TASC General Mathematics Level 3 common mistakes
TASC General Mathematics Level 3 · Understand the traps and how to avoid them.
Solving a linear equation such as 3(x − 2) = 5x + 4 by applying the 3 only to the x and not to the −2, producing 3x − 2 = 5x + 4.
The distributive property is learned as a rule about brackets but is not yet linked to the idea that the bracket represents a single quantity multiplied as a whole; under time pressure the eye reads 3x and skips the second term.
What to do instead
Before solving, write an intermediate line that expands the bracket explicitly and check it by substituting a simple number (e.g. x = 1) into the original and expanded forms to confirm they match. Practise five expansions a day where the bracket contains a negative term.
When solving inequalities, multiplying or dividing by a negative number without reversing the inequality sign, so −2x > 6 becomes x > −3 instead of x < −3.
The procedure for inequalities is copied from the procedure for equations, where the direction of the relation is never affected; the sign reversal feels like an arbitrary extra rule rather than a consequence of order.
What to do instead
Test the answer by substituting a value from the claimed solution set into the original inequality. Also derive the rule once by hand: start from 2 < 3, multiply both sides by −1, and observe that −2 > −3, then write that observation at the top of the working page.
Finding the gradient of a line through two points by computing (x₂ − x₁)/(y₂ − y₁), that is, inverting rise over run.
The formula is memorised as a fraction of two differences without a mental image attaching the vertical change to the numerator, and both differences look equally plausible when written quickly.
What to do instead
Always sketch the two points on a small grid and count the vertical and horizontal steps physically before writing the fraction. Write the formula as rise/run with the words above and below the fraction bar each time until the sketch alone is enough.
Reading a value from a graph or table and reporting the y-value when the question asks for the x-value (or vice versa), for example giving the maximum height when asked for the time at which it occurs.
Attention locks onto the number that is easiest to read on the axis being scanned, and the distinction between input and output is not held in mind while the eyes move to the graph.
What to do instead
Underline the quantity requested in the question and label it 'input' or 'output' before touching the graph. Then trace from the correct axis: for an output, start on the vertical axis; for an input, start on the horizontal axis. State the answer with units and a phrase such as 'at t = 4 seconds'.
Assuming a relationship shown on a graph is linear over its whole range because two plotted points happen to lie on a straight segment, then extrapolating that line beyond the data.
Two points always determine a straight line, so a linear model fitted to a small window looks convincing; candidates rarely check whether a third point deviates or whether the context implies a turning point.
What to do instead
Before fitting any model, plot or list at least three points and check the first differences or the shape for curvature. Mark the region covered by data and refuse to extend a line past the last observed point without an explicit statement that the trend is assumed to continue.
Interpreting the gradient of a real-world graph as a bare number, giving 'the gradient is 3' when the correct answer is 'the cost rises by $3 per item'.
Gradient is practised in purely algebraic contexts where a number is the complete answer, so the habit carries over into modelling questions where the units and meaning are the whole point.
What to do instead
Adopt a fixed sentence frame for every gradient: 'For each additional [unit of x], [y] changes by [amount] [units of y].' Fill in the frame in writing before writing the final answer, and check that the units of the gradient are units of y divided by units of x.
Computing a mean of grouped or summarised data by averaging the category values without weighting by frequency, for example averaging 70, 80 and 90 equally when the group sizes are 2, 30 and 1.
The word 'average' triggers a single memorised procedure (add and divide by how many), and the frequency column is perceived as extra information rather than as the weighting that defines the calculation.
What to do instead
Before any average, ask 'how many values does each number stand for?' and write the frequency beside each value. Then compute the total by multiplying value by frequency, sum the frequencies separately, and divide. Compare the weighted and unweighted results so the difference becomes visible.
Treating dependent events as independent, for example giving the probability of drawing two red counters without replacement as (3/8) × (3/8) rather than (3/8) × (2/7).
The multiplication rule for independent events is the only combined-probability rule many candidates hold, and the phrase 'and' cues it automatically without checking whether the first outcome changes the second.
What to do instead
For every two-stage probability, write out the composition of the set after the first event ('8 counters, 3 red; after one red, 7 counters, 2 red'). Only after that written restatement decide whether the second denominator stays the same or changes.
Calculating a percentage increase by adding the percentage of the original to the original percentage rather than to the amount, or applying a 15% discount and then a 10% discount and reporting a 25% total reduction.
Percentages are treated as additive quantities like dollars, and successive percentage changes feel as though they should simply combine; the multiplicative structure is invisible when the base is not written down.
What to do instead
Write the base amount explicitly for every percentage step and use multipliers: a 15% discount is ×0.85, a further 10% is ×0.90, so the combined effect is ×0.765, a 23.5% reduction. Check by computing the money amounts at each stage on a sample value such as $100.
Mixing up simple and compound interest, applying the compound formula when the question specifies simple interest, or using the number of years as the exponent when interest is compounded monthly.
Both formulas contain principal, rate and time, and the visual similarity causes the wrong one to be retrieved; the compounding period is often stated in a subordinate clause that is skimmed.
What to do instead
Before substituting, write the compounding frequency as a word ('annually', 'monthly') and adjust rate and time to match that period: divide the annual rate by 12 and multiply the years by 12 for monthly compounding. Then compute the first year by hand with both formulas to see how they diverge.