100 topics · Read, check sources and follow your curiosity.

A little room to wonder

Mathematics

Find patterns, represent ideas and explain why a result should hold.

How could you convince someone that your answer works?

Four coloured six-sided dice showing different arrangements of dots.
Dice offer a way to compare possible outcomes with the assumptions behind a probability model.Diacritica, via Wikimedia Commons. CC BY-SA 3.0. · Source · CC BY-SA 3.0

An answer has a reason

Mathematics studies relationships in number, shape, change and uncertainty. Calculating an answer is one part of it. Another is explaining the method, checking assumptions and deciding whether the result fits the question.

A diagram, a table, words and an equation can describe the same relationship. Switching between them can reveal something you missed. If one representation becomes confusing, another may give you a useful starting point.

Examples suggest; reasoning explains

A pattern that works for several examples may suggest a conjecture: a claim you think could be true. It is not yet proof. A single counterexample is enough to show that a claim about every case is false.

For example, “every odd number is prime” fails at 9, because 9 = 3 × 3. Finding that counterexample improves the investigation. You can then ask a more precise question instead of defending the original guess.

Make assumptions visible

For two independent fair six-sided dice, there are 36 equally likely ordered outcomes. A total of 7 has six of those outcomes, while a total of 2 has one. The totals are not equally likely just because each is a possible total.

A model is useful within its assumptions. Real dice might be uneven, and a small set of rolls need not match the theoretical proportions exactly. Keep the calculation, the observations and the limits of the model separate.

Only if you feel like trying

Build a possibility table

  1. Draw a six-by-six table. Label each edge with the numbers 1 to 6.
  2. Write the sum of the row and column numbers in each square.
  3. Count the squares for each possible total. Explain why 7 and 2 have different probabilities under the fair-dice model.

Optional paper activity. You do not need dice, a gambling game or any money.