Maths as discovery
A mathematical investigation starts with a question and explores it, without a single fixed method. It’s how mathematicians actually work: experimenting, spotting patterns, making conjectures and proving them.
The investigation cycle
| Stage | What you do |
|---|---|
| 1. Question | Understand what you’re exploring. |
| 2. Explore | Try specific cases, systematically. |
| 3. Organize | Record results in tables or diagrams. |
| 4. Notice | Look for patterns. |
| 5. Conjecture | Predict a rule. |
| 6. Test | Check the rule on new cases. |
| 7. Generalize | Write the rule algebraically and explain why it works. |
| 8. Extend | Ask a new question. |
Example investigation: handshakes
Question: If everyone in a room shakes hands with everyone else exactly once, how many handshakes are there?
Explore and organize:
| People (n) | Handshakes |
|---|---|
| 2 | 1 |
| 3 | 3 |
| 4 | 6 |
| 5 | 10 |
| 6 | 15 |
Notice: differences 2, 3, 4, 5. These are the triangular numbers.
Conjecture: handshakes = n(n − 1)/2.
Test: n = 10 → 45. Listing confirms.
Explain: Each of n people shakes n − 1 hands, but that counts every handshake twice, so divide by 2.
Extension: How many diagonals does an n-sided polygon have? (Same idea: n(n − 3)/2.)
Example investigation: squares on a grid
Question: How many squares of all sizes are on an n × n grid?
| n | Squares |
|---|---|
| 1 | 1 |
| 2 | 5 |
| 3 | 14 |
| 4 | 30 |
Pattern: 1², 1² + 2², 1² + 2² + 3², … The total is the sum of the first n square numbers.
Being systematic
- Change one variable at a time.
- Start with the simplest cases.
- Use a table to avoid missing or repeating cases.
Writing up an investigation
- State the question.
- Show your data and diagrams.
- Describe patterns and your conjecture.
- Test and justify it.
- State the general rule.
- Suggest an extension.
Worked examples
Example 1: How many line segments join 7 points on a circle? 7 × 6 ÷ 2 = 21.
Example 2: What’s a good first step in any investigation? Try small, simple cases.
Example 3: Squares on a 5 × 5 grid? 30 + 25 = 55.
Common mistakes
- Jumping to a rule after two cases.
- Unorganized data, which hides patterns.
- Stating a rule without explaining why it works.
Try it yourself
- Continue the handshake table for 7 and 8 people.
- Investigate: how many diagonals does a hexagon have? Use n(n − 3)/2.
- What is the sum of the first 10 odd numbers? Make a conjecture first using smaller cases.
- Suggest one extension to the squares-on-a-grid investigation.
Answers: 1) 21, 28 2) 9 3) 100 (the sum of the first n odd numbers is n²) 4) e.g. count rectangles, or use a non-square grid
Quick recap
- An investigation explores an open question.
- Work systematically, organize data and form a conjecture.
- Generalize, justify and extend.