HSC Mathematics Standard 2 common mistakes
HSC Mathematics Standard 2 · Understand the traps and how to avoid them.
When solving a linear equation or inequality, the candidate moves a term across the equals sign but forgets to change its sign, or fails to reverse the inequality symbol when multiplying or dividing by a negative number.
The balancing procedure is learned as a set of symbol-shuffling moves rather than as 'do the same operation to both sides', so the sign change on transposition and the flip rule for negatives are memorised separately and drop out under time pressure.
What to do instead
Write each step as a full operation on both sides (e.g. 'subtract 3 from both sides') rather than as an arrow move, and circle any negative multiplier or divisor before applying it to an inequality; then test the final inequality with one substituted value to confirm the direction is correct.
The candidate substitutes a value into a formula such as a linear model or a simultaneous-equation context but mixes up which variable is the input and which is the output, or solves for the wrong unknown after correctly forming the equation.
Word problems are translated in one pass and the candidate latches onto the first relationship found, without explicitly naming what each pronumeral represents before algebra begins.
What to do instead
Before writing any equation, write a one-line 'let' statement defining each pronumeral with its units, and underline the quantity the question actually asks for; after solving, restate the answer as a sentence with units to check it answers that quantity.
When finding a perimeter, area or volume, the candidate uses a diameter where a radius is required, or mixes units within one calculation (for example centimetres with metres) without converting.
Formulas are recalled as strings of letters without attached meaning, and the habit of checking unit consistency has not been automated, so the substitution step is done quickly and unchecked.
What to do instead
Adopt a fixed routine: write the formula, list each quantity with its unit beside it, convert everything to one unit before substituting, and label the result with its correct unit (cm, cm², cm³) to expose mismatches.
The candidate computes a total surface area or composite area by adding all faces of a solid without subtracting the faces that are joined or hidden, or by treating a composite shape as a single standard shape.
Standard formula recall dominates, and the candidate does not sketch the actual object, so overlapping or internal surfaces are never visualised.
What to do instead
Draw and label a net or exploded sketch of the object, shade only the exposed surfaces, and add the shaded regions separately, writing a short list of each face's area before summing.
The candidate applies a percentage increase or decrease to the wrong base amount, or adds successive percentage changes by simply adding the percentages (for example treating a 10% rise then a 10% fall as no change).
Percentages are treated as additive quantities rather than as multipliers, and the 'of what' in each percentage is not identified, which is especially tempting when two changes appear in the same sentence.
What to do instead
Convert every percentage change to a multiplier (1.10, 0.90) and chain them by multiplication, and always write the base amount next to each percentage before calculating; check that a rise then equal fall must end below the start.
In compound interest and depreciation questions, the candidate uses the simple interest formula, or applies the interest rate per month when the rate is stated per annum without adjusting the number of periods.
Both formulas look similar and the simple-interest one is practised first, while the compounding period is stated in prose and is easy to skim past.
What to do instead
Before substituting, write down the compounding period and the number of periods explicitly (e.g. 'monthly, 3 years = 36 periods; rate per period = annual rate ÷ 12'), then select the formula and check that the exponent equals the period count.
The candidate reads a mean, median or standard deviation from a grouped frequency table or histogram by using class midpoints as if they were exact data values, or by counting frequencies rather than the total number of scores when locating the median.
Grouped data is treated as raw data because the distinction is procedural rather than conceptual, and cumulative frequency is not tabulated before locating the middle value.
What to do instead
Build a cumulative frequency column first, find the position of the median as (n+1)/2, and state explicitly that midpoints give only an estimate for grouped data; for standard deviation, confirm whether the question expects the population or sample form before computing.
The candidate calculates a z-score or compares scores from two datasets by comparing raw values directly, ignoring that the datasets have different means and standard deviations.
The z-score is memorised as a formula to evaluate rather than as a tool for putting scores on a common scale, so the comparison is made before standardising.
What to do instead
Convert each score to a z-score with (x − mean)/standard deviation, show both z-values side by side, and interpret the sign and size before naming the better relative performance.
The candidate uses a shortest-path or minimum-spanning-tree procedure but stops before all vertices are connected, or includes a cycle when finding a minimum spanning tree, and does not check that the final network satisfies the definition.
The algorithms are executed step by step without a stated stopping condition, and the definitions of 'spanning' and 'tree' are not used as checks at the end.
What to do instead
State the stopping condition before starting (all vertices included and n−1 edges for a spanning tree; start and end vertex reached for a path), then verify the finished network against that condition and recount edges.
The candidate counts the degree of a vertex incorrectly by counting edges that pass through rather than each edge ending at the vertex, or misapplies Euler's rule for traversable networks by checking the wrong number of odd-degree vertices.
Degree is counted visually from a diagram, and loops and edges meeting at a vertex are not treated consistently, while Euler's condition is recalled as 'two odd vertices' without the full statement.
What to do instead
List each vertex and tally its incident edges one at a time, counting a loop twice, then use the rule explicitly: a network has an Eulerian trail if exactly zero or two vertices have odd degree, and an Eulerian circuit only if all degrees are even.