Number Sense and Operations
Domain overview
Number Sense and Operations on the MTLE Mathematics (Grades 5–12) exam tests how flexibly you move among fractions, decimals, percents, integers, exponents, radicals, and the real and complex number systems. Expect items that require both procedural fluency and number-sense reasoning: estimating, comparing, and choosing efficient representations.
Core rules
- Place value and scientific notation: multiplying by 10^k shifts the decimal; 0.0042 = 4.2 × 10^-3.
- Fraction division: a/b ÷ c/d = a/b × d/c; keep the dividend and multiply by the reciprocal.
- Integer exponents: x^-n = 1/x^n; x^(m/n) = (n√x)^m for x ≥ 0 when n is even.
- Rational vs. irrational: a rational number can be written p/q with integers p, q and q ≠ 0; √2 and π are irrational.
- Proportional reasoning: percent change = (new − old)/old × 100%; distinguish 'percent of' from 'percent more than'.
- Number theory: GCF × LCM = product for two positive integers.
Worked example
Evaluate 27^(2/3) − 16^(1/4). Rewrite 27 as 3^3: 27^(2/3) = (3^3)^(2/3) = 3^2 = 9. Rewrite 16 as 2^4: 16^(1/4) = 2. So 9 − 2 = 7. The reasoning is to convert to a common base before applying the power rule, rather than multiplying 27 by 2/3.
Traps
- Applying a negative exponent to only the numerator.
- Treating (a + b)^2 as a^2 + b^2.
- Confusing percent decrease with the percent of the original remaining.
- Assuming √(a + b) = √a + √b.
- Forgetting that division by zero is undefined, not zero.