SAT Math common mistakes
SAT Math · Understand the traps and how to avoid them.
When clearing fractions in an equation like x/3 - (x-2)/4 = 5, the student multiplies only the terms that visibly contain a fraction by 12, producing 4x - 3(x-2) = 5 instead of 4x - 3(x-2) = 60.
The student has memorized 'multiply by the common denominator' but treats it as an operation on individual fractions rather than on both entire sides of the equation, so non-fraction terms get skipped.
What to do instead
Before multiplying, draw one big parenthesis around each whole side and write the factor in front: 12(x/3 - (x-2)/4) = 12(5). Then distribute to every term, including the standalone number, and count terms afterward to confirm none were missed.
Given a system such as 3x + 2y = 17 and x - 2y = 3, the student solves the long way for x and y and then adds them, and under time pressure stops after finding x and enters that as the answer even though the question asks for x + y.
School habits train students to 'solve the system' as an end in itself, so the target quantity in the prompt is never isolated as a separate step.
What to do instead
Before solving, underline the exact expression requested and circle it in the margin. Then test whether adding or subtracting the two equations immediately produces that expression; if it does, stop there. If not, solve fully but re-read the circled target before entering the answer.
Given a quadratic with roots 3 and -5, the student writes the factored form as (x + 3)(x - 5), and conversely reads (x - 7)(x + 2) as having roots 7 and 2.
Students pattern-match 'root' to 'the number that appears in the factor' and forget that the factor contains the root with the opposite sign.
What to do instead
Verify every factor by substituting the claimed root: if x = 3 is a root, then (3 - 3) must equal 0, which forces the factor to be (x - 3). Do this one-second substitution check on each root before moving on.
The student simplifies (2x^3)^2 as 2x^6 and rewrites 3x^-2 as -3x^2 or as 1/(3x^2), applying the outside exponent or the negative sign to only the variable.
The visible variable dominates attention, so the coefficient and the exponent's sign are processed as decorations rather than as separate factors subject to the same rule.
What to do instead
Expand the power explicitly before simplifying: (2x^3)^2 = (2x^3)(2x^3) = 4x^6. For negative exponents, write the reciprocal first (3x^-2 = 3/x^2) and only then simplify, so the coefficient stays in the numerator.
For 'a population of 8,000 increases by 6% each year,' the student writes P = 8000(0.06)^t or P = 8000(6)^t instead of 8000(1.06)^t, and for decay writes (0.94)^t as 0.06^t.
The percent number is the most salient quantity in the problem, so it gets placed directly into the model without converting it to a growth factor.
What to do instead
Build a two-row table by hand before choosing the model: t = 0 gives 8000, t = 1 gives 8000 + 0.06(8000) = 8480. The base must be 8480/8000 = 1.06, and the table itself reveals any wrong base instantly.
In a two-way table, the student answers 'what fraction of the students who chose math are seniors?' by dividing the cell by the grand total instead of by the math-column total.
The phrase 'of the students who...' is processed as background wording rather than as a signal that the sample space has been restricted to one row or column.
What to do instead
Rewrite the question as a conditional statement: 'given math, probability of senior.' Then circle the group named immediately after 'of the' or 'given' and use that group's total as the denominator every time.
For a price that rises 20% and then falls 20%, the student concludes the price returns to its original value, or computes the net change as 0% instead of -4%.
Percent changes feel additive because the same number, 20, appears twice, and the student reasons about the percentages rather than about the dollar amounts they act on.
What to do instead
Assign the starting value 100 and compute each step numerically: 100 to 120 to 96. Then convert back to a percent change (96 versus 100 means a 4% decrease). Never combine successive percents by adding or subtracting.
In a rate problem such as 'a machine fills 3/4 of a tank in 2/5 of an hour,' the student sets up (3/4)/(2/5) but then multiplies straight across, or inverts the wrong fraction, landing on 15/8 tanks per hour instead of 15/8 checking out wrong because the setup was actually hours per tank.
The student picks an arrangement that 'looks like' a fraction without tracking which quantity is the numerator and which is the denominator in the desired ratio.
What to do instead
Write the answer's units first, e.g., tanks/hour, then build the fraction so the units cancel to that: (3/4 tank)/(2/5 hour) = (3/4)(5/2) = 15/8 tanks per hour. If the units of the computed fraction are not the units requested, invert and redo.
In a right triangle the student uses sine when the known and unknown sides are the two legs, or uses cosine for the side opposite the given angle; separately, the student computes sin(30) on a calculator left in radian mode and records an answer near -0.988 instead of 0.5.
The mnemonic is recalled as a chant without the side labels, and calculator mode is invisible on screen, so a plausible-looking decimal is accepted without a sanity check.
What to do instead
Before writing any equation, mark the hypotenuse and label the two remaining sides 'opposite' or 'adjacent' relative to the angle in the figure. Then choose the ratio only after the labels are on the page, and check that the trig value matches a known reference angle (sin 30 = 0.5) to confirm degree mode.
Converting x^2 + y^2 + 6x - 4y = 12 to standard form, the student reports center (6, -4) and radius 12, reading the numbers straight out of the expanded equation.
The student half-remembers '(x - h)' but does not carry the sign flip or the balancing constant through the completing-the-square step.
What to do instead
Complete the square explicitly and write the final form as (x - h)^2 + (y - k)^2 = r^2 before extracting anything: here (x + 3)^2 + (y - 2)^2 = 25, so the center is (-3, 2) and r = 5. Always take the square root of the right side for the radius, then expand back to check.