Word problems and remainders
Many children who can calculate perfectly well still lose marks on word problems. The trouble is usually deciding what to do, not doing it. Bar models, estimating and reading the question to the end solve most of it, and remainders need their own explicit teaching.
Why word problems are hard
A word problem asks children to do three things: understand the situation, choose the operations, and calculate. Calculation is the part we practise most, but it's rarely where marks go. Children react to a keyword (“altogether”, “left”, “each”) and pick an operation without picturing the problem, or they stop after the first step of a two-step question.
Bar models
A bar model is a simple drawing of the quantities in a problem: a long bar for the whole and shorter bars for the parts, or equal bars for equal groups. It makes the structure visible and shows which operation is needed. For example:
- Tickets cost £12 each. 28 pupils go. Total? → 28 equal bars of £12 → 28 × 12 = £336Equal groups: multiply.
- £336 was shared equally between 28 pupils. Each? → one bar split into 28 → 336 ÷ 28 = £12Sharing a whole: divide.
Once children are used to drawing bars, multi-step problems become a series of small, visible decisions instead of a guessing game.
A routine for every problem
- Read the whole question, including the last line that says what to find.
- Draw it: a bar model or a quick sketch.
- Estimate the answer roughly.
- Calculate, showing each step.
- Check: does the answer match the estimate, and does it answer the actual question?
Remainders in real life
When a division doesn't work out exactly, the context decides what happens to the remainder. This is a separate skill from dividing, and it's tested directly in the SATs.
- 50 children, minibuses hold 12. How many minibuses? 50 ÷ 12 = 4 r 2 → 5Round up: the 2 extra children still need a seat.
- 50 eggs, boxes of 12. How many full boxes? 50 ÷ 12 = 4 r 2 → 4Round down: only full boxes count.
- 50 sweets shared between 12 children. How many are left? → 2The remainder is the answer.
- £50 shared between 12 people → £4.17 each (to the nearest penny)The remainder becomes a decimal.
- 50 ÷ 12 = 4 r 2 minibusesHalf a minibus isn't possible; the answer must be a whole number.
The quickest check is to write the answer in a full sentence: “We need 4 r 2 minibuses” obviously makes no sense.
In the SATs
Word problems fill much of the two reasoning papers. They're often multi-step, involve money, measures or time, and include more information than is needed. Some carry two marks, with one available for correct working, so showing the steps matters. Remainder questions ask children to decide whether to round up, round down, or give the remainder as a fraction or decimal.
Common mistakes
- Choosing an operation from a keyword without understanding the problem.
- Answering only the first step of a two-step problem.
- Dividing the wrong way round (small number by large).
- Giving a remainder answer such as “4 r 2” where the context needs a whole number.
- Not converting units before calculating, such as mixing pounds and pence.
Teaching ideas
Numberless problems
Give a word problem with the numbers removed: “Some children went on a trip. Each coach held some children. How many coaches were needed?” Pupils discuss the structure and operation first. When the numbers are revealed, the calculation is the easy bit.
Same numbers, different answers
Give one division (50 ÷ 12) and four contexts that need rounding up, rounding down, the remainder and a decimal. Pupils match each context to its answer and explain why.
Write the question
Give a bar model and ask children to write a word problem it could represent. Writing problems is one of the best ways to understand them.
Highlight the question
Ask pupils to underline exactly what the question is asking for before they start. It's a small habit that stops many “right calculation, wrong answer” mistakes.
Try it: five quick questions
SATs-style questions to use as a starter or a quick check. Tap to see each answer.
A school has 185 pupils. Each table seats 8. How many tables are needed?
Show answer
24 (185 ÷ 8 = 23 r 1, so round up).
Pencils come in packs of 12. How many full packs can be made from 150 pencils?
Show answer
12 (150 ÷ 12 = 12 r 6).
A book costs £6.75. How much do 4 books cost?
Show answer
£27.00
Sam has £20. He buys 3 drinks at £1.45. How much change?
Show answer
£15.65
£100 is shared equally between 8 people. How much each?
Show answer
£12.50
Free worksheets for this topic
Printable A4 PDFs with answers. Free to use in class or at home.
