OST Algebra I common mistakes
OST Algebra I · Understand the traps and how to avoid them.
When simplifying an expression such as (3x^2y^3)^2, the candidate squares the coefficient but multiplies the exponents by 2 only on the first variable, writing 9x^4y^3 instead of 9x^4y^6.
The power-of-a-power rule is recalled as 'multiply the exponents' but is applied to the first factor encountered and then abandoned, so the rule is half-executed. The coefficient is handled separately and correctly, which masks the incomplete exponent work.
What to do instead
Before writing anything, count the factors inside the parentheses and write that count above each one. Then apply the outer exponent to every factor, including the coefficient, in a single left-to-right pass. Check by expanding a small case: (a^2b^3)^2 means a^2b^3 times a^2b^3, which is a^4b^6.
The candidate adds or subtracts rational expressions such as 1/(x+2) + 1/(x-3) by adding numerators and denominators straight across, producing 2/(2x-1).
Fraction addition rules from arithmetic are over-generalized to algebraic fractions. The candidate sees matching structure (two fractions, a plus sign) and reaches for the familiar-looking shortcut without checking whether the denominators are equal.
What to do instead
Write the one-line test before every rational addition: denominators equal or not? If not, find the least common denominator by listing each denominator's factors, rewrite each fraction, then combine numerators only. Verify with x = 1 numerically to see the two forms disagree.
When solving a multi-step linear equation like 3(x - 4) = 2x + 5, the candidate distributes to get 3x - 4 = 2x + 5, dropping the multiplication on the second term.
Attention lands on the variable term inside the parentheses, and the constant is copied through unchanged because it 'looks finished.' The distributive property is known as a phrase but not applied term-by-term.
What to do instead
Draw an arrow from the outside factor to each term inside the parentheses before simplifying, and write the product under each term. Adopt the habit of substituting a number back into the original equation after solving to confirm both sides match.
Given a system of two equations, the candidate solves for x correctly but then substitutes the value back into the equation it just manipulated rather than the other one, or forgets to solve for the second variable and reports a single number as the answer.
The substitution step feels like the finish line, so the second variable is treated as optional. Substituting into the same equation also produces a true-but-useless identity that looks like confirmation.
What to do instead
Treat a system answer as an ordered pair and leave a blank for each coordinate until both are filled. Substitute into the equation that was not rearranged, and check the pair in both original equations. On multiple-choice items, plug the pair into both equations rather than trusting the algebra.
The candidate factors x^2 - 5x - 6 as (x - 3)(x + 2) and moves on without multiplying back, even though the middle term is wrong.
The candidate searches for factor pairs of the constant whose sum matches the middle coefficient but stops at the first pair that multiplies to the constant, skipping the sum check. Sign rules for the constant are also applied loosely.
What to do instead
Always run the two-second expansion check: outer plus inner must equal the middle coefficient. Write the target sum above the constant before listing pairs, so the pair is chosen against both conditions at once rather than sequentially.
When identifying the slope and y-intercept of a line given as 2x + 3y = 12, the candidate reports slope 2 and intercept 12, reading coefficients directly off standard form.
Slope-intercept form is over-applied: any equation with an x-term and a constant is assumed to display m and b in place. The candidate does not isolate y first.
What to do instead
Convert to y = mx + b before naming slope or intercept, showing the division of every term by the y-coefficient. As a guard, note that in standard form Ax + By = C the slope is -A/B, and confirm the converted equation matches that value.
The candidate graphs f(x) = (x - 2)^2 + 3 with the vertex at (-2, 3), shifting left instead of right.
The minus sign inside the parentheses is read literally as 'negative two,' and horizontal shifts are intuitively mapped in the same direction as the sign, which is the opposite of the actual behavior.
What to do instead
Rewrite the inside as (x - h) and read h as the shift, so (x - 2) means h = 2, a shift right. Confirm with one test point: at x = 2 the squared term is zero, so the vertex must sit at x = 2.
Asked for the domain of f(x) = 1/(x - 5), the candidate answers all real numbers, or excludes x = 5 from the range instead of the domain.
Domain restrictions are associated with square roots only, so rational functions are assumed unrestricted. Domain and range are also conflated because both are described as 'all real numbers except...' patterns.
What to do instead
Run a fixed checklist on every function: division by zero, even roots of negatives, and logs of nonpositives. For each, state the excluded input value explicitly and write the domain in set notation before moving on.
The candidate computes a mean of 14 for a data set, then adds a new value of 40 and reports the new mean as 27 by averaging 14 and 40.
The mean is treated as a property of the summary rather than of the underlying total. The candidate does not track how many values produced the original mean, so the weighting is lost.
What to do instead
Convert means into totals immediately: mean times count equals sum. Add the new value to the total, increment the count, then divide. Keep the running total and count as a two-column side note for every mean problem.
On a two-way table question, the candidate computes P(A and B) by multiplying the marginal probabilities of A and B even though the table shows the events are not independent.
The multiplication rule for independent events is remembered as the general 'and' rule. The candidate does not check whether the table's conditional probabilities differ from the marginals.
What to do instead
Read joint probability directly from the table cell over the grand total, and reserve multiplication for cases where independence is stated or verified. To test independence, compare P(A) with P(A given B); if they differ, the events are dependent and multiplication is invalid.