Key Stage 3 maths · Working mathematically

KS3 maths problem solving: modelling and financial maths

Problem solving is the third thread running through Key Stage 3 maths. Pupils in Years 7, 8 and 9 use what they know to tackle multi-step problems, including ones about money, and problems they haven’t seen before. They begin to model real situations with equations, tables and graphs, and they judge whether the answer they get makes sense in the real world.

Solving problems, explained

In primary school, many problems practise a method that has just been taught. In secondary school, pupils are increasingly asked to decide for themselves which maths to use. The National Curriculum says pupils should develop their mathematical knowledge partly through solving problems and evaluating the outcomes, including multi-step problems, where one answer feeds into the next calculation. Problem solving isn’t a separate unit. It is how number, algebra, geometry, ratio and statistics are practised in Years 7, 8 and 9.

Financial maths is a clear example of formal knowledge put to use. Pupils work out discounts and price increases with percentages, compare best buys by finding the price per 100 grams or per item, and use simple interest, where the same amount of interest is paid each year. For example, £500 at 3% simple interest earns £15 a year, so £60 in 4 years. They also handle exchange rates, which are proportion in disguise, and work out wages, budgets and the cost of plans that combine a fixed charge with a charge per use.

Modelling means turning a real situation into mathematics, working with it, and then checking the result against reality. A pupil picks out the quantities that matter, makes simple assumptions such as “prices stay the same”, and expresses the situation as a table, an equation, a formula or a graph. The model is then tested and sometimes improved. If the model says a bus journey takes 2.4 hours, the pupil converts that to a sensible time (2 hours 24 minutes), and asks whether traffic or a stop should be allowed for.

The hardest problems are unfamiliar and don’t announce which method to use. Pupils are expected to select appropriate concepts, methods and techniques for these. General strategies help: read the problem and say in your own words what is being asked, draw a diagram or table, try simple numbers first, work backwards from the answer, list the possibilities in an organised way, or write an equation for the unknown. At the end a good problem solver checks by estimating, using an inverse operation, or asking whether the answer is realistic. Being stuck is a normal part of the process, not a sign of being bad at maths.

Words your child will hear

Multi-step problem: A problem that needs more than one calculation, where the result of one step is used in the next.
Model: A mathematical description of a real situation, such as an equation or a graph, used to predict or compare.
Assumption: Something taken to be true to keep a model simple, such as “the price doesn’t change”.
Unit price: The cost of one unit, such as pence per 100 grams, used to compare best buys.
Percentage change: An increase or decrease written as a percentage of the original amount.
Simple interest: Interest worked out each year on the original amount only, so the same sum is added every year.
Evaluate: Judge whether an answer is sensible, accurate and fit for the question.

What the National Curriculum says

In Key Stage 3, 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 mathematics
Begin to model situations mathematically and express the results using a range of formal mathematical representations
Select appropriate concepts, methods and techniques to apply to unfamiliar and non-routine problems

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

A worked example

Club A charges a £15 joining fee plus £22 a month. Club B has no joining fee and charges £27 a month. After how many months does Club A become the cheaper choice overall?

Understand the problem. We want the number of months at which the total cost of A drops below the total cost of B. Assume the monthly prices never change and there are no other costs.
Build the model. Let m be the number of months, and write the total cost in pounds for each club.
```
Club A: 15 + 22m
Club B: 27m
```
Find when the two costs are equal by setting them equal and solving. Take 22m from both sides.
```
15 + 22m = 27m
15 = 5m
m = 3
```
Interpret it. At 3 months A costs 15 + 66 = £81 and B costs £81. At 4 months A is £103 and B is £108. B costs £5 more each month, so after the £15 start A is cheaper from the fourth month.
Evaluate. Over 12 months A costs 15 + 264 = £279 and B costs £324, which is £45 less. This agrees with 12 × 5 − 15 = 45. The answer only holds if prices stay the same, so a pupil might note that.

The two clubs cost the same (£81) after 3 months. Club A is cheaper from the fourth month onwards.

Common mistakes and how to help

Picking the numbers out of the question and combining them without thinking

Pupils are used to routine questions where the operation is obvious from the lesson, so they grab the numbers and look for a calculation that seems to fit.

What helps: Ask “what is the question really asking?” and “roughly what size should the answer be?” before any calculation. Drawing the situation often shows which operation is needed.

Thinking that a percentage decrease and an increase of the same size cancel out

20% up and 20% down sound like opposites. But the second percentage is of a different amount. £50 reduced by 20% is £40, and £40 raised by 20% is £48.

What helps: Always work out each step in turn with real numbers, and ask “20% of what?”. Using a multiplier makes it clear: ×0.8 and then ×1.2 is ×0.96, not ×1.

Not checking the answer against the real situation

A calculator or a successful method gives a number, and the pupil stops. Answers like 4.5 buses, −2 months or £12.5 get written down without a second look.

What helps: Make “does it make sense?” a final step every time. Money is written to 2 decimal places, whole objects are rounded in the sensible direction, and units must match.

Giving up because the problem doesn’t look like one from class

Pupils expect to match a question to a method they have just practised, and an unfamiliar one feels like failure.

What helps: Ask “what do you know?” and “can you try a smaller or simpler case?”. Resist giving the method away. Struggling for a few minutes is how problem-solving skill grows.

How to help at home

Best buy in the shop. Compare 500 g for £1.80 with 750 g for £2.40. The price per 100 g is 36p against 32p, so the larger pack is better value. Let your child work it out and decide whether you would actually use all of it.
Sale or fraction off. Which is the better deal on a £60 item: 30% off or one third off? Taking 30% off gives £42, and one third off leaves £40, so one third off is the better deal because a third is more than 30%. Real offers are full of problems like this.
Plan with a budget. Give your child a fixed amount and a real task, such as a small party, a day out or a set of new stationery. Ask them to list the assumptions, work out the costs and say what they would change if the budget dropped.
Compare two plans. Taxi fares, some phone tariffs and gym memberships with a joining fee combine a fixed charge with a charge per mile, minute or month. Ask when each plan is cheaper and how they know.
Be a good coach when they’re stuck. Ask questions instead of showing the method: “What do you know? What could you draw? What if the numbers were smaller?”. Praise the strategy as well as the answer.

Leads on to

Key Stage 4: Fluency in number and algebra
Key Stage 4: Reasoning and proof in maths
Key Stage 4: Problem solving and modelling
Key Stage 4: Surds, indices and proof (higher)