MTTC Mathematics (Elementary) (089) common mistakes
MTTC Mathematics (Elementary) (089) · Understand the traps and how to avoid them.
When dividing fractions, the candidate flips the first fraction instead of the divisor, or flips both, then computes straight across and reports the result without checking its size.
The 'keep-change-flip' mnemonic gets memorized as a rhythm without the underlying meaning that division asks how many divisors fit into the dividend, so the candidate cannot tell which fraction is being inverted and does not notice an answer far from the expected magnitude.
What to do instead
Before computing, rewrite the problem as a question about how many groups of the divisor fit in the dividend and estimate the answer (for example 3/4 ÷ 1/8 should be around 6). Then apply keep-change-flip only to the divisor, and verify the result by multiplying it back by the divisor to recover the dividend.
The candidate adds fractions by adding numerators and adding denominators, so 1/2 + 1/3 becomes 2/5, and produces the same error on mixed numbers and on algebraic fractions with letter denominators.
Whole-number addition is additive across every part of the numeral, and nothing in the candidate's prior experience forces a break with that habit; the idea that denominators name the size of the parts, so unlike parts must be renamed before counting, was never made explicit.
What to do instead
Rebuild the concept with a concrete model: split a strip into halves and into thirds, observe that the pieces differ in size, and only count pieces once both strips are cut into sixths. Practice converting to a common denominator and always state the unit ('sixths') aloud before adding. Extend the same reasoning to variables by treating x as a unit label.
When multiplying or dividing by a decimal or a fraction, the candidate predicts the direction of the result from the operation symbol alone, claiming multiplication always makes numbers bigger and division always makes them smaller.
This generalization is true for the whole numbers the candidate practiced for years, so it feels like a rule rather than a coincidence; the actual determinant is whether the multiplier exceeds or falls below one, which is rarely emphasized.
What to do instead
Test the rule against counterexamples the candidate generates personally (0.5 × 8, 3 ÷ 0.5) and record both the operation and the size of the second factor next to each answer. Build a reference table showing that multiplying by a factor less than one shrinks the product, and use estimation to catch violations.
The candidate computes a percent change or percent-of problem by subtracting or dividing the wrong pair of numbers, for example finding the percent increase from 50 to 60 as 10/60 rather than 10/50, or adding the tax to the wrong base.
Percent problems contain two or three quantities and the candidate has not identified which one is the whole, so the phrase 'percent of' is matched to whichever number appears last in the sentence rather than to the reference amount.
What to do instead
Require a written setup before any arithmetic: state the part, the whole, and the percent, and use part = percent × whole as the single governing equation. For change problems, identify the original amount as the base and divide the change by it, then sanity-check that a 20% increase from 50 gives 60, not some other value.
The candidate computes a mean, median, or range for a data set but reports the wrong measure because the question asks for one statistic and the candidate supplies the one easiest to calculate, or forgets to order the data before finding the median.
The three measures are taught together and share vocabulary, so under time pressure the candidate pattern-matches on the word 'average' or on the look of the data rather than on the specific statistic requested; the median's dependence on order is easily overlooked.
What to do instead
Underline the statistic named in the question, then write its definition in one line before touching the numbers, and sort the data explicitly for median and range problems. For even-sized data sets, average the two middle values and check that the median lies between them. Verify the mean by checking that the total of deviations from it is zero.
The candidate confuses permutations with combinations, computing nPr when order does not matter or nCr when it does, and often multiplies by an extra factor for arrangements that are actually identical.
Both formulas look similar and both are triggered by the word 'choose', so the candidate selects by keyword rather than by asking whether rearranging the selected items produces a new outcome; repeated elements in a set are also routinely missed.
What to do instead
Before choosing a formula, write one sample outcome and then swap two of its elements: if the swapped version counts as different, use permutations; if not, use combinations. For arrangements with repeated letters, divide the total factorial by the factorial of each repeat count and test the answer on a short word such as 'book'.
The candidate solves a linear equation or inequality by performing an operation on one side only, or by distributing a negative sign across a parenthesis incorrectly, producing a sign error in the final answer.
The work is often done mentally or in compressed lines, so an operation applied to one side is invisible in the written record; the rule that multiplying or dividing an inequality by a negative reverses the symbol is also frequently dropped.
What to do instead
Write one operation per line and apply it explicitly to both sides, drawing a vertical line down the equals sign as a visual check. When a negative multiplier precedes a parenthesis, distribute it to every term and rewrite the line before continuing. For inequalities, circle the symbol whenever a negative multiplier or divisor appears and confirm the reversal.
The candidate substitutes a value into an expression or function but mishandles signs and exponents, for example evaluating −x² as 16 when x = 4, or treating f(2) as f times 2.
The notation −x² is read left to right as 'negative x, squared', and function notation resembles multiplication, so both errors come from applying ordinary reading order to symbolic conventions that have their own precedence rules.
What to do instead
Drill order of operations with the exponent bound to its base: rewrite −x² as −(x·x) before substituting, and use parentheses around substituted negative values. For function notation, practice reading f(2) aloud as 'the output of f when the input is 2' and evaluate from a table or rule rather than treating f as a variable.
The candidate answers a geometry question using a perimeter formula where area is required, or applies the area formula for a rectangle to a triangle or parallelogram without the factor of one-half or the height adjustment.
The formulas are memorized as a list of strings tied to shape names rather than derived from the meaning of area as covering and perimeter as distance around, so the units in the question do not trigger a check.
What to do instead
Attach units to every formula: area answers are in square units, perimeter answers in linear units, and the question's units reveal which is wanted. Rebuild the triangle formula by showing that two congruent triangles form a parallelogram, so the one-half is earned rather than memorized. Always confirm the height is perpendicular to the chosen base.
The candidate works a multi-step word problem by grabbing the two most prominent numbers and applying an operation, ignoring the remainder of the text and never checking whether the answer is reasonable in context.
Timed conditions reward fast pattern-matching on numbers, and the candidate has not built the habit of translating the situation into a structure before computing; the final answer is also never compared back to the story.
What to do instead
Use a three-pass routine: first read for the situation and restate the goal in one sentence, then list the given quantities with their labels, then choose operations from the relationships rather than the numbers. Finish by substituting the answer back into the story and asking whether the size and units make sense.