FTCE Mathematics 6-12 common mistakes
FTCE Mathematics 6-12 · Understand the traps and how to avoid them.
When simplifying or solving, multiplying both sides of an equation by an expression containing the variable (e.g., clearing denominators) without later checking whether any solution makes that expression zero.
Fraction-clearing is practiced as a mechanical step, and candidates treat it as reversible without tracking the hidden restriction that the multiplier must be nonzero.
What to do instead
Before clearing denominators, list the excluded values (where each denominator equals zero). After solving, discard any candidate on that list and state the restriction explicitly.
Applying exponent rules across a sum or difference, e.g., writing (a+b)^2 = a^2 + b^2 or (a+b)^(1/2) = a^(1/2) + b^(1/2).
The distributive habit from multiplication is overgeneralized to operations (squaring, radicals) that are not linear, and the correct expansion is not mentally verified.
What to do instead
Test every exponent or radical manipulation with numbers: substitute a=3, b=4 and confirm both sides. If the numeric check fails, expand or rationalize properly.
Solving an inequality and keeping the same direction after multiplying or dividing by a negative number.
The procedure is memorized as identical to equation-solving, and the sign-flip rule is the one step that differs, so it drops out under time pressure.
What to do instead
Solve inequalities by isolating the variable while moving all variable terms to the side that keeps the coefficient positive; then no division by a negative occurs. Verify with a test point in the final interval.
Reading a graph's features from the wrong axis or confusing the roles of x and y, e.g., reporting a function's zeros as its y-intercept, or misreading a transformation's direction.
Graph-reading is done by visual impression rather than by mapping coordinates, and transformation rules (f(x-h) shifts right) are recalled as slogans rather than derived.
What to do instead
For each feature, write the coordinate pair explicitly (zero = point where y=0; intercept = point where x=0). For transformations, evaluate f at two sample inputs to confirm the shift direction.
Using degrees and radians interchangeably in a trig or calculus problem, e.g., differentiating sin(x) or using arc length s = rθ with θ in degrees.
Degree measure dominates prior coursework, and the radian requirement is an implicit convention in the formulas rather than a stated condition.
What to do instead
Adopt a default: convert every angle to radians at the start of any problem involving derivatives, arc length, or angular velocity. If an answer looks off by a factor near 57, suspect a degree/radian mix-up.
Mixing up permutations and combinations, or forgetting to divide by repeats, when counting arrangements.
Both are 'choose r from n' situations, and the distinguishing question (does order matter?) is skipped in favor of matching keywords.
What to do instead
Before computing, answer two questions in writing: does order matter, and are items repeated? Use nPr when order matters, nCr when it does not, and divide by the factorial of each repeat count for multiset arrangements.
Confusing P(A|B) with P(B|A), or treating dependent events as independent by multiplying P(A)P(B).
Conditional probability is read in the same direction as the sentence is spoken, and independence is assumed whenever events are described separately.
What to do instead
Write the conditional in set notation and check whether the given condition narrows the sample space. Test independence explicitly by comparing P(A∩B) to P(A)P(B) before multiplying.
Computing a mean, median, or standard deviation from grouped or summarized data without accounting for frequencies, or choosing a measure of center that the distribution's shape makes misleading.
Summary statistics are computed from raw-data formulas by habit, and the effect of skew on mean versus median is treated as trivia rather than a decision rule.
What to do instead
For grouped data, weight each value by its frequency before dividing by the total. Before reporting a center, sketch the distribution's shape; report the median for skewed data and state why.
Computing a limit by direct substitution when substitution yields 0/0, and concluding the limit does not exist.
Substitution is the first tool learned, and the indeterminate form is misread as a final answer rather than a signal to transform the expression.
What to do instead
Treat 0/0 or ∞/∞ as a prompt, not a result: factor and cancel, rationalize, or apply L'Hôpital's rule, then re-evaluate. Confirm with a nearby numeric value or a one-sided check.
Applying a formula with mismatched units or dimensions, e.g., mixing centimeters and meters in an area or volume calculation, or using a perimeter formula for area.
Formulas are recalled symbolically without attaching units, and conversion is deferred to the end (or skipped) rather than done up front.
What to do instead
Convert all given quantities to a single unit before substituting. Track units through the computation (cm × cm = cm²) and reject any answer whose units do not match what was asked.