GA Milestones Algebra EOC common mistakes
GA Milestones Algebra EOC · Understand the traps and how to avoid them.
When solving a multi-step linear equation such as 3(x - 4) + 2x = 17, the candidate distributes to the first term only, writing 3x - 4 + 2x = 17 instead of 3x - 12 + 2x = 17, then carries that error to a wrong final answer.
The distributive property is learned as a quick 'multiply the outside number by the first thing inside' shortcut, and under time pressure the second multiplication is dropped. The candidate often does not re-check because the resulting equation still looks solvable.
What to do instead
Adopt a written habit: after distributing, draw an arrow from the outside factor to every term inside the parentheses before simplifying. Verify by substituting the solution back into the original equation, not the simplified one, so distribution errors surface.
The candidate solves 2x + 5 > 11 as x > 3 but then solves -2x + 5 > 11 as x > -3, forgetting to reverse the inequality symbol when dividing by a negative number.
The reversal rule is memorized as a phrase rather than understood as a property of multiplying by a negative, so it is applied inconsistently. The candidate also stops checking once an algebraic step 'looks right'.
What to do instead
Test every inequality answer with a sample value. For -2x > 6, pick x = -4 and x = -2 and see which satisfies the original. Also rewrite subtraction of a negative as addition before dividing, which makes the sign change visible.
Given a system such as y = 2x + 1 and 3x + y = 11, the candidate substitutes and writes 3x + 2x + 1 = 11, then reports only x = 2 as the solution without finding y or writing the answer as an ordered pair.
The candidate treats 'solve the system' as 'find x', a habit carried over from single-variable equations. The ordered-pair convention is not reinforced during practice.
What to do instead
Before starting any system, write the target format at the top of the workspace: '(x, y)'. After finding the first variable, immediately substitute it into the simpler original equation and box the pair. Check by substituting the pair into both equations.
When asked for the equation of a line through (2, 5) and (6, 13), the candidate computes slope as 8/4 = 2 but then writes y = 2x + 5, using the y-coordinate of the point as the intercept.
The candidate confuses a point on the line with the y-intercept. Because the slope is correct, the answer feels finished and is not verified.
What to do instead
Always finish with point-slope form: y - y1 = m(x - x1), then convert. Verify by plugging the second point into the final equation; if (6, 13) does not satisfy it, the intercept is wrong.
The candidate factors x^2 - 5x - 6 as (x - 6)(x + 1) but then states the solutions are x = 6 and x = 1, dropping the sign change required by the zero product property.
Setting each factor to zero is done mentally, and the candidate reads the constant inside each factor as the answer instead of solving x - 6 = 0.
What to do instead
Write the zero product step explicitly: (x - 6)(x + 1) = 0, so x - 6 = 0 or x + 1 = 0. Then solve each mini-equation on its own line. Substitute both answers into the original quadratic to confirm.
Given f(x) = x^2 - 3x + 2, the candidate evaluates f(-2) as -4 + 6 + 2 = 4, squaring -2 incorrectly or dropping the sign in the middle term.
Function notation is read as multiplication, and negative-number arithmetic is rushed. The candidate writes (-2)^2 as -4 because the exponent is applied before the negative sign is considered.
What to do instead
Substitute with parentheses every time: f(-2) = (-2)^2 - 3(-2) + 2. Work one operation per line and keep the parentheses visible until the exponent is evaluated. Practice a short set of negative-input evaluations until the pattern is automatic.
The candidate graphs y = -2(x - 3)^2 + 4 with the vertex at (3, 4) but opens the parabola upward, or shifts the vertex to (-3, 4) because the sign inside the parentheses is read literally.
The vertex form y = a(x - h)^2 + k is memorized without the sign logic: the horizontal shift is the opposite of the sign shown, and a negative leading coefficient reflects the parabola. Both rules are easy to invert under pressure.
What to do instead
Anchor each graph with three checks: sign of a for direction, the value inside the parentheses solved for zero to get h, and the constant for k. Plot the vertex and one symmetric pair of points before drawing the curve.
When simplifying (x^2 + 5x + 6) / (x + 2), the candidate cancels the x^2 with the x and the 6 with the 2, writing x + 3 without factoring, and gets a correct-looking answer by luck or an incorrect one otherwise.
Cancellation is treated as deleting matching symbols rather than dividing common factors. Because the expression contains matching letters and numbers, the move feels legitimate.
What to do instead
Enforce a rule: factor the numerator and denominator completely and write them as a product before cancelling anything. Cancel only whole factors, drawing a line through the entire factor. Check with a numeric substitution such as x = 1 on both the original and simplified forms.
The candidate answers a 'which expression represents the area' word problem by writing an equation to solve for a number instead of building an expression in terms of the given variable, producing a numeric answer where an algebraic expression was required.
Most prior practice involved solving for a value, so the candidate defaults to equation mode. The distinction between 'evaluate' and 'represent' is not noticed in the prompt.
What to do instead
Before writing anything, underline the question's final words and label the task as 'expression' or 'equation'. If the prompt asks for a representation, no equals sign should appear in the answer. Practice sorting a mixed set of prompts into the two categories.
The candidate computes the slope of a line perpendicular to one with slope 3/4 as -3/4, flipping the sign but not taking the reciprocal.
The perpendicular rule is half-remembered as 'make it negative', and the reciprocal step is dropped. Parallel and perpendicular rules get blended.
What to do instead
Write both rules side by side before answering: parallel means same slope; perpendicular means negative reciprocal. For 3/4, compute the reciprocal first (4/3), then apply the sign (-4/3), and verify the product of the slopes is -1.