Why do a mathematical inquiry?
In class you usually answer questions someone else wrote. In an inquiry, you choose the question and find the answer yourself.
- You see how maths explains real things (sport, money, games, nature).
- You practise reasoning, using data and explaining clearly.
- You learn the way real mathematicians and scientists work.
The procedure is a cycle of six stations: Question → Read → Plan → Do → Analyse → Reflect.
Choosing a good topic and question
A good question is:
- Interesting to you (a hobby, a game, a daily puzzle).
- Clear and narrow: not "Is maths in sport?" but "Does the angle of a basketball shot change the chance of scoring?"
- Possible with your time, tools and data.
- Mathematical: you can measure, count, model or prove something.
Write the question in one sentence, and guess an answer first (a hypothesis).
Methods of inquiry
- Literature research (reviewing prior studies): read books, articles and trusted websites about the topic. Record author, title, year and link. Summarise in your own words.
- Case study: look deeply at one real example, such as how a mobile tariff is calculated, or the maths in a famous building.
- Mathematical experiment: change one thing, measure the result many times, and look for a pattern (dice, paper folding, pendulum, simulations on a computer).
- Development research: design and build something (a game, a puzzle, a spreadsheet tool, an app), then test and improve it.
Carrying out the plan and analysing data
Follow your plan and keep a log: date, what you did, data, problems. Repeat measurements to reduce chance errors.
Then use maths: tables, graphs, mean and percentages, formulas, a model, or a proof. In the dice experiment, the theory says a sum of 7 has probability 6/36 = 1/6 ≈ 16.7%. With 60 rolls you expect about 10 sevens. Your real count may differ; with more rolls it usually gets closer.
Reflecting, evaluating and reporting
Ask: Did I answer the question? Was my method fair? What were the limits (too few trials, measuring errors)? What would I change? What new question appeared?
A report usually has: question and reason, prior studies, method, results (with graphs), analysis, conclusion, reflection, and a list of sources.
Using AI tools honestly
AI and computer tools can simulate thousands of dice rolls, draw graphs, find patterns in data or check algebra. Rules:
- Say which tool you used and for what.
- Check AI answers yourself; they can be wrong.
- The question, the reasoning and the conclusion must be your own work.
Try it
Roll two dice (or use the 3D) 36 times and tally the sums. Which sum came most? Compare with the theory. Then write one new question this raised.
Key formulas and definitions
- Inquiry cycle: Question → Read → Plan → Do → Analyse → Reflect
- Hypothesis: your guess before testing
- P(sum = 7 with two dice) = 6/36 = 1/6 ≈ 16.7%
- Expected count = probability × number of trials
- Experimental probability = times it happened ÷ total trials
- Relative error = |experiment − theory| ÷ theory
Worked examples
1. Improve the question: "Is there maths in music?"
Too wide. Better: "How do the lengths of guitar strings for the notes of one octave compare?" It is narrow, measurable and mathematical.
2. You roll two dice 60 times. How many sevens do you expect?
Expected = 1/6 × 60 = 10 sevens.
3. In 60 rolls you got 13 sevens. Find the experimental probability and the relative error from theory.
Experimental = 13/60 ≈ 0.217 (21.7%). Theory = 1/6 ≈ 0.167. Relative error = |0.217 − 0.167| ÷ 0.167 ≈ 0.30, i.e. 30%. More rolls should reduce it.
4. Which method fits: "Design a board game where every player has a fair chance"?
Development research: build the game, test it many times, measure win rates, and improve the rules.
5. You have 200 rolls; sum 2 appeared 7 times. Compare with theory.
P(2) = 1/36. Expected = 200/36 ≈ 5.6. Observed 7 is close; the difference is due to chance.
6. Order the report parts.
Question → prior studies → method → results → analysis → conclusion → reflection → sources.
Common mistakes
- Choosing a question that is too broad to answer.
- Doing too few trials and trusting the result.
- Copying text from sources or AI without citing it.
- Skipping reflection: not saying the limits of the method.