MTEL Mathematics (63) common mistakes
MTEL Mathematics (63) · Understand the traps and how to avoid them.
On number-sense items, the candidate rounds intermediate values (for example, rounds 3.47 to 3.5 before multiplying) and then selects an answer choice that is close to but not equal to the exact result.
Mental arithmetic habits from timed practice push candidates toward 'good enough' approximations, and the presence of a distractor near the rounded value rewards that habit.
What to do instead
Work exact fractions and decimals to the final step, then round only if the stem asks for an approximation. Before choosing, redo the computation a second way (e.g., fraction versus decimal) and confirm both land on the same choice.
The candidate treats 'a divides b' as interchangeable with 'b divides a' or assumes division is commutative when simplifying integer expressions involving factors and remainders.
Everyday phrasing ('divide 12 by 3') blurs the direction of the relation, and the notation a|b is easily read backwards under time pressure.
What to do instead
Rewrite each divisibility statement in words with an explicit arrow ('a is a factor of b') before manipulating it, and test with a concrete counterexample pair such as 2 and 6 to check direction.
When simplifying rational expressions or solving rational equations, the candidate cancels terms across an addition sign (e.g., cancels x in (x+3)/x to get 3) or multiplies through without noting excluded x-values.
Cancellation is over-learned from fraction multiplication, and the excluded-value condition feels like a formality rather than part of the answer.
What to do instead
Factor numerator and denominator completely and cancel only whole factors; after solving, list every value that makes any original denominator zero and reject those solutions explicitly.
The candidate solves an absolute-value equation or inequality by dropping the bars and solving only the positive case, losing the second solution or reversing the inequality direction.
The piecewise definition of absolute value is memorized as a procedure rather than understood, so the negative branch is forgotten under time pressure.
What to do instead
Always write the two cases (expression = +k and expression = -k, or the compound inequality for < and >) before solving, and check each candidate in the original equation.
On function items, the candidate confuses f(g(x)) with g(f(x)) or evaluates a composition at the wrong input, and on inverse-function items substitutes into f rather than f-inverse.
Composition notation reads left-to-right in everyday language but applies right-to-left mathematically, and inverse notation looks like a reciprocal.
What to do instead
Work compositions inside-out with explicit substitution, and verify inverses by confirming f(f-inverse(x)) = x symbolically for a general x, not just one numeric test.
The candidate transforms a graph (shift, stretch, reflect) using the sign or order that feels natural rather than the one the equation specifies, e.g., shifting y = f(x+3) three units right.
Horizontal transformations act opposite to the sign shown, and multiple transformations are applied in the wrong sequence.
What to do instead
Rewrite the function in the form a·f(b(x-h))+k, read h as the horizontal shift (right if positive), and apply transformations in the order reflections/stretches, then shifts; verify with two tracked points.
The candidate computes a probability of 'at least one' by adding individual probabilities instead of using the complement, producing a value greater than 1 or double-counting overlaps.
The phrase 'at least one' suggests addition, and the complement method is not the first strategy that comes to mind.
What to do instead
For any 'at least one' event, immediately compute 1 minus P(none) and check that the result lies in [0,1]; for overlapping events, apply inclusion-exclusion explicitly.
The candidate reads a two-way table or conditional-probability stem and divides by the grand total when the condition restricts the sample space to a row or column.
The denominator in a conditional probability is easy to overlook, and the grand total is the most visually prominent number in a table.
What to do instead
Underline the conditioning phrase ('given that', 'among those who'), then set the denominator to the size of that restricted group before dividing; write the fraction before computing.
In geometry, the candidate applies a formula with mismatched units (mixing radius and diameter, or degrees and radians) or uses the wrong area/perimeter formula for the figure described.
Formulas are recalled as strings of symbols without the associated diagram, and unit conversion is treated as an afterthought.
What to do instead
Sketch the figure, label every given quantity with its unit, convert to a single unit system first, and state which formula applies and why before substituting numbers.
On calculus items, the candidate differentiates or integrates a product or composite by applying the power rule term-by-term, skipping the product, quotient, or chain rule.
Power-rule drills dominate early practice, and the inner function in a composite is easy to miss when the expression is not written with parentheses.
What to do instead
Before differentiating, classify the expression as sum, product, quotient, or composite and name the rule required; for chain-rule problems, identify the inner function explicitly and check the result by differentiating back or testing a value.