NVO Grade 10 Mathematics common mistakes
NVO Grade 10 Mathematics · Understand the traps and how to avoid them.
When solving an equation, the candidate moves a term across the equals sign but keeps its sign unchanged, for example rewriting 3x + 5 = 2x − 7 as 3x − 2x = −7 + 5 and then reporting x = 2 instead of the correct x = −12.
The candidate has memorised 'move it over' as a physical action rather than understanding that subtracting the same quantity from both sides preserves equality. Because the manipulation feels routine, no check is performed, and the sign error survives to the final answer.
What to do instead
Require the candidate to write the operation on both sides explicitly for one full week of practice (e.g., '3x + 5 − 2x = 2x − 7 − 2x'), then substitute the final value back into the original equation as a mandatory last step. Track how often substitution catches an error; the goal is to make verification automatic rather than optional.
The candidate expands (x + 3)² as x² + 9, or factorises x² + 5x + 6 as (x + 5)(x + 1) by picking numbers that add to 5 without checking that they also multiply to 6.
The candidate pattern-matches squaring to the same operation used for multiplication by a constant, and treats factorisation as a single-condition search. The distributive structure of a binomial square and the dual constraint of factor pairs have not been consolidated.
What to do instead
Have the candidate draw the two-by-two area diagram for (x + 3)(x + 3) and physically label the four regions until the middle term appears on its own. For factorisation, enforce a two-column check: list factor pairs of the constant, then test the sum. Do ten mixed items and require the candidate to expand the proposed factorisation back as proof.
The candidate solves a linear inequality such as −2x < 8 and reports x < −4, failing to reverse the inequality when dividing by a negative number.
The candidate treats inequalities as equations with a different symbol, and the reversal rule is remembered as an arbitrary exception rather than as a consequence of multiplying by a negative quantity. Under time pressure, the rule is dropped.
What to do instead
Demonstrate with numbers the candidate already trusts: 2 < 3 but −2 > −3. Then have them test a value from their answer interval in the original inequality; x = −5 gives 10 < 8, which is false, exposing the error immediately. Build the habit of always testing one endpoint value.
Given a straight-line graph or two coordinate pairs, the candidate computes the gradient as the change in x divided by the change in y, or ignores negative signs, producing a slope of the wrong magnitude or sign.
The formula is recalled as 'difference over difference' without anchoring which difference goes on top. The candidate also reads graphs left-to-right inconsistently, so rise and run are taken in mismatched directions.
What to do instead
Anchor the gradient as 'rise over run' with a physical gesture: vertical change on top, horizontal on the bottom. Have the candidate compute gradients of both an increasing and a decreasing line from a drawn grid, then verify by asking whether the line goes up or down as x increases. Require the sign to be stated before the number.
The candidate uses the wrong circle relationship, calculating circumference as πr² or area as 2πr, or applies the diameter where the radius is required in a problem that gives the diameter.
The two formulas are visually similar and are stored as a single undifferentiated memory. The candidate also skims the problem statement and grabs the first number given without checking whether it is a radius or a diameter.
What to do instead
Have the candidate derive circumference by unwrapping a labelled circle and area by cutting it into sectors, so the formulas carry distinct meanings. Enforce a habit of writing 'r = ___' at the top of every circle problem before touching a formula, and label units (cm versus cm²) to catch swaps.
When finding a missing side with Pythagoras, the candidate always adds the squares, so a problem giving the hypotenuse and one leg produces a side longer than the hypotenuse.
The theorem is memorised as a² + b² = c² without the understanding that c is specifically the longest side opposite the right angle. The candidate does not identify which side is the hypotenuse before substituting.
What to do instead
Require a labelled sketch with the right angle marked and the hypotenuse identified before any calculation. Ask the candidate to predict whether the answer will be shorter or longer than the known sides. Include mixed problems where the hypotenuse is unknown and where a leg is unknown, and check that the answer is always shorter than the hypotenuse when a leg is sought.
The candidate computes the volume of a composite solid by adding the surface areas of its parts, or uses a perimeter formula where an area is required, because the question mentions a shape and the candidate grabs the most familiar formula.
Formula selection is driven by the shape name rather than by the quantity requested. The candidate does not pause to ask what is being measured, and units are not used as a check.
What to do instead
Train a two-question routine before any calculation: 'What quantity is asked for?' and 'What units should the answer have?' Have the candidate write the target unit first (cm², cm³, cm) and reject any formula whose output does not match. For composite solids, require a labelled decomposition showing which volumes are added or subtracted.
The candidate simplifies (2x³)² as 2x⁶, squaring the coefficient incorrectly, or adds exponents when multiplying terms with different bases, writing 3x² · 4x³ as 12x⁶ instead of 12x⁵.
The candidate applies the exponent rule to the variable but forgets that the coefficient is also inside the bracket, and generalises 'add the exponents' to cases where the bases differ. The rules are recalled as isolated slogans rather than as consequences of repeated multiplication.
What to do instead
Have the candidate expand (2x³)(2x³) in full to see the coefficient 4 appear, and expand 3x² · 4x³ as 3·x·x·4·x·x·x to count five x's. Insist on full expansion for any item the candidate gets wrong, then re-attempt the shortcut. Test with a numerical substitution such as x = 2 to verify both sides match.
The candidate reports the mean of a data set as the median, or claims the mean increases when an outlier is added to the low end, without recomputing. The candidate also reads a histogram bar height as the number of values in a wider interval.
The three averages are treated as interchangeable labels, and the candidate does not connect each measure to the question being asked (typical value versus middle value versus most common). Reading grouped data requires interpreting interval widths, which is a separate skill that is often skipped.
What to do instead
Give the candidate a small data set and require all three averages to be computed side by side, then add an extreme value and record which averages move and by how much. For histograms, have them write the frequency and the interval width on each bar and compute totals explicitly. Discuss which average best represents skewed data and why.
The candidate states that the probability of drawing a red counter from a bag containing 3 red and 5 blue counters is 3/5, using the number of red counters over the number of blue counters rather than over the total.
The candidate identifies the favourable outcomes but omits the sample space, because the denominator is not visually present in the problem. The concept of probability as part over whole has not been tied to a physical model.
What to do instead
Have the candidate list or draw the entire sample space before writing any fraction, and physically count the total. Enforce the check that all outcome probabilities sum to 1. Use bags of real or drawn counters so the total is counted, not inferred, and repeat until the denominator is written first without prompting.
After computing a numerical answer to a word problem, the candidate stops without returning to the question, so a problem asking for the cost of one item is answered with the total cost, or a problem asking 'how many weeks' is answered with a decimal such as 7.4 weeks.
The candidate treats the calculation as the goal rather than the question, and does not re-read the final sentence. Contextual constraints such as whole numbers of objects or rounding up to cover a cost are not considered.
What to do instead
Require the candidate to underline the actual question in the prompt before starting and to write a one-line answer sentence at the end ('Each item costs ___'). For real-world counts, ask whether a fraction of an object makes sense and whether rounding up or down is required. Practise with problems where the intermediate value differs from the requested value.