Mathematical Processes and Problem Solving
Mathematical Processes and Problem Solving is the reasoning domain: translating a multi-step situation into symbols, selecting a strategy, estimating to test reasonableness, extending patterns, and justifying or critiquing a solution. Items may ask which representation is equivalent or where a given line of reasoning breaks down.
Core rules
- Translate in a fixed order: define the variable in words, write the equation, solve, then answer the question actually asked - often one step past the equation's solution.
- Work rates add. A job finished in a hours alone proceeds at 1/a per hour; together the rates sum and time = 1 ÷ (sum of rates).
- Distance = rate × time. In "meet" problems the times are equal; in "catch-up" problems the distances are equal.
- Consecutive integers are n, n + 1, n + 2; consecutive even or odd integers are n, n + 2, n + 4.
- Before assuming a pattern rule, check first differences, then ratios.
Worked example
Ana paints a room in 6 hours; Ben paints the same room in 4 hours. Working together, how long will it take? Add rates, never times. Ana's rate is 1/6 room per hour and Ben's is 1/4, so combined = 2/12 + 3/12 = 5/12 room per hour. Time for one room = 1 ÷ 5/12 = 12/5 = 2.4 hours, or 2 hours 24 minutes. The result must be smaller than either individual time, which confirms the arithmetic; averaging 6 and 4 to get 5 hours would violate that check.
Traps that cost marks
- Averaging times in a work-rate problem instead of adding rates.
- Following keywords blindly: "5 more than twice a number" is 2x + 5, not 2(x + 5).
- Answering an intermediate value, such as the number of hours already worked, rather than the quantity requested.
- Accepting an answer without a reasonableness check - a painter finishing in 24 hours signals an inverted rate.