Mathematics Basic common mistakes
Mathematics Basic · Understand the traps and how to avoid them.
When computing a percent change, dividing the difference by the new value instead of the original value (e.g., a price rising from 40 to 50 reported as a 20% increase rather than 25%).
The word 'change' pulls attention to the two numbers being compared, and the new value is the most recent number read, so it feels like the natural base. Candidates also memorize 'difference over original' as a phrase without connecting it to the idea that percent change is measured relative to where you started.
What to do instead
Before dividing, write the sentence 'The change is ___% of the starting value' and circle the starting value. Then compute difference ÷ starting value. Sanity-check direction: an increase must give a positive percent and a decrease a negative one, and a value that doubles must be a 100% increase, not 50%.
Applying exponent rules additively, writing x^3 · x^4 = x^12 or (x^3)^4 = x^7, or cancelling terms across a plus sign such as reducing (x + 6)/6 to x.
Multiplication of powers and power-of-a-power look similar in notation, and the addition rule for like terms (x^3 + x^4 is not x^7 either) gets overgeneralized. Cancelling across a sum is tempting because the 6s are visually adjacent and the expression 'looks simplifyable'.
What to do instead
Rewrite each expression in expanded form once: x^3 · x^4 = (x·x·x)(x·x·x·x) = x^7, and (x^3)^4 = x^3·x^3·x^3·x^3 = x^12. For fractions, test with a number: let x = 1, then (1 + 6)/6 = 7/6, not 1, which proves the cancellation is invalid. Only factors connected by multiplication may cancel.
Solving a linear equation by moving a term to the other side and changing its sign but forgetting to apply the same operation to the other side, or distributing a negative sign to only the first term inside parentheses.
The shortcut 'move it over and flip the sign' is taught as a rule about individual terms rather than as subtracting the same quantity from both sides, so the balance of the equation is lost from view. With a leading minus, attention lands on the first term and the rest of the parentheses are processed automatically.
What to do instead
Write each step as an operation on both sides, e.g. 'subtract 3x from both sides' rather than 'move 3x'. For negatives, expand explicitly: −(2x − 5) = −2x + 5, and verify by substituting a value such as x = 1 into both the original and the simplified form.
Confusing slope and intercept when writing the equation of a line, producing y = 3x + 2 for a line with slope 2 through (0, 3), or reading the y-intercept off a graph as the x-value where the line crosses the horizontal axis.
In y = mx + b the letters m and b carry no meaning on their own, and both numbers in a problem are usually presented as plain integers, so candidates assign them by order of appearance rather than by role. Slope as 'rise over run' is also easily inverted to run over rise.
What to do instead
Label every number before writing: 'this is the rate of change per unit (slope)' versus 'this is the starting value when x = 0 (intercept)'. Compute slope as (y2 − y1)/(x2 − x1) with the same order in numerator and denominator, then substitute the intercept point into y = mx + b to confirm b.
Treating a function's value and its input as interchangeable, so that f(3) is read as 'f times 3', or f(x + 1) is simplified to f(x) + 1.
Parentheses normally signal multiplication in earlier arithmetic, and function notation is introduced as a small notational change rather than a new concept. The idea that f names a rule applied to the whole input inside the parentheses is abstract and easy to skip.
What to do instead
Read f(3) aloud as 'the output of f when the input is 3' every time. For composite inputs, substitute the entire expression: if f(x) = x² then f(x + 1) = (x + 1)², not x² + 1. Check with a concrete function and number to see the two results differ.
Applying area and volume scale factors linearly, assuming that doubling the side of a square doubles its area or that doubling the radius of a sphere doubles its volume.
The formulas are recalled but the exponents are not connected to physical meaning, so 'twice as long' is assumed to mean 'twice as much'. Perimeter, which does scale linearly, is the most recent experience and gets generalized to area.
What to do instead
Anchor the rule: lengths scale by k, areas by k², volumes by k³. Test on the simplest case - a 1×1 square has area 1, a 2×2 square has area 4 - and compute the actual formula rather than reasoning by analogy whenever the numbers are available.
Mixing units or using the diameter in place of the radius, for example computing the area of a circle with radius 5 as π·10², or adding a length in centimetres to one in metres without converting.
Diagrams label the full width, and the number that spans the figure feels like the primary measurement. Unit conversions are treated as a separate topic rather than as part of every calculation, so they are skipped under time pressure.
What to do instead
Write the formula with the quantity named before substituting: A = πr², r = ___ (half the given width). Convert all measurements to one unit in a first pass and record the conversion factor (1 m = 100 cm) explicitly. Check that the answer's unit matches what the question asks for.
Reporting the mean of a data set as the 'typical' value when the distribution is skewed or contains an outlier, or reading a histogram bar height as a count when the axis is a frequency density.
Mean, median, and mode are learned as three parallel procedures, and the mean is practiced most, so it becomes the default. Axis labels are scanned rather than read, and the tallest bar draws the eye regardless of what the vertical axis measures.
What to do instead
Before choosing a measure, look at the shape: with outliers or skew, state the median and say why. When reading any graph, read both axis labels and their units first, and note whether bars represent counts or density (area then gives the count).
Treating dependent events as independent, computing P(both red) for two draws without replacement as (5/10)·(5/10) instead of (5/10)·(4/9).
The multiplication rule is memorized in its independent form, and the phrase 'without replacement' is easy to pass over in a word problem. The sample space genuinely changes after the first draw, which is invisible unless the candidate stops to update it.
What to do instead
Draw the two-stage tree or write the second probability only after stating the new totals: 'after one red is removed, 4 red remain out of 9'. Ask explicitly whether the first outcome changes the second; if yes, the denominators must change.
Concluding that a probability near 0.5 or a small sample means 'due to happen', for example expecting tails after five heads, or judging a survey of 10 people as reliable evidence about a population.
Randomness is intuitively modeled as a self-correcting balance rather than as independent trials, and sample size is not connected to the spread of an estimate. Everyday phrases like 'it's overdue' reinforce the first error.
What to do instead
State that past independent outcomes do not change the next probability, and that the long-run proportion settles only because many more trials are added, not because early results are compensated. For samples, ask how much the result could vary with only 10 observations before drawing a conclusion.