Mixing up the units. Adding pence to pounds, or giving an answer in the wrong unit. write the unit beside every number in your working.
  6. Not checking the answer is sensible. 166 boys out of 324 pupils is fine; 482 boys would be impossible. ask: is it bigger or smaller than I expected? Then check with the inverse.

Quick checklist

Tick each one when you can do it without help.

Download

Print these and work through them away from the screen.

Year 4 practice paper PDF · 11+ style, with answer sheet and answers Year 5 practice paper PDF · 11+ style, with answer sheet and answers Year 6 practice paper PDF · 11+ style, with answer sheet and answers

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