MTEL 65 common mistakes
MTEL 65 · Understand the traps and how to avoid them.
Given a ratio of 3:5 for boys to girls in a class of 32, the candidate multiplies 32 by 3/5 (or sets up 3/5 = x/32) instead of using 3/8, treating a part-to-part ratio as a part-to-whole fraction.
English phrasing such as "ratio of boys to girls" is silently rewritten as "boys out of everyone" because the fraction-of-a-whole model is more familiar and requires no extra step.
What to do instead
Force a tape diagram before any computation: draw 3 boxes and 5 boxes, write 3 + 5 = 8 total parts, then 32 ÷ 8 = 4 students per part, so boys = 3 × 4 = 12. Whenever a ratio appears, sum the parts first and label what the denominator represents.
Adds or subtracts successive percent changes - concluding that a 20% increase followed by a 20% decrease leaves the price unchanged - or applies a discount and then a tax both to the original price.
Percent is treated as a fixed quantity to be added rather than a multiplicative factor applied to a base that has already changed after the first event.
What to do instead
Convert every percent change to a multiplier and chain them: 1.20 × 0.80 = 0.96, a net 4% decrease. Start from $100 to check, and write the sentence "the base for the second percent is the result of the first" next to the work.
Applies exponent rules additively, writing x³ · x⁴ = x¹² and (x³)⁴ = x⁷, and evaluates a negative exponent as a negative number, e.g. 2⁻³ = −8.
The symbols are memorized as surface patterns without the underlying meaning of repeated multiplication or reciprocal, so multiplication of powers collapses into the addition rule.
What to do instead
Expand once in full: (x³)(x⁴) = (xxx)(xxxx) = x⁷, but (x³)⁴ = x³·x³·x³·x³ = x¹². Derive negative exponents from the division pattern 2³/2⁵ = 1/2², giving 2⁻³ = 1/2³ = 1/8, never a negative value.
Expands (2x + 3)² as 4x² + 9, and drops a negative across a parenthesis by writing 5 − (x − 3) = 5 − x − 3.
The exponent is read as "apply to each term separately," and a leading minus sign is treated as belonging to the first term only; distribution was practiced only with positive multipliers.
What to do instead
Rewrite every square as an explicit product and multiply term by term: (2x+3)(2x+3) = 4x² + 12x + 9. For subtraction, distribute −1 first: 5 + (−1)(x − 3) = 5 − x + 3, then verify by substituting x = 1 into both the original and rewritten expressions.
Solves −2x > 6 as x > −3, keeping the inequality direction after dividing by a negative coefficient.
The procedure is stored as "do the same operation to both sides," and dividing by a negative does not feel like an operation that changes the solution set.
What to do instead
Always test a point after solving: x = −4 satisfies x < −3, and substituting gives −2(−4) = 8 > 6, which is true, so the direction must reverse. Reinforce with the number-line picture that multiplying by a negative reflects every point across zero.
Writes f⁻¹(x) = 1/f(x), or evaluates f(g(2)) when the item asks for g(f(2)), and assumes the inverse has the same domain and range as the original function.
The superscript −1 is visually confused with the exponent −1, and function composition is read left to right like ordinary text rather than inside out.
What to do instead
Interpret f⁻¹ as "undo f": swap x and y, then solve. Work composition from the innermost parentheses outward. Verify numerically - if f(x) = 2x + 1, then f⁻¹(5) = 2 and f(2) = 5 - and state that the inverse's domain equals the original's range.
Doubles the side lengths of a rectangle and concludes the area doubles, or triples a cylinder's radius and height and claims the volume triples.
Length, area, and volume all register as "size," so the separate square and cube relationships between the scale factor and the resulting measure are never tracked.
What to do instead
Attach units to the scale factor: a length ratio of k produces an area ratio of k² (m·m) and a volume ratio of k³ (m·m·m). Confirm with concrete numbers - a 2-by-3 rectangle has area 6, while doubling to 4-by-6 gives 24, four times larger.
Converts 1 m² to 100 cm², or a rate of 3 m³/min to 300 cm³/min, applying the linear conversion factor exactly once.
The well-drilled fact 1 m = 100 cm is applied mechanically without noticing the exponent on the unit, even though each dimension needs its own conversion.
What to do instead
Convert one dimension at a time: 1 m² = (100 cm)(100 cm) = 10,000 cm², and 1 m³ = 100³ = 1,000,000 cm³. For rates, carry the exponent inside a unit fraction, (100 cm/1 m)², so the square applies to the whole conversion factor.
Computes P(A or B) for two overlapping events by adding P(A) + P(B), sometimes returning a probability greater than 1 or double-counting the shared outcomes.
"Or" is translated directly into addition using a rule memorized only for mutually exclusive events, and the overlap region of the sample space is never visualized.
What to do instead
Sketch two circles and label the intersection with a count before computing. Use P(A∪B) = P(A) + P(B) − P(A∩B), or add the three disjoint regions. Check that the answer is at most 1 and that the complement P(Aᶜ ∩ Bᶜ) is consistent.
Given a two-way table, answers P(plays a sport | is in grade 8) by dividing the cell count by the grand total rather than by the grade-8 row total, and similarly swaps P(A|B) with P(B|A).
The conditioning event is not treated as a reduced sample space, so the early habit of "divide by the total" carries over unchanged, and the reversed conditional looks like the same statement.
What to do instead
Circle the given group first and write "out of ___" beside it - that row or column total is the denominator, not the grand total. Practice on a table where the two conditionals differ sharply, and state that P(A|B) = P(B|A) only when the events are independent.