What is a problem, and what is a good solver?
An exercise is a question where you already know the method. A problem is a question where you do not yet know the method. You must think, try and find a way. Problems that do not look like anything you have seen before are called non-routine problems.
A good problem solver is not someone who sees the answer at once. A good solver has a plan and does not give up after the first try. This lesson gives you a four-step plan you can use for any problem in maths, science or daily life.
- Understand the problem.
- Make a Plan (choose a strategy).
- Do the plan.
- Look back and check.
Step 1: Understand and analyse the problem
Most mistakes come from solving the wrong problem. Before you calculate, ask:
- What is asked? Underline the question word (how many, how long, which day).
- What is given? List the numbers, units and conditions.
- Is anything missing or extra? Some problems have too little information, some have too much.
- Can I say it in my own words? Tell it to a friend or write one clear sentence.
- Can I see it? A quick sketch often shows the structure.
In the handshake problem: given 6 friends, each pair shakes hands once. Asked: the total number of handshakes. Note the word pair: A shaking hands with B is the same handshake as B with A.
Step 2: Plan with heuristics (strategies)
A heuristic is a helpful strategy that often works but does not promise an answer. Keep this toolbox:
- Draw a picture or diagram: make the problem visible.
- Try small cases: solve the problem for 2, 3, 4 first.
- Make a table or list: organise what you find.
- Find a pattern: look for what repeats or grows by the same rule.
- Guess and check: try a number, see how far off you are, improve it.
- Work backwards: start from the final result and undo each step.
- Make it simpler: remove a condition, or use smaller numbers, then bring it back.
- Break into parts: solve pieces, then join them.
- Think of cases and use symmetry or algebra.
If one strategy fails, that is information, not a failure. Switch to another. This is the non-routine approach: try, look, adapt.
Step 3 and 4: Do it, then look back and reflect
Do the plan neatly. Write each step so that you can find a mistake later. If the plan is not working after a fair try, go back to step 2.
Look back is the step most students skip. It is where you learn the most.
- Check: does the answer fit the question and the units? Is it a sensible size?
- Test another way: count again, put the answer back in the problem, or solve by a second method.
- Generalise: could the method work for other numbers? (6 friends gave 15; what about n friends?)
- Reflect: what worked, what wasted time, what would I try first next time?
Handshakes: the pattern 1, 3, 6, 10, 15 adds 1+2+3+4+5. In general n friends give 1 + 2 + … + (n − 1) = n(n − 1) ÷ 2. Why ÷ 2? Each of n friends shakes n − 1 hands, but every handshake is counted from both sides.
Strategies in action: more worked problems
Work backwards. A number is doubled, then 6 is added, then the result is halved, giving 10. Undo each step from the end: 10 × 2 = 20, 20 − 6 = 14, 14 ÷ 2 = 7.
Guess and check. A farm has hens and cows: 20 heads, 56 legs. Try 10 cows: legs = 40 + 20 = 60, too many. Try 8 cows: 32 + 24 = 56. Correct.
Draw a picture. A snail is at the bottom of a 10 m well. By day it climbs 3 m, by night it slips back 2 m. Draw heights each morning and evening: it reaches 10 m on day 8, before slipping. The trap is to answer 10 days.
Small cases and pattern. How many squares (all sizes) in a 4 × 4 grid? 1×1 grid: 1. 2×2: 4 + 1 = 5. 3×3: 9 + 4 + 1 = 14. 4×4: 16 + 9 + 4 + 1 = 30.
Try it at home
Predict then check: in your class there are 12 students. Predict how many pairs can be made. Use the 3D slider (up to 10 friends) to see the pattern, then use n(n − 1) ÷ 2 for 12. Keep a solver's diary: after each hard problem write one line: "I was stuck when… I tried… it worked because…".
Key formulas and definitions
- Four steps: Understand → Plan → Do → Look back
- Heuristics: draw, small cases, table, pattern, guess and check, work backwards, simpler problem, break into parts
- Handshakes among n people: n(n − 1) ÷ 2
- 1 + 2 + 3 + … + (n − 1) = n(n − 1) ÷ 2
- Squares in an n × n grid: 1² + 2² + … + n²
Worked examples
1. Eight people at a meeting each shake hands once with everyone else. How many handshakes?
Understand: pairs of people. Plan: small cases give 1, 3, 6, 10, 15, 21, so use n(n − 1) ÷ 2. Do: 8 × 7 ÷ 2 = 28. Look back: 1+2+3+4+5+6+7 = 28 as well.
2. I think of a number, double it, add 6, then divide by 2. The result is 10. What is my number?
Work backwards: 10 × 2 = 20; 20 − 6 = 14; 14 ÷ 2 = 7. Check: 7 × 2 = 14, + 6 = 20, ÷ 2 = 10. The number is 7.
3. A farm has hens and cows. There are 20 heads and 56 legs in all. How many cows are there?
Guess and check with a table. 10 cows: 40 + 20 = 60 legs (too many). 8 cows, 12 hens: 32 + 24 = 56 legs. There are 8 cows.
4. A snail is at the bottom of a 10 m well. Each day it climbs 3 m and each night it slips back 2 m. On which day does it get out?
Draw heights. It gains 1 m a day, so each morning it starts at 0, 1, 2, … metres. At the start of day 8 it is at 7 m, climbs 3 m and reaches 10 m, so it is out on day 8, before slipping.
5. How many squares of all sizes are there in a 4 × 4 grid of small squares?
Small cases: 1×1 grid has 1; 2×2 has 4 + 1 = 5; 3×3 has 9 + 4 + 1 = 14. For 4×4: 16 small + 9 (2×2) + 4 (3×3) + 1 (4×4) = 30.
Common mistakes
- Starting to calculate before reading the whole problem. Underline what is asked first.
- Giving up after one failed idea. Switch to another strategy.
- Believing a pattern after only two or three cases. Test it on one more case before trusting it.
- Skipping "Look back". Always put your answer back into the problem to check it.