NYS Math common mistakes
NYS Math · Understand the traps and how to avoid them.
Treating a negative exponent as making the base negative, writing 2^(-3) = -8 or (-2)^2 = -4, and applying the negative sign to the base instead of the exponent.
Students memorize 'negative means opposite' and transfer it to the base. The exponent rules for negative exponents and for odd/even powers get compressed into one fuzzy rule, and the sign of the base is visually confused with the sign of the exponent.
What to do instead
Rewrite every negative exponent as a reciprocal first: 2^(-3) = 1/2^3 = 1/8. For powers of a negative base, always parenthesize the base and ask whether the exponent is even or odd before computing. Drill a mixed set where the negative sign sits in different positions: -2^2, (-2)^2, 2^(-2), (-2)^(-2), and write the value of each.
Adding or subtracting radicals and rational expressions by combining non-like parts, e.g. writing sqrt(9) + sqrt(16) = sqrt(25) = 5, or 1/x + 1/y = 1/(x+y).
The distributive habit from integer arithmetic (3 + 4 = 7) is over-generalized to operations that are not linear, like square roots and reciprocals. Students do not test with numbers to see the rule fails.
What to do instead
Before simplifying, substitute easy numbers to check whether the proposed identity holds: sqrt(9) + sqrt(16) = 3 + 4 = 7, not 5. For rational expressions, always find a common denominator. Keep a short 'false identities' list (sqrt(a+b) ≠ sqrt(a)+sqrt(b), 1/a + 1/b ≠ 1/(a+b), (a+b)^2 ≠ a^2 + b^2) and check against it.
Solving a linear equation or inequality by moving terms across the equals sign but forgetting to flip the inequality when multiplying or dividing by a negative, and losing the boundary point as open vs. closed.
The mechanics of solving equations are practiced heavily, so students go on autopilot; the sign-flip rule is a small exception that is easily skipped under time pressure. Open/closed endpoint notation is a separate visual convention that gets dropped.
What to do instead
Every time you divide or multiply an inequality by a negative, write 'FLIP' and reverse the symbol in the same step. Circle the coefficient's sign before dividing. After solving, test one value on each side of the boundary in the original inequality to confirm the direction, and mark open circles for strict inequalities and closed for ≤/≥.
Factoring or expanding quadratics without checking the middle term, e.g. factoring x^2 + 5x + 6 as (x + 6)(x + 1) because the factors multiply to 6, ignoring that they must also add to 5.
Students focus on one condition (product equals the constant) and forget the second condition (sum equals the coefficient of x). The two constraints are stored separately and not applied together.
What to do instead
Write the two target numbers explicitly before factoring: 'multiply to c, add to b.' List factor pairs of c and test the sum. After factoring, expand mentally or on paper to verify the middle term equals b. Practice with c positive and negative, since the sign pattern changes which pairs are candidates.
Misreading function notation, treating f(x + 2) as f(x) + 2, or confusing f(3) with f(x) = 3, and mixing up domain restrictions with range values.
The parentheses in function notation look like multiplication grouping, so students apply algebraic distribution to the input. The distinction between an input value and an output value is abstract and gets blurred when the same letter x is reused.
What to do instead
Read f(x + 2) aloud as 'the output when the input is x + 2' and substitute the whole expression for every x in the rule. For f(3), replace x with 3 everywhere and evaluate to a number. Practice a table of inputs and outputs and label the domain (allowed inputs) and range (resulting outputs) as separate columns.
Applying transformations in the wrong order or direction, e.g. shifting y = f(x) left for f(x - 3) instead of right, or stretching vertically when the coefficient is inside the function.
The inside-versus-outside distinction is counterintuitive: changes inside the parentheses act horizontally and in the opposite direction to the sign shown, while outside changes act vertically in the expected direction. Students rely on intuition rather than tracking which variable is affected.
What to do instead
Classify the change first: inside the function = horizontal, outside = vertical. For inside changes, reverse the sign (f(x - 3) shifts right 3). Test with a single point: take a known point on the parent graph, apply the transformation by hand, and see where it lands. Sketch both graphs to confirm.
Using the wrong trigonometric ratio or forgetting that inverse trig returns an angle, e.g. using sine when the sides given are adjacent and hypotenuse, or reporting sin(theta) = 0.5 as the answer when theta was asked.
SOH-CAH-TOA is memorized as a string without being tied to which sides are involved. Students also lose track of whether the unknown is a side (use a ratio) or an angle (use an inverse ratio).
What to do instead
Label the triangle's sides relative to the angle in question (opposite, adjacent, hypotenuse) before choosing a ratio. Write the ratio equation with the unknown isolated, then decide: unknown side → use sin/cos/tan; unknown angle → use sin⁻¹/cos⁻¹/tan⁻¹. Check that the answer's units match (length vs. degrees).
Confusing surface area with volume formulas and using the wrong dimension, e.g. computing volume of a cylinder as 2πrh, or using diameter where radius belongs in area and volume formulas.
The formulas look similar and are often memorized as symbols without units attached. Students also read 'radius' and 'diameter' as interchangeable when a diagram labels the full width.
What to do instead
Attach units to every formula: area is square units, volume is cubic units. Write the formula from a reference sheet each time and substitute r = d/2 explicitly when diameter is given. Sanity-check magnitude: a volume answer should be much larger than a surface-area answer for the same solid.
Computing a probability as if events were independent when they are not, e.g. P(A and B) = P(A)·P(B) for drawing two cards without replacement, or adding probabilities for overlapping events without subtracting the intersection.
The multiplication rule for independent events is the first one learned and becomes a default. The phrase 'without replacement' changes the sample space, but students do not update the denominator for the second draw.
What to do instead
Before multiplying, ask whether the first event changes the second. If sampling without replacement, adjust the count: P = (a/n)·((a-1)/(n-1)). For 'or' with overlap, use P(A or B) = P(A) + P(B) - P(A and B). Draw a two-way table or tree diagram to see the changing totals.
Reporting the mean as the measure of center for a skewed distribution, or reading a histogram's y-axis as the count of individual data values rather than the frequency of a bin.
Mean is over-practiced, so it is chosen by default even when the distribution is skewed and the median is more appropriate. Histogram axes are read quickly, and bar height is confused with the raw data values rather than the number of observations in an interval.
What to do instead
Check shape first: symmetric → mean and median close; skewed or with outliers → report the median and IQR. For histograms, state what each axis represents before answering: x = interval of values, y = frequency in that interval. Estimate the median by locating the bin containing the middle observation.