Mathematical Thinking and Reasoning
This domain tests multi-step problem solving and modeling: translating a real situation into mathematics, choosing a representation, carrying out a plan, and justifying the result. Questions often combine two domains, such as a linear cost model with an inequality or a volume calculation with a rate.
Unpack the situation
Identify the quantities, their units, and the question being asked. Assign variables with meaning, including units. Decide whether the relationship is linear, an inequality, a system, a geometric measurement, or a rate. Write an equation or inequality before computing; this keeps the final answer tied to the question.
Use multiple representations
Tables, graphs, equations, and verbal descriptions should agree. A graph can show when one plan becomes cheaper; a table can check discrete values; an equation can give an exact break-even point. When comparing models, compare the same quantity at the same input.
Reason and justify
Explain why a step is valid, not just what you computed. Check whether the answer is reasonable in context: a negative time, a fractional number of people, or a length that exceeds the container may signal an error. Round only at the end, and state units. For “at least” and “at most” constraints, confirm the inequality direction.
Worked example
Gym A charges $25 to join plus $10 per month. Gym B charges $55 to join plus $5 per month. After how many months is the total cost the same? Let m be the number of months. Write 25 + 10m = 55 + 5m. Subtract 5m: 25 + 5m = 55. Subtract 25: 5m = 30. Divide: m = 6. Check: Gym A costs 25 + 10(6) = $85; Gym B costs 55 + 5(6) = $85. The answer is 6 months; interpret that for fewer than 6 months Gym A costs less, and for more than 6 months Gym B costs less.
Traps that cost marks
- Answering the value of the variable instead of the quantity the question asks for.
- Mixing units, such as minutes and hours, before computing.
- Rounding intermediate values and changing the final answer.
- Reversing an inequality when the context says “at least” or “no more than.”
- Giving a solution that is mathematically valid but impossible in context.
- Failing to check the solution in the original equation or model.