CEMC common mistakes
CEMC · Understand the traps and how to avoid them.
When solving an inequality, multiplying or dividing both sides by an expression containing the variable and dropping the case where that expression is negative, so the direction of the inequality is never reversed for part of the domain.
Students learn 'flip the sign when multiplying by a negative' as a rule about literal negative numbers, and treat a symbolic factor like (x-2) as if its sign were fixed, forgetting it changes sign across its root.
What to do instead
Before dividing by any variable expression, split into the cases expression > 0 and expression < 0, reverse the inequality in the second case, and intersect each solution with its case condition; alternatively move everything to one side and use a sign chart on the factored form.
Solving an absolute-value equation by removing the bars and keeping only the positive branch, e.g. writing |2x-5| = 7 as 2x-5 = 7 and reporting a single solution.
The bars look like parentheses that can simply be deleted; students do not internalize that |A| = k encodes two equations, and the second branch is easy to forget under time pressure.
What to do instead
Immediately rewrite |A| = k as the pair A = k or A = -k (with k >= 0, no solution if k < 0), solve both, then check each answer in the original equation.
Applying a logarithm to a sum, writing log(a+b) = log a + log b, or cancelling logs by writing log a = log b implies a = b without first confirming both arguments are positive.
The product and power laws are heavily drilled while the absence of a sum law is not, and the domain restriction of the logarithm is treated as a formality rather than a constraint that can invalidate a solution.
What to do instead
Keep a written list of valid log laws with no sum/difference entry, and after solving any log equation test each candidate in the original expression, discarding any that makes an argument zero or negative.
Assuming a triangle is right-angled or that a diagram's apparent right angle is given, then applying Pythagoras or SOHCAHTOA to a triangle where the needed angle is not actually 90 degrees.
Diagrams are drawn to look plausible and students pattern-match the visual shape to the familiar right-triangle toolkit instead of checking the stated data.
What to do instead
Before choosing a tool, list what is given (sides and angles) and select the law of sines or law of cosines unless a 90-degree angle is explicitly stated or provable; use the converse of Pythagoras to test a claimed right angle.
Using the law of sines to find an angle and reporting only the acute value from the calculator, ignoring the obtuse supplement that also satisfies the equation.
arcsin returns only values in [-90, 90], and students treat the calculator output as the unique answer without recalling that sin(theta) = sin(180 - theta).
What to do instead
After any inverse-sine step in a triangle, explicitly check whether the supplementary angle is geometrically possible given the angle sum constraint, and state both candidates before eliminating one.
Computing a count of arrangements or selections with a single formula without checking whether order matters and whether repetition is allowed, e.g. using permutations where combinations are required.
Students memorize nPr and nCr as interchangeable 'counting formulas' and skip the modelling step of deciding what constitutes a distinct outcome.
What to do instead
Before computing, write one sentence describing a single outcome and ask whether swapping two chosen items produces a different outcome; if not, use combinations, and if repetition is permitted use the multiplication principle with replacement.
Adding probabilities of overlapping events, P(A or B) = P(A) + P(B), without subtracting the intersection, or multiplying P(A and B) = P(A)P(B) for events that are not independent.
The addition and multiplication rules are remembered in their simplest special-case form, and independence is assumed rather than verified from the problem description.
What to do instead
Always test independence or overlap first: for 'or' use P(A)+P(B)-P(A and B); for 'and' use P(A)P(B|A) unless independence is established, and check by computing the conditional probability directly.
Treating a geometric or telescoping series as if it were arithmetic, or applying the infinite geometric sum formula a/(1-r) when |r| >= 1 or when the series is finite.
The geometric sum formula is the most practised one, so it is reached for by default; the convergence condition |r| < 1 is memorized as a footnote rather than a gate.
What to do instead
Classify the series first by computing the ratio or difference of consecutive terms; only use a/(1-r) after confirming the ratio is constant and its absolute value is strictly less than 1, otherwise use the finite sum or state divergence.
Manipulating a recurrence or limit by substituting the limit L into both sides without justifying that the sequence converges, or dividing by a quantity that could be zero in a limit.
The substitution trick is fast and usually works on textbook problems, so the convergence precondition is dropped, and algebraic division by a vanishing factor is not noticed.
What to do instead
Before substituting, show the sequence is monotone and bounded (or cite a convergence theorem); for limits, factor or rationalize to remove the vanishing factor before evaluating, and check one-sided behaviour.
Answering the quantity asked for in an intermediate step rather than the quantity requested, e.g. reporting the number of terms or a partial value when the question asks for the sum, or giving a count when a probability is required.
Multi-step problems create a strong sense of completion at the last computed number, and the final sentence of the question is often read before the setup and not re-read at the end.
What to do instead
Underline the requested quantity when first reading the problem, and make the last line of the solution an explicit sentence of the form 'the required value is ___', checking units and the form (count vs. probability vs. sum) against the underlined phrase.