Junkao Mathematics - independent shared foundations common mistakes
Junkao Mathematics - independent shared foundations · Understand the traps and how to avoid them.
The candidate applies a function to both sides of an equation or inequality without checking whether the function is injective or order-preserving: squaring both sides of an equation and keeping only the positive root, or dropping the reversal of an inequality when multiplying by a negative quantity or a variable of unknown sign.
Early algebra taught 'do the same to both sides' as a universal licence, and the fine print about one-to-one functions and sign was never made explicit. Squaring feels like a clean simplification, and the spurious roots or lost negative solutions are not visible in the final line, so the error goes undetected.
What to do instead
Adopt a two-line habit: after squaring, write 'check for extraneous roots' and substitute each answer back. For inequalities, never multiply by an expression containing the variable; instead move everything to one side, factor, and use a sign table or sketch. When solving trigonometric equations, state the general solution with the period (for example x = 30 degrees + 360k or x = 150 degrees + 360k) rather than a single value.
The candidate treats a sequence as if it were a series: uses the arithmetic mean formula for the sum of n terms, confuses the nth term with the sum to n terms, or answers a 'sum to infinity' question for a geometric series whose common ratio has modulus greater than or equal to one.
The notation a_n, S_n and the summation symbol look similar under time pressure, and both the nth-term and sum formulas contain the same parameters a and d or r. The convergence condition |r| < 1 is memorised as a phrase but not checked against the actual value, so a divergent series is summed anyway.
What to do instead
Annotate every question with 'term' or 'sum' before selecting a formula, and write the formula from memory each time rather than recalling a shape. For geometric problems, compute r explicitly and write '|r| = ... < 1, so the sum to infinity exists' as a visible step; if |r| >= 1, write 'diverges' and stop. Keep a one-line card of the four formulas (arithmetic term, arithmetic sum, geometric term, geometric sum) and rehearse them until they are distinct.
The candidate substitutes into a formula for the equation of a line, tangent or normal without checking the sign of the gradient or the position of the point: using the negative reciprocal for a parallel line, or writing the tangent at a stationary point with a nonzero gradient.
The perpendicular gradient rule is memorised as 'flip and change sign' but is applied indiscriminately, and the candidate does not pause to consider whether the gradient is zero or undefined. The point-gradient form is then used mechanically with the wrong m.
What to do instead
Before writing the line, state the gradient explicitly and justify it: 'gradient of tangent = f'(a) = ...; gradient of normal = -1/that'. If f'(a) = 0, write 'tangent is horizontal, y = ...' and 'normal is vertical, x = ...' rather than forcing the formula. For parallel and perpendicular questions, write the two gradients side by side and check the relationship.
The candidate solves a logarithmic or exponential equation by 'cancelling' the log or the base: writing log(a + b) = log a + log b, or taking logs of a sum term by term, or solving e^x = 5 by writing x = ln 5 - ln e.
The logarithm laws for products and quotients are over-generalised to sums because the notation looks distributive. The inverse relationship between exp and ln is understood, but the candidate applies it to a sum instead of isolating the exponential term first.
What to do instead
Write the three valid laws (product, quotient, power) on the top of the page and treat any sum inside a log as a warning sign. For equations, isolate the exponential or logarithmic term completely before applying the inverse, then solve the resulting linear equation. Verify by substitution: log 2 + log 3 = log 6, whereas log(2 + 3) is not log 2 + log 3.
In a probability problem, the candidate uses the binomial formula for a situation where draws are without replacement from a small population, or treats events as independent when the question describes sampling without replacement, and multiplies probabilities that should be conditional.
The binomial is the most recent and most drilled model, so any 'number of successes in n trials' phrasing triggers it, and the distinction between with and without replacement is buried in a single word of the statement that the candidate reads past.
What to do instead
Before choosing a model, answer two questions in writing: are the trials independent (with replacement, or a large population), and is the number of trials fixed? If sampling is without replacement from a small population, use the hypergeometric/combinatorial count, not the binomial. When events are not independent, write the conditional probability P(B|A) explicitly and use the multiplication rule P(A∩B)=P(A)P(B|A). Circle the words 'with/without replacement' and 'independent' in the statement before starting.
When working with the normal distribution, the candidate uses the raw value x in the standard normal table instead of standardising with z=(x−μ)/σ, or misuses the symmetry/complement to obtain a probability greater than one or a negative probability, and in inference problems confuses the standard deviation of the sample with the standard error σ/√n.
The standardisation step is treated as an optional preliminary, and the table is consulted from memory of 'look up the number'; the sample-versus-population distinction is blurred because both are called 'standard deviation' in the statement.
What to do instead
Write the standardisation as an explicit line every time: P(X≤a)=P(Z≤(a−μ)/σ), and only then consult the table. Draw the bell curve and shade the required region before reading the table, so the complement or symmetry is decided visually. In inference, write the standard error as σ/√n in the first line and label it 'standard error', keeping σ (population) and s (sample) visually distinct. Check that every probability lies in [0,1].
When asked for the relative position of two lines or a line and a plane in space, the candidate computes the determinant of the three direction/connecting vectors and stops, declaring 'they intersect' or 'they are skew' from that single number without checking the parallel/contained cases separately.
The candidate collapses the classification into one test, forgetting that the determinant being zero covers three different situations (parallel distinct, coincident, coplanar-intersecting), and that the determinant being nonzero only rules out coplanarity.
What to do instead
Use a decision tree and write it out each time: compare direction vectors first (proportional or not). If proportional, test whether a point of one satisfies the other (coincident vs parallel distinct). If not proportional, compute the mixed product; zero means they intersect in a point (find it), nonzero means they are skew. For line–plane: check n·v; if zero, test whether the line's point lies in the plane (contained vs parallel). State the case name explicitly in the answer.
When computing a distance from a point to a plane or between skew lines, the candidate substitutes into the formula without normalising, using n·PQ instead of |n·PQ|/|n|, or forgetting the absolute value, and reports a negative distance or one scaled by the norm of the normal vector.
The formula is recalled as 'dot product over something' and the denominator (the modulus of the normal, or the cross product for skew lines) is dropped because it feels like a formality; the absolute value is dropped because the arithmetic 'came out positive anyway' in practice exercises.
What to do instead
Memorise the distance formula as a fraction with a visible denominator: d = |Ax₀+By₀+Cz₀+D| / √(A²+B²+C²), and for skew lines d = |(P₂−P₁)·(v₁×v₂)| / |v₁×v₂|. After computing, sanity-check the magnitude against the coordinates (a distance cannot exceed the distance between the two named points). Write the absolute value bars in the very first line of working, before substituting numbers, so they cannot be omitted later.
When computing a limit that leads to an indeterminate form, the candidate applies L'Hôpital directly without first checking the form, or applies it to a quotient that is not 0/0 or ∞/∞, and differentiates numerator and denominator as if differentiating a quotient (using the quotient rule instead of separate derivatives).
L'Hôpital is over-practised as a reflex, and the quotient rule is so recently drilled that it contaminates the procedure; the candidate also skips the substitution step that would reveal the form.
What to do instead
Adopt a fixed routine: substitute the value first and write the resulting form (0/0, ∞/∞, 0·∞, ∞−∞, 1^∞…) explicitly. Only for 0/0 and ∞/∞ apply L'Hôpital, and write the derivatives of numerator and denominator as two separate expressions, never as a quotient. For other forms, transform first (common denominator, conjugate, or take logarithms for 1^∞). Verify the final limit by a quick numerical check with a value close to the limit point.
In an optimisation problem, the candidate finds f′(x)=0 and reports those critical points as the answer, without verifying that the point lies in the feasible interval, without classifying it as a maximum or minimum, and without answering the actual question asked (the dimensions, the maximum area, the minimum cost).
The algebraic step (solve f′=0) is the part that feels like 'the maths', while translating back to the problem statement and checking endpoints is treated as cosmetic; the candidate also forgets that a constrained problem can have its optimum at a boundary.
What to do instead
End every optimisation with a four-line checklist written in the answer: domain of the variable from the geometry; critical points inside that domain; classification via the second derivative or a sign table of f′; evaluation of f at the critical point and at the endpoints. Then write a sentence in words giving the requested quantity with units. If the question asks for dimensions, the final line must state the dimensions, not just the value of the function.