TSIA2 Math common mistakes
TSIA2 Math · Understand the traps and how to avoid them.
Comparing two rates or prices by subtracting ("Store A is $3 cheaper per item, Store B is only $2 cheaper, so A is the better deal") instead of building a common per-unit rate before deciding which is larger.
Everyday shopping language encourages additive comparison, and the numbers in the problem are already in familiar dollar amounts, so the natural move is to subtract the two totals rather than divide each by its quantity.
What to do instead
Force a rate before comparing: write both quantities as a fraction with the same denominator (price per 1 item, or items per 1 dollar), reduce both, then compare the single numbers. If the two fractions do not share a denominator, cross-multiply rather than subtract.
Chaining percent changes by adding or subtracting the percents (treating "20% off, then an extra 10% off" as 30% off), or applying a percent increase by multiplying by the percent itself (0.15 instead of 1.15).
The word "off" reads as a subtraction, so the percents get treated as amounts that can be combined. The multiplier form of a percent change is a formula step that is easy to skip when working quickly.
What to do instead
Always name the base first ("percent of what?"), then write new = old x (1 + r) or old x (1 - r). For stacked changes, multiply the multipliers (0.80 x 0.90 = 0.72, i.e., 28% off) and never add them.
Setting up every proportion as a direct proportion, e.g., writing 3 workers / 4 days = 5 workers / x days and cross-multiplying, when more workers means fewer days.
Cross-multiplication is drilled as a universal tool, and the setup looks structurally identical whether the relationship is direct or inverse. The reasoning step about which way the quantities move is skipped.
What to do instead
Before writing the proportion, do a one-sentence sanity check: "if this quantity goes up, does the other go up or down?" If it goes down, invert one side (3 x 4 = 5 x x) or reason in total work units (worker-days) instead of a ratio.
Distributing a coefficient or a minus sign to only the first term inside parentheses: writing 3(x + 2) = 3x + 2, or -(x - 5) = -x - 5.
The eye locks onto the first term after the parenthesis, and the multiplication gets performed once and then abandoned. With a leading minus sign the invisible coefficient of 1 makes the second term feel unaffected.
What to do instead
Draw an arrow from the outside factor to each term inside before doing any arithmetic, and write the intermediate line with both products visible. Then check numerically by substituting x = 1 into the original and simplified forms; unequal results expose the missed term instantly.
Solving a linear inequality through a division or multiplication by a negative and leaving the inequality symbol pointing the original way.
The flip rule is the one step of inequality solving that has no analogue in equation solving, so the muscle memory built from equations overrides it. It is also a silent error: the algebra looks finished.
What to do instead
Circle any negative multiplier or divisor the moment it appears and write "FLIP" next to it. After getting a solution, test one value from your answer interval in the original inequality; if the test fails, the symbol is backwards.
Computing slope from two points with the coordinates subtracted in inconsistent order, e.g., (y2 - y1) over (x1 - x2), producing a sign error, then carrying that wrong slope into the line equation.
The formula is recalled as a shape rather than as a rule about matching order, and under time pressure the second coordinate pair is copied in whatever order it sits on the page.
What to do instead
Label the points (x1, y1) and (x2, y2) in writing before substituting, keep the same point first in both numerator and denominator, and finish by checking that both original points satisfy the final equation.
Solving a system or a multi-step equation correctly for an intermediate variable and then selecting that number as the answer, when the question asked for the other variable or for a combined quantity.
Getting a clean number produces a strong feeling of completion, and the question stem was read before the algebraic work rather than after it.
What to do instead
Underline the exact quantity requested in the stem at the start, and box it again after solving. Before choosing an answer, restate the request as a question ("they want y, not x") and substitute the found value back to get the requested one.
Substituting the diameter into a circle formula where the radius belongs, or computing circumference when the problem asks for area (or the reverse).
Both formulas contain the same constants and get stored as one blurred memory, and a diameter is often the number actually printed in the problem, so it gets used without the halving step.
What to do instead
Write the needed formula from scratch before touching the numbers, then write "r = d/2 = ___" as its own line. Track units as a check: area answers carry square units, perimeter and circumference answers carry linear units.
Scaling the area or volume of a similar figure by the same factor as its side lengths (doubling the sides and doubling the area), instead of using the square or cube of the scale factor.
The scale factor is the only number explicitly given, so it feels like the multiplier for everything. Area and volume are computed last, after the ratio has already been mentally fixed as "the" factor.
What to do instead
Set up a three-row table (length : area : volume) and fill it with k : k^2 : k^3 before computing anything. Sanity-check with a simple case: a 1-by-1 square doubled to 2-by-2 visibly has four times the area, not two.
Reading a conditional probability from a two-way table as a joint probability, dividing by the grand total when the question says "of the students who play a sport, what fraction also..."
The grand total sits at the bottom-right corner and is the most visually prominent number in the table, and the conditional phrase is buried in the middle of the sentence.
What to do instead
Before dividing, circle the group named after "given" or "of those" and use that row or column total as the denominator. Write the fraction as (cell / circled total) so the denominator is chosen deliberately rather than by habit.