NGSA Mathematics common mistakes
NGSA Mathematics · Understand the traps and how to avoid them.
When multiplying or dividing by a negative number while solving an inequality, the candidate keeps the inequality sign unchanged (e.g., solving -2x > 8 and writing x > -4).
The candidate memorizes the addition/subtraction and multiplication properties of equality and transfers them to inequalities without tracking the direction-reversing rule, which has no analogue in equations.
What to do instead
Before solving, mark every step where a negative multiplier or divisor is applied with a small arrow pointing left. Then test a value: substitute x = -5 into -2x > 8 to see whether it satisfies the original inequality, confirming the correct direction. Practice five mixed inequalities per session until the arrow becomes automatic.
The candidate adds fractions by adding numerators and denominators separately (1/2 + 1/3 written as 2/5) or adds mixed numbers without carrying the fractional part.
Whole-number addition is additive over both components, and the candidate overgeneralizes that pattern to fractions whose denominators represent different-sized units.
What to do instead
Draw a fraction bar model or use a common-denominator ladder before computing. For each addition, explicitly write the equivalent fractions with a shared denominator (3/6 + 2/6) and only then combine numerators. Check the answer is larger than each addend when both are positive.
The candidate confuses area and perimeter formulas, using 2(l + w) when asked for area, or applies the circle circumference formula when the area is requested.
Both pairs are taught together, share the same dimensions, and are recalled as a single verbal cluster; the candidate retrieves the first formula that surfaces rather than selecting by the question's units and wording.
What to do instead
Before computing, write the requested quantity and its unit (square units for area, linear units for perimeter). Match the formula to the unit, not to the shape name. Keep a one-line reference card and do three identification-only drills where the candidate names the formula without calculating.
The candidate computes a mean, median, or range from a frequency table by averaging the listed values rather than weighting each value by its frequency.
The candidate reads the table as a plain list of numbers and does not treat the frequency column as a multiplier, a skill distinct from computing statistics from raw data.
What to do instead
Rewrite the table as an expanded list (or use a value-times-frequency column) before calculating. For the median, cumulate frequencies to locate the middle position. Always state the total count n explicitly, since the denominator must equal the sum of frequencies.
The candidate answers a multi-step word problem using only the first numbers encountered, ignoring the final question asked (e.g., computing total cost when the question asks for change from a payment).
Working memory is consumed by the arithmetic, and the candidate stops at the first intermediate result that looks like a plausible answer instead of mapping operations to the actual target quantity.
What to do instead
Underline the question sentence and write a one-line plan naming the unknown before calculating. After computing, restate the answer as a full sentence with units and check it against the question. Use a two-column layout: plan on the left, computation on the right.
The candidate mishandles place value when multiplying or dividing decimals, misplacing the decimal point by one or more positions (2.5 times 0.4 answered as 1.0 instead of 1.00, or 0.36 divided by 0.4 answered as 9 rather than 0.9).
The candidate applies a memorized 'count the decimal places' rule without estimating the magnitude, so an off-by-one placement error goes undetected.
What to do instead
Estimate first: 2.5 times 0.4 is about 2.5 times 0.5, roughly 1.25, so the answer must be near 1. For division, convert to an equivalent problem with a whole-number divisor (3.6 divided by 4) and check the quotient against the estimate. Compare the final answer's magnitude to the estimate before writing it down.
The candidate solves for x but substitutes back into the transformed equation rather than the original, or omits the check entirely, so extraneous or sign-error solutions survive.
Once a value is found, the candidate treats the task as complete and sees substitution as redundant busywork, especially under time pressure.
What to do instead
Make substitution into the original equation a mandatory final line of every solution, written out explicitly. If both sides do not match, return to the first step where signs or distribution changed. Track this in a log to reveal whether errors cluster at distribution, sign changes, or division.
The candidate computes the probability of a compound event by adding individual probabilities when the events are independent (P(A and B) written as P(A) + P(B)), or fails to adjust for sampling without replacement.
The candidate conflates the 'or' and 'and' rules and does not distinguish independent from dependent draws, because both scenarios use similar spinner or bag wording.
What to do instead
Before computing, write whether the question asks for 'and' or 'or' and whether the second event depends on the first. For 'and' with independence, multiply; for 'or' with mutually exclusive outcomes, add. For draws without replacement, reduce the denominator on the second draw and verify the result is between 0 and 1.
The candidate measures or draws angles with a protractor using the wrong scale, reading 130 degrees as 50 degrees, or aligns the baseline with the wrong mark.
Protractors carry two numbered scales, and the candidate picks the one that happens to sit under the ray without checking whether the angle is acute or obtuse.
What to do instead
Classify the angle as acute or obtuse before reading, then select the scale consistent with that classification. Place the center hole exactly on the vertex and the zero line on one ray, and read where the second ray crosses the same scale. Verify by checking the reading against the visual estimate.
The candidate treats a proportional relationship as additive, extending a table by adding a constant difference instead of multiplying by a constant ratio (e.g., 3 items cost 6, so 6 items cost 9).
Additive reasoning is established earlier and is intuitive, while multiplicative reasoning requires recognizing a constant ratio, which the candidate has not yet automatized.
What to do instead
For each table, test the ratio y/x across rows; if it is constant, use multiplication. Write the unit rate explicitly (cost per item) and multiply by the new quantity. Contrast a paired additive and multiplicative table side by side so the candidate can distinguish them by testing the ratio.