Mathematical Reasoning and Problem Solving
Mathematical Reasoning and Problem Solving ties the other domains together. It asks you to solve multi-step problems, choose representations, estimate, and explain why an answer is reasonable. Marks often depend on showing the reasoning, not only the final number.
Unpack the problem
- Identify the question, the known quantities, the unknown, and the units.
- Bar models and strip diagrams show part-whole and comparison relationships.
- Translate words into an equation. “Of” often signals multiplication; “per” often signals a rate.
Multi-step planning
- Break the problem into subgoals and solve them in order, labelling each intermediate result.
- Work backwards when the final result and the operations are known.
- Estimate first with friendly numbers to predict the size of the answer.
Justify and communicate
- State the strategy, show substitutions, and include units.
- Check with an inverse operation or a second estimation.
- Explain using mathematical properties or relationships, not “I did it in my head.”
Worked example
A class has 30 students. Two-thirds go on a trip, and three-fifths of those bring a lunch. How many bring a lunch?
First find who goes: 30 × 2/3 = 20 students. Then find who brings lunch: 20 × 3/5 = 12 students. Check: 12 is more than half of 20 and less than 20, so it is reasonable. A second method multiplies the fractions first: 2/3 × 3/5 = 6/15 = 2/5, and 30 × 2/5 = 12. Both methods agree.
Traps that cost marks
- Stopping at an intermediate answer, such as the 20 who go on the trip.
- Using the wrong base for a fraction of a quantity.
- Ignoring units or mixing them within one calculation.
- Rounding early and carrying the error into later steps.
- Answering a different question than the one asked.
- Giving a number without the reasoning the problem requires.