MCAS Grade 10 Math common mistakes
MCAS Grade 10 Math · Understand the traps and how to avoid them.
When subtracting polynomials such as (3x² − 5x + 2) − (x² − 4x + 7), the student changes the sign of only the first term in the second polynomial and writes 2x² − 9x + 9 instead of 2x² − x − 5.
The minus sign is read as belonging to the leading term rather than to the whole parenthesized quantity, so the remaining terms keep their original signs. This is a habit carried over from arithmetic, where a single subtraction usually touches a single number.
What to do instead
Force a rewrite step: replace the subtraction with addition of the opposite and write every sign change explicitly before combining. Then check the result numerically by substituting x = 2 into the original and simplified expressions and confirming both give the same value.
The student solves an equation such as x² = 6x by dividing both sides by x, reports only x = 6, and loses the solution x = 0; similarly, a quadratic is factored but only one factor is set to zero.
Dividing by a variable feels like legitimate simplifying, and once the equation looks linear the student stops looking for additional solutions. Zero-product reasoning is often memorized as a phrase without being applied to every factor.
What to do instead
Adopt a rule: never divide both sides by an expression containing the variable; factor and set each factor equal to zero instead. Train the check of solution counts against the degree - a quadratic has at most two solutions - and verify by substitution rather than by inspection.
Given f(x) = x² − 3, the student evaluates f(x + 2) as x² − 1, or reads f(4) as f multiplied by 4, rather than substituting the whole input into the rule.
Function notation is visually similar to multiplication and the parentheses look like grouping rather than an input slot. The student substitutes into the simplified result instead of into the original rule.
What to do instead
Use a placeholder method: rewrite the rule as f(□) = □² − 3, then drop the entire input, parentheses included, into every box. Practice three or four mixed evaluations per session (numeric input, variable input, expression input) and state aloud that f is a name, not a factor.
On a real-world linear model, the student identifies the slope as the starting amount and the y-intercept as the rate of change, or accepts a domain that includes negative time or fractional items because the equation allows them.
The student matches numbers to labels by position in the story rather than by units. Slope and intercept are learned as formula components, so units (dollars per month versus dollars) are never attached to them.
What to do instead
Annotate every parameter with units before answering, and test the interpretation by evaluating at input zero (which yields the intercept) and at input one (whose change from the first value is the slope). For the domain, list the constraints the context imposes - non-negative, whole numbers for countable items - and state them in interval or inequality form.
The student simplifies √(9 + 16) as 3 + 4 = 7, or rewrites √50 as 25√2, treating the radical as if it distributed over addition and mis-pulling the largest perfect square.
The distributive property is overgeneralized from multiplication and addition of ordinary numbers. Radical simplification is often remembered as 'take something out' without a reliable procedure for finding the largest square factor.
What to do instead
Counter the misgeneralization with a numeric check: √25 = 5 while √9 + √16 = 7, so the split must be false. For simplification, factor the radicand into primes or into the largest perfect square times the remainder, and confirm by squaring the answer back to the original radicand.
In percent and unit problems, the student computes a percent increase by dividing the change by the new value instead of the original, or converts rates by multiplying where a reciprocal is needed, producing answers with plausible but inverted magnitudes.
The word 'percent' triggers a memorized divide-by-a-number routine, and the reference quantity is never identified. Unit conversion is done by pattern-matching the units written on the page rather than by tracking them through the operation.
What to do instead
Write the setup as (change) ÷ (original) with the original explicitly labeled before computing, and sanity-check direction: a 50% increase followed by a 50% decrease must land below the start. In conversions, carry the units through as fractions and cancel them so only the target unit survives.
For a cone or pyramid, the student substitutes the slant height for the perpendicular height, or scales a solid's volume by the same factor used for its side lengths - tripling the dimensions and claiming the volume triples.
Textbook diagrams draw the slant height prominently and the perpendicular height as an interior dashed line that looks like an auxiliary mark. Linear and volumetric scaling are stored as separate formulas without the underlying relationship.
What to do instead
Before substituting, label the right triangle inside the figure and mark which segment is perpendicular to the base. For scale factors, memorize the ladder: lengths scale by k, areas by k², volumes by k³, and verify with a unit cube doubled in every dimension.
The student finds the slope of a perpendicular line by keeping the same slope, or computes distance where midpoint was requested, and applies origin-based rotation rules like (x, y) → (−y, x) to figures centered elsewhere.
Perpendicular and parallel are distinguished only by a vague sense of 'different,' and the negative reciprocal is recalled imprecisely. Coordinate formulas are recalled by shape rather than by what they return, and transformation rules are memorized without their center.
What to do instead
Build a one-line reference card: parallel slopes equal, perpendicular slopes are negative reciprocals whose product is −1, distance returns a length, midpoint returns a point. For transformations not centered at the origin, translate the center to the origin, apply the rule, then translate back.
Reading a two-way table, the student answers a conditional question such as 'what fraction of the students who play a sport also take music?' with a marginal total divided by the grand total, ignoring the stated subgroup.
Every fraction in the table looks like a valid proportion, and the denominator is chosen by whichever total is easiest to spot. The conditional phrasing is skimmed rather than parsed.
What to do instead
Circle the restricting phrase and write 'given ___' before touching the numbers; the given group fixes the denominator and the other condition fixes the numerator. Check plausibility by confirming the denominator equals the row or column total of that subgroup, and practice restating each question as 'out of the ___'.
For a skewed distribution, the student reports the mean and standard deviation as the best summary, or concludes that a strong correlation between two variables proves that one causes the other.
Mean and standard deviation are the most rehearsed statistics, so they are applied by default without checking shape. Observational data and experiments are not distinguished, so association is read as a causal mechanism.
What to do instead
Always inspect shape first: symmetric calls for mean and standard deviation, skewed or outlier-prone calls for median and IQR, and state that reason in the answer. For association claims, ask how the data were collected - random assignment supports causation, observation does not - and name a plausible confounding variable.