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Learn Index · Mathematical Thinking and Problem Solving Grade 9–10

Mathematical investigations

Open questions where you collect data, look for patterns and explain what you find.

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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

  1. State the question.
  2. Show your data and diagrams.
  3. Describe patterns and your conjecture.
  4. Test and justify it.
  5. State the general rule.
  6. 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

  1. Continue the handshake table for 7 and 8 people.
  2. Investigate: how many diagonals does a hexagon have? Use n(n − 3)/2.
  3. What is the sum of the first 10 odd numbers? Make a conjecture first using smaller cases.
  4. 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.