Mathematical Processes and Number Sense
What to know
Number sense on this exam means fluency with the real number system and the operations on it, plus the reasoning habits that let you check whether an answer is plausible. Be solid on place value and scientific notation, the hierarchy of operations, integer and rational arithmetic, factors/multiples, primes, GCF and LCM, divisibility rules, and the distinction between rational and irrational numbers. Also expect proportional reasoning: unit rates, percent change, and scaling.
Traps to watch
- Order of operations with negatives and exponents: $-3^2 = -9$ but $(-3)^2 = 9$. Work inside grouping symbols first, then exponents, then multiply/divide left to right, then add/subtract left to right.
- Percent increase vs. decrease are not symmetric: a 20% increase then a 20% decrease does not return to the original value ($1.2 \times 0.8 = 0.96$).
- Absolute value and distance: $|a-b|$ is distance, always nonnegative.
- Rational vs. irrational: a decimal that repeats is rational; a non-repeating, non-terminating decimal is irrational. $\sqrt{9}$ is rational (3); $\sqrt{10}$ is irrational.
- LCM vs. GCF confusion in word problems about repeating events versus dividing into groups.
Worked example
Find the least number of minutes until two bells ring together if one rings every 18 minutes and the other every 24 minutes. This is an LCM problem. Prime factorizations: $18 = 2 \cdot 3^2$, $24 = 2^3 \cdot 3$. Take the highest power of each prime: $2^3 \cdot 3^2 = 72$ minutes. Rule of thumb: together again → LCM; largest equal groups → GCF.