A toolbox of strategies
Not every problem can be solved the same way. Good problem solvers have a toolbox of strategies and choose the right one.
Pólya’s four steps
Mathematician George Pólya described problem solving in four steps:
- Understand the problem.
- Plan: choose a strategy.
- Carry out the plan.
- Look back: check, and think about what you learned.
The strategies
| Strategy | Use when… |
|---|---|
| Draw a diagram | The problem involves shapes, positions or movement |
| Make a table or list | There are several cases or combinations |
| Look for a pattern | Numbers grow or repeat |
| Guess and check | A few tries can narrow down the answer |
| Work backwards | You know the end result, not the start |
| Solve a simpler problem | The numbers are big or the problem is complex |
| Write an equation | Relationships between quantities are clear |
| Act it out / use objects | It helps to see the situation |
| Use logical reasoning | Clues eliminate possibilities |
Examples
Work backwards: Mia spent half her money, then $5 more, and has $10 left. How much did she start with?
- Undo the $5: 10 + 5 = 15
- Undo the half: 15 × 2 = $30
Guess and check: Two numbers add to 20 and multiply to 96.
- 10 and 10 → 100 (too big)
- 12 and 8 → 96 ✓
Look for a pattern: How many handshakes if 10 people each shake hands once?
| People | 2 | 3 | 4 | 5 |
|---|---|---|---|---|
| Handshakes | 1 | 3 | 6 | 10 |
Each new person shakes hands with everyone already there: add 4, then 5, … For 10 people: 1 + 2 + … + 9 = 45.
Simpler problem: What is the sum of 1 to 100? Try 1 to 10: pair 1 + 10, 2 + 9, … five pairs of 11 = 55. For 1 to 100: 50 pairs of 101 = 5,050.
Choosing a strategy
Ask:
- Do I know the end and need the start? → Work backwards.
- Are there lots of combinations? → Make a list.
- Is there a sequence? → Look for a pattern.
- Can I describe it with algebra? → Write an equation.
Often, more than one strategy works. Try another if you get stuck.
Worked examples
Example 1: A farmer has chickens and cows: 20 heads, 56 legs. How many of each?
Guess and check (or equation): 12 chickens (24 legs) and 8 cows (32 legs) → 12 chickens, 8 cows.
Example 2: A number is doubled, then 7 is added, giving 31. Find it.
Work backwards: 31 − 7 = 24; 24 ÷ 2 = 12.
Example 3: How many squares on a 3 × 3 grid (all sizes)?
List: 9 small + 4 medium + 1 large = 14.
Common mistakes
- Starting to calculate before understanding the question.
- Sticking with a strategy that isn’t working.
- Skipping the "look back" step.
Try it yourself
- Two numbers add to 15 and differ by 3. Find them.
- Ali has some stickers. He gives away 8, then half the rest, and has 6 left. How many did he start with?
- How many diagonals does a hexagon have? (Hint: try a square and pentagon first.)
Answers: 1) 9 and 6 2) 20 3) 9
Quick recap
- Follow Pólya‘s steps: understand, plan, carry out, look back.
- Choose a strategy to fit the problem.
- Switch strategies when stuck.