Mathematical Processes and Reasoning - Ngā Tukanga Pāngarau
Mathematical processes and reasoning reward clear thinking, not just a correct number. You need to select a strategy, show the steps, justify choices and check that the answer fits the context.
Core rules and concepts
- Use a problem-solving cycle: understand the question, plan a strategy, carry it out, then check reasonableness. Identify what is given, what is unknown and what units apply.
- Representations help: tables for patterns, diagrams for geometry, equations for relationships, and estimates to sense-check.
- Show working line by line, keep units attached, and finish with a sentence answering the actual question.
- Interpret remainders according to context: dividing people into vans means rounding up; dividing money into whole items means rounding down.
- Justify with evidence: state the rule or calculation used, and identify assumptions such as constant speed or no extra fees.
- Check by substitution, inverse operations or estimation.
Worked example
A school hires a bus for a fixed $450 plus $12 per student. The budget is $1,200. What is the greatest number of students who can go? Set up 450 + 12s ≤ 1200. Subtract 450: 12s ≤ 750. Divide by 12: s ≤ 62.5. Students must be whole people, so the greatest number is 62, not 63. Check: 62 × 12 + 450 = 744 + 450 = $1,194, within budget. For 63 students the cost is $1,206, over budget.
Traps
- Giving the raw calculator answer 62.5 when the context requires a whole number.
- Rounding up when the context needs rounding down.
- Omitting the concluding sentence or the units.
- Not checking whether the answer is plausible.
- Assuming information that is not stated.