Key Stage 4 maths · Working mathematically

GCSE maths problem solving: modelling and money problems

Much of GCSE maths is problem solving: questions that are multi-step, set in real life, or unlike anything the pupil has practised. Pupils in Years 10 and 11 must decide for themselves which method to use, link different parts of maths together, use maths in financial settings, and say what their answer means in context. They also learn to model a situation and to comment on how their assumptions might affect the result.

Problem solving and modelling, explained

In earlier years most questions told pupils what to do: “add these fractions”, “find the area”. GCSE problem-solving questions give a situation and leave the method to the pupil. A typical question might ask which of two phone deals is cheaper, how many tiles are needed to cover an awkwardly shaped floor, or whether a ladder is long enough. The first job is to work out what is being asked and which pieces of information matter. Teachers encourage pupils to write down what they know and what they need, draw a diagram, try simpler numbers, work backwards from the end, or try a few cases systematically.

Financial contexts come up often. Pupils work with percentage increase and decrease, VAT, profit and loss, best-buy comparisons, exchange rates and wages. They also use simple and compound interest. With compound interest, £1,000 at 3% a year for two years grows by a multiplier of 1.03 each year, so it becomes 1,000 × 1.03² = £1,060.90, because the second year’s interest is paid on the first year’s interest too. Instalment plans, deposits and sale prices are all everyday examples of multi-step arithmetic.

The curriculum also asks pupils to make and use connections between different parts of mathematics. A single question might need algebra to describe the perimeter of a shape, an equation to find an unknown length, then Pythagoras’ theorem or trigonometry to find a diagonal, and finally ratio to share a cost. Recognising which topic is hiding inside a question is a skill that grows with practice, and it is why mixed practice helps more than doing one topic at a time.

Modelling means turning a real situation into a mathematical one, simplifying as necessary. A pupil might treat a can as a cylinder, ignore air resistance on a thrown ball, or assume a savings account keeps the same interest rate for ten years. Every simplification is an assumption, and GCSE questions often ask pupils to say how the answer would change if the assumption were false. For example, “the model assumes a constant speed, so the real journey is likely to take longer because of traffic”.

Finally, pupils interpret their solution in context. An answer of 5.19 minibuses for 83 people with 16 seats each really means 6 minibuses. A length of −2 cm from an equation must be rejected. A result of 0.35 hours becomes 21 minutes. Checking that an answer is sensible, includes the right units and answers the actual question is part of the marks, and it is a habit worth building from Year 10.

Words your child will hear

Multi-step problem: A question that needs several calculations or ideas in a sequence to reach the answer.
Mathematical model: A simplified version of a real situation that can be described with numbers, formulae or graphs.
Assumption: Something taken to be true to make a model simple, such as “the speed stays constant”.
Multiplier: The number you multiply by for a percentage change. Increasing by 20% uses 1.2; decreasing by 20% uses 0.8.
Compound interest: Interest that is added each year to a total that already includes earlier interest.
Interpret: Say what a number or result means in the context of the question, including units.
Estimate: A rough answer found by rounding, used to check that a full calculation is sensible.

What the National Curriculum says

In Key Stage 4, pupils should be taught to:

Develop their mathematical knowledge, in part through solving problems and evaluating the outcomes, including multi-step problems
Develop their use of formal mathematical knowledge to interpret and solve problems, including in financial contexts
Make and use connections between different parts of mathematics to solve problems
Model situations mathematically and express the results using a range of formal mathematical representations, reflecting on how their solutions may have been affected by any modelling assumptions
Select appropriate concepts, methods and techniques to apply to unfamiliar and non-routine problems; interpret their solution in the context of the given problem

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

A sofa costs £800 cash. A shop offers it for a 20% deposit and then 12 monthly payments of £55. Which option is cheaper, and by what percentage of the cash price? Give one assumption you have made.

Read the question and pull out the facts: cash price £800, deposit 20%, 12 payments of £55. Find the deposit first.
```
20% of £800 = 0.2 × 800 = £160
```
Find the total of the monthly payments.
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12 × £55 = £660
```
Add the deposit and the payments to get the total cost of the payment plan.
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£160 + £660 = £820
```
Compare with the cash price and write the difference as a percentage of the cash price.
```
£820 − £800 = £20
20 ÷ 800 × 100 = 2.5%
```
State an assumption and interpret the result. This assumes there are no other charges, such as delivery, arrangement or late-payment fees, and that every monthly payment is made on time. The extra £20 is the cost of spreading the payments.

Paying cash is cheaper by £20. The payment plan costs 2.5% more than the cash price.

Common mistakes and how to help

Grabbing the numbers and calculating without working out what is asked

Under time pressure it feels safer to do something with the numbers than to stop and think. Pupils then answer a different question from the one set.

What helps: Ask your child to say in their own words what the answer will be, for example “a percentage” or “a number of coaches”. A rough estimate before calculating also catches many wrong turns.

Rounding in the middle of a multi-step calculation

Rounding feels tidy, so pupils round each stage. Small errors then grow, and a compound interest answer can end up several pence or pounds out.

What helps: Keep the full calculator display or store it in memory, and round only at the end, to the accuracy the question asks for. Money is usually rounded to the nearest penny.

Believing a percentage increase followed by the same percentage decrease gets back to the start

The numbers 20% and 20% look like opposites, but they apply to different amounts. £100 up 20% is £120, and 20% of £120 is £24, so the second step lands on £96.

What helps: Use multipliers: 1.2 then 0.8 gives 0.96, a 4% fall overall. Test it with £100 so the idea is concrete. It matters in shop sales and in interest calculations.

Leaving the answer as a bare number with no link to the context

Once the calculation is finished, pupils feel the job is done. They may write 5.19 minibuses, a negative length or an answer with no unit.

What helps: Make “does this make sense?” a final habit. Reread the question, check the units, and decide whether to round up, round down or reject a solution.

How to help at home

Compare two real deals. Look together at two broadband, gym or phone deals and work out which is cheaper over a year, including any set-up cost. Let your child decide which calculations are needed before you help.
Estimate before you calculate. Before a bill is totalled or a discount worked out, ask for a rough answer by rounding. If the final calculation is far from it, something has gone wrong.
Plan something with a budget. Ask your child to plan a family meal, a day out or a small decorating job within a set amount, listing assumptions such as prices, quantities and waste. Then ask what would change if one assumption were wrong.
Let them struggle a little. When your child is stuck, try “what do we know?” and “what could we draw?” rather than showing the method. Getting started on an unfamiliar problem is the skill being tested.

Builds on

Key Stage 3: Developing fluency
Key Stage 3: Reasoning mathematically
Key Stage 3: Solving problems