MTTC Mathematics (Secondary) (022) common mistakes
MTTC Mathematics (Secondary) (022) · Understand the traps and how to avoid them.
Simplifies a radical over a sum, writing √(a+b) as √a+√b, or cancels an added term, reducing (x+3)/x to 3.
The candidate overextends distributive and cancellation patterns that are valid only for multiplication or division, then stops before checking equivalence.
What to do instead
Classify the operation before simplifying: radicals distribute over products and quotients, not sums, and cancellation requires a common factor in every term. Test with a numerical substitution such as a=9 and b=16.
Classifies numbers by memorized labels rather than definitions, calling 0 irrational, π rational because 22/7 approximates it, or omitting complex roots because the prompt does not say 'complex.'
Number-system definitions are stored as examples instead of closure and property criteria, and previous 'no real solution' habits carry over.
What to do instead
Use a decision sequence from natural to integer to rational to real to complex, and justify each placement by a defining property. When solving, state the intended solution set and include nonreal roots unless the problem excludes them.
Finds an inverse by swapping x and y and solving, but does not restrict the original domain or check that the result is a function; may also write f^{-1}(x)=1/f(x).
The swap procedure is memorized without the one-to-one condition, and inverse notation is visually confused with reciprocal notation.
What to do instead
Apply the horizontal-line test, restrict the domain where needed, and verify both f(f^{-1}(x))=x and f^{-1}(f(x))=x on the stated domains. Explicitly distinguish inverse function from reciprocal.
Solves radical or rational equations by squaring or clearing denominators and accepts every algebraic result without substituting back into the original equation.
Squaring and multiplying by variable expressions can introduce extraneous solutions, but checking feels optional once an answer is produced.
What to do instead
Record excluded values before multiplying, note where squaring may introduce solutions, and substitute every candidate into the original equation. Reject candidates that make a denominator zero or a radicand invalid.
Assumes a diagram proves congruence or similarity, uses AAA as a congruence criterion, or applies CPCTC before proving triangles congruent.
Visual appearance and familiar theorem names substitute for checking the exact hypotheses of a proof.
What to do instead
Write the given and prove statements first, mark only explicitly given information, and match the evidence to SSS, SAS, ASA, AAS, or HL before using CPCTC. Treat the diagram as illustrative, not as evidence.
Mishandles coordinate geometry formulas, such as using the midpoint formula for distance, taking a negative reciprocal of the wrong slope, or misreading signs in a circle equation.
Similar-looking formulas are recalled as a cluster, and sign errors in completing the square go unnoticed.
What to do instead
Derive or recall each formula from its meaning, label the two points or the center and radius before substituting, and complete the square step by step. Check the final circle equation by expanding it back.
Confuses P(A|B) with P(B|A), treats independent events as mutually exclusive, or adds probabilities for overlapping events without subtracting the intersection.
Everyday language blurs condition and causation, and the formulas for independence and mutual exclusivity look similar when memorized.
What to do instead
Restate the condition in words and build a two-way table or Venn diagram before computing. Use P(A|B)=P(A∩B)/P(B), test independence with P(A∩B)=P(A)P(B), and reserve mutually exclusive for P(A∩B)=0.
Adds instead of multiplies choices in counting problems, or uses permutations when order does not matter and combinations when it does.
The candidate does not explicitly decide whether order matters or whether repetition is allowed before selecting a formula.
What to do instead
Draw a decision tree for the choices and label order and repetition before computing. Use the multiplication principle for sequential choices and adjust for overcounting when identical items or unordered selections occur.
Uses degree mode in a radian context, gives only the principal value of an inverse trigonometric expression, or misses periodic solutions when solving sin x=k, cos x=k, or tan x=k.
Calculator mode and the restricted ranges of inverse functions are treated as afterthoughts, and the unit circle is recalled only for first-quadrant angles.
What to do instead
Set the calculator mode to match the problem, sketch the unit circle, and list all solutions in the requested interval using the period. State inverse-trig answers within the standard range and then extend if needed.
Differentiates or integrates by pattern while omitting the inner derivative, such as differentiating e^{3x} as e^{3x} or integrating 1/(3x) as ln|3x| without the chain-rule factor.
The outer function is recognized and the chain rule is applied only when the inside is a simple x, so composite structure is overlooked.
What to do instead
Name the outer and inner functions before differentiating or integrating, write the chain-rule factor explicitly, and check by differentiating the antiderivative. For limits, substitute first and identify indeterminate forms before choosing a technique.