Problem-Solving Approach

Some OCR GCSE Mathematics (J560) questions tell you exactly what to do; others drop you into a situation and leave you to work out the route yourself. These multi-step, "no signposted method" questions are the heart of AO3 — solving problems within mathematics and in real contexts — and on Higher tier they carry, together with AO2, the majority of the marks. The good news: problem-solving is not a mysterious talent. It is a repeatable process — decode, plan, execute, check — that you can practise until it is second nature. This lesson gives you that process and applies it to "show that" problems, unit tracking and checking.

This is the AO3 lesson above all. Planning a multi-step solution is AO3; the reasoning and "show that" work within it is AO2; and the individual calculations along the way are AO1. If problem-solving feels like the hardest part of the exam, that is normal — and it is exactly why having a process matters: when the path is not obvious, you fall back on the routine rather than on inspiration.

A Reliable Problem-Solving Process

When a question has no obvious method, work through these four stages rather than freezing or guessing.

flowchart TD
  A[Read the whole question] --> B[Decode: what is given? what is asked?]
  B --> C[Plan: what must I find first? what method links them?]
  C --> D[Execute: carry out each step, showing working]
  D --> E[Check: units, magnitude, does it answer the question?]
  E --> F{Sensible?}
  F -->|Yes| G[State the final answer in the form asked]
  F -->|No| C

Stage	What you actually do
Decode	Underline the numbers and the quantity asked for; note the units; re-read so you answer that question
Plan	Identify the missing middle — what you need to find before you can find the goal — and the method linking them
Execute	Work step by step, one line each, keeping full accuracy and showing working for method marks
Check	Verify units, sign and magnitude; confirm you answered what was asked; estimate to sanity-check

The reason a written process helps is that it stops the panic of "I don't know where to start". Under exam pressure, an unfamiliar question can feel impossible — but almost every problem yields to the same four moves, and simply starting the decode step (underlining what is given and what is asked) usually reveals the first calculation. You rarely need to see the whole solution before you begin; you need only see the next step. Find the next step, take it, and the step after that becomes visible. This is why writing down a first relevant line — even on a question you cannot yet fully see — is both a mark-earning habit and a way to unstick your thinking.

Decoding the Question

Most problem-solving marks are lost at the very first stage — answering a slightly different question from the one set. Slow down and ask three things:

1. What am I actually asked to find? (The final quantity, and its units.)
2. What am I given? (Every number, and what it represents.)
3. What is the hidden middle step? (What must I work out before I can reach the goal?)

Spending a few extra seconds on this decoding stage is almost always worth it. The most expensive mistakes in problem-solving are not arithmetic slips — they are solving the wrong problem entirely, which wastes all the working that follows. A clear answer to "what exactly is asked?" and "what must I find first?" prevents that whole category of error before it starts.

Worked Example: spotting the hidden step

A rectangular lawn measures $12$12 m by $8$8 m. Turf costs £4.50 per square metre. Work out the total cost of turfing the lawn.

• Asked for: total cost, in pounds.
• Given: dimensions $12$12 m and $8$8 m; price £4.50 per m².
• Hidden middle step: you cannot multiply a length by a price — you first need the area.

Plan: area $\to$→ cost. Area $= 12 \times 8 = 96\text{ m}^{2}$=12×8=96 m2. Cost $= 96 \times 4.50 = 432$=96×4.50=432, i.e. £432. Answer: £432. The whole question hinges on realising the area must come first.

Planning Multi-Step Problems

Once decoded, sketch the chain of steps before you start calculating. A short plan like "find area $\to$→ find scale factor $\to$→ multiply" keeps you from getting lost halfway and makes your working easy to mark.

Worked Example: a three-step problem

A recipe for 4 people needs $600$600 g of flour. Jaya is cooking for 10 people. Flour is sold in $1.5$1.5 kg bags. How many bags must she buy?

• Plan: find flour per person $\to$→ flour for 10 $\to$→ number of bags (rounding up).
• Step 1: per person $= 600 \div 4 = 150$=600÷4=150 g.
• Step 2: for 10 people $= 150 \times 10 = 1500$=150×10=1500 g $= 1.5$=1.5 kg.
• Step 3: each bag is $1.5$1.5 kg, so she needs $1.5 \div 1.5 = 1$1.5÷1.5=1 bag.

Answer: 1 bag. Here the rounding happens to be exact; if she had needed $1.6$1.6 kg she would round up to 2 bags, because you cannot buy part of a bag. Deciding which way to round in context is a classic AO3 judgement.

"Show That" Problems

"Show that" questions are a special problem-solving type: the destination is given, and your job is to build the road to it. Because the answer is printed, you cannot fudge it — every step must be genuine and lead exactly to the stated result.

Worked Example

A number $n$n is even. Show that $n^{2} + 2n$n2+2n is always a multiple of 4.

Since $n$n is even, write $n = 2k$n=2k for some integer $k$k. Then:

$n^{2} + 2n = (2k)^{2} + 2(2k) = 4k^{2} + 4k = 4(k^{2} + k).$n2+2n=(2k)2+2(2k)=4k2+4k=4(k2+k).

Because $k^{2} + k$k2+k is an integer, $4(k^{2} + k)$4(k2+k) is a multiple of 4 for every even $n$n, as required. The marks are for the substitution ($n = 2k$n=2k), the algebra, and the factor of 4 made explicit — present all three.

A general tip for "show that" and "prove": represent the general object algebraically before you start. "An even number" becomes $2k$2k; "an odd number" becomes $2k+1$2k+1; "three consecutive integers" become $n$n, $n+1$n+1, $n+2$n+2; "a multiple of 5" becomes $5m$5m. Once the general form is on the page, the algebra usually does the work, and the factor or property you need (like the $4$4 above) falls out. Trying to argue from a single numerical example is not a proof — one example shows the statement is plausible, but the marks require an argument that holds for every case.

Working Backwards

Sometimes the cleanest plan is to work from the answer towards the start. "Reverse" or "inverse" problems give you the result of a process and ask for the input.

Worked Example: reversing a calculation

A number is doubled, then 5 is added, giving 23. Work out the original number.

Reverse each step in turn: undo "add 5" by subtracting 5 ($23 - 5 = 18$23−5=18), then undo "double" by halving ($18 \div 2 = 9$18÷2=9). Answer: 9. (Check forwards: $9 \times 2 = 18$9×2=18, $18 + 5 = 23$18+5=23 — correct.) Recognising that a problem can be unwound from its result is a powerful planning move — and it gives you a built-in check by running the steps forwards.

Tracking Units