Key takeaways
- Use Pólya's four steps: understand, plan, solve, look back.
- Write down what you know and what you need before choosing a formula.
- When stuck, draw a picture, try a simpler case or work backwards.
- Always check your answer in the original problem and ask if it makes sense.
In this guide
What is mathematical problem solving?
Problem solving is what you do when you don't immediately know the answer or the method. It is different from an exercise, where you just repeat a method you already know.
In 1945 the mathematician George Pólya wrote a short book called How to Solve It. He described four steps that good problem solvers follow. Teachers around the world still use them today.
Why do problem-solving strategies matter?
Most students who get stuck do know enough maths. What they lack is a way to start. A clear routine turns a scary problem into a list of small, doable steps.
Most wrong answers come from misreading the question, not from bad arithmetic.
Pólya's four steps
| Step | Key questions to ask yourself |
|---|---|
| 1. Understand | What do I know? What am I asked to find? Are there any conditions? |
| 2. Plan | Have I seen a similar problem? Which formula links what I know to what I need? |
| 3. Solve | Am I following my plan carefully? Can I check each step? |
| 4. Look back | Does the answer fit the question? Are the units right? Is there a faster way? |
How to solve a word problem step by step
- 1Read the problem twice. The first time for the story, the second time for the numbers.
- 2List the knowns and the unknown. Give the unknown a letter, like x.
- 3Draw a sketch or table. Pictures make relationships easy to see.
- 4Write an equation that connects the knowns to the unknown.
- 5Solve it one step at a time.
- 6Check the answer in the original words of the problem.
Equations like 4w + 6 = 30 can be checked with the linear equation solver, and the area with the rectangle area calculator.
Strategies for when you are stuck
- Draw a picture. Especially for geometry, motion and distance problems.
- Try a simpler case. Replace big numbers with small ones, solve that, then look for the pattern.
- Work backwards. Start from the answer you want and ask what step comes just before it.
- Guess and check. Make a sensible guess, test it, and adjust. Two consecutive numbers that add to 45? Try 20 and 21 (41, too small), then 22 and 23 (45).
- Look for a pattern. Write the first few cases in a table.
- Split the problem. Solve one part at a time.
Common mistakes to avoid
- Jumping to a formula too early. Understand the situation first.
- Ignoring units. Mixing minutes and hours, or cm and m, gives wrong answers that look right.
- Not checking. A negative length or a speed of 5,000 km/h for a bicycle is a sign to look again.
- Giving up too soon. Try at least two different strategies before asking for help.
Tips and tricks
- Key words help: 'total' often means add, 'difference' means subtract, 'per' means divide, 'of' often means multiply.
- Right triangles appear everywhere. Ladders, ramps and diagonal distances often need the Pythagorean theorem calculator.
- Lines and rates: 'how steep' or 'how fast does it change' often means slope. Try the slope calculator.
Where you'll use it in real life
- Budgeting: planning how many months to save for a goal.
- Travel: working out when to leave to arrive on time.
- Work: estimating materials, time or cost for a project.
- Everyday decisions: comparing deals, like which phone plan is cheaper for your usage.
Quick summary
Good problem solvers follow the same four steps: understand the problem, make a plan, carry it out carefully, and look back to check. When you are stuck, draw a picture, try a simpler case, work backwards or guess and check. Use the free step-by-step calculators linked above to confirm each part of your working.
Frequently asked questions
What if I don’t know which formula to use?
Write down what you know and what you need to find. Then look for a formula that contains both. Browsing the calculator categories can help you match your values to a formula.
How do I check if my answer is correct?
Put your answer back into the original problem and see if everything fits. Also ask whether it makes sense: a negative length or a huge speed is a warning sign.
What are Pólya’s four steps?
Understand the problem, devise a plan, carry out the plan, and look back to check the answer and the method.
How can I get better at word problems?
Practise turning sentences into equations. Read each problem twice, name the unknown with a letter, and draw a sketch before you calculate.
