Problem Solving and Mathematical Reasoning
Core ideas
This domain rewards strategy over memorization. Develop the habits of reading a problem twice, restating it in your own words, identifying what is given and what is asked, and choosing a representation (equation, diagram, table, small cases). Use techniques such as working backward, case analysis, symmetry, invariants, and extreme cases. Justify each step so a reader can follow the logic, and check answers for reasonableness.
Traps to watch
- Answering a related question instead of the one asked (e.g., finding x when 2x + 1 was requested).
- Missing cases, especially with absolute values, signs, or "at least" conditions.
- Trusting a pattern from too few examples; test a fourth case.
- Arithmetic slips under time pressure; estimate to catch them.
- Overcomplicating: many problems yield to a small substitution or symmetry argument.
Worked example
If a clock shows 3:00, at what time do the hands first overlap after 3:00? The minute hand moves 6 degrees per minute, the hour hand 0.5 degrees per minute; at 3:00 the gap is 90 degrees. Closing speed is 5.5 degrees per minute, so time = 90/5.5 = 180/11 minutes, about 16 minutes 22 seconds, i.e., roughly 3:16:22. Setting up a relative-rate equation is cleaner than guessing.
Rule of thumb
When stuck, try a smaller version of the problem, look for symmetry, or work backward from the desired answer. Verify the final result against the original wording.