IB Mathematics: Analysis and Approaches HL common mistakes
IB Mathematics: Analysis and Approaches HL · Understand the traps and how to avoid them.
Solving a logarithmic equation by combining or expanding logs incorrectly - writing log(a + b) = log a + log b, or dropping the base change - and then accepting a root that makes an original argument zero or negative.
Students memorise the product and quotient laws as visual patterns and over-generalise them to sums; because the algebra 'looks finished' after solving the resulting quadratic or linear equation, the check-back step is skipped entirely, and the calculator gives no warning when it substitutes an invalid x.
What to do instead
Before solving, write the domain restrictions from every log argument (argument > 0) as an explicit line of working. After obtaining candidate roots, substitute each back into the ORIGINAL unsimplified equation and reject any that fails - state the rejection in words, since IB mark schemes award a mark for it. Drill the three laws as log(xy), log(x/y), log(x^n) only, and practise rewriting log base 2 as ln/ln to confirm the base-change formula.
Rounding intermediate values to 3 significant figures on the calculator paper and carrying those rounded numbers forward (for example storing 1.41 instead of √2, or using 9.8 for g mid-calculation), producing a final answer that falls outside the accepted range.
Handwritten work looks tidier with short decimals, and students transcribe calculator displays into their written lines because they believe the marker needs to see every value; they do not realise IB accepts answers within a narrow tolerance and that early rounding compounds through powers and exponentials.
What to do instead
Adopt a fixed routine: keep exact surd/fraction forms or use the calculator's memory/ANS key for every intermediate value, and only round on the final answer line, writing '= 4.72 (3 s.f.)'. Practise by computing the same quantity two ways - once with stored values and once with rounded ones - so the size of the discrepancy becomes visible. Where a question says 'give your answer to 3 s.f.', that instruction applies to the final answer only, not to intermediate working.
Treating a function as invertible or composable without checking domains - writing down f⁻¹(x) for a quadratic such as f(x) = x² − 4x defined on all real numbers, or evaluating (f∘g)(x) at an x that lies outside g's domain.
Students learn the swap-x-and-y procedure as a mechanical algorithm and apply it to any function; the horizontal line test and the 'restricted domain' idea feel like optional theory rather than the condition that makes the inverse a function. Composite domain questions are also read as pure algebra.
What to do instead
Before inverting, complete the square to identify the vertex and state whether the given domain makes the function one-to-one; if not, say so and give the largest valid restriction. For composites, write the domain of the inner function first, then test the outer function at the output values, and express the composite domain as an inequality. Get in the habit of sketching both the function and its inverse as reflections in y = x to confirm the range of f becomes the domain of f⁻¹.
Applying transformations to a graph in the wrong direction or order - stretching horizontally by a factor of b rather than 1/b, translating left when the bracket reads (x − 3), or performing a stretch after a translation when the function is written as af(b(x − h)) + k.
The notation f(x − 3) is read left-to-right as 'minus means left', which contradicts the actual rightward shift; similarly 2x inside the bracket is read as a stretch by 2 rather than a compression by ½. Order errors arise because students never fix a convention for reading the form.
What to do instead
Anchor the transformation to one or two tracked points rather than the whole curve: choose a point such as the vertex or an intercept, apply the operations in the order dictated by the bracket, and check where it lands. Write every transformed function in the standard form af(b(x − h)) + k before starting, and state each transformation in words and in order as a numbered list. Verify with a graphing display by plotting both the original and the image.
Working in the wrong angle mode - computing arc length, sector area, or the derivative of a trigonometric function while the calculator is in degree mode, or leaving an answer in degrees for a question posed in radians.
The calculator retains its mode between sessions, and nothing on screen prominently flags a mismatch; the formulas s = rθ and A = ½r²θ are quoted without the caveat that they are radian-only, so the error passes unnoticed until the answer is wildly small.
What to do instead
Make 'check mode' the first action after switching on, and re-check after any reset or battery change. Train a reflex: if the question contains π or says 'radians', the mode must be radians. For derivatives and integrals of sin, cos and tan, memorise that the standard results only hold in radians and write 'x in radians' at the top of that part. If an answer looks off by a factor of about 57, suspect the mode immediately.
Applying the sine rule where the cosine rule is needed, and when the sine rule does apply, reporting only the acute solution and missing the obtuse possibility in the ambiguous SSA case.
Both rules look similar and both contain a side and its opposite angle, so pattern-matching happens on surface features rather than on which information is given. The ambiguous case is under-practised because most textbook questions are set up so the acute answer is the only one offered.
What to do instead
Classify the given data first, in writing: SSS or SAS means cosine rule; AAS or SSA means sine rule. For the SSA case, immediately compute the candidate angle θ and then also list 180° − θ, testing each against the requirement that the angles sum to less than 180°. Sketch the triangle roughly to scale before calculating so that an obviously obtuse angle is not reported as acute.
Treating 'independent' and 'mutually exclusive' as interchangeable, and computing P(A|B) as P(B|A) or as P(A ∩ B) without dividing by P(B).
Both phrases describe a relationship between two events, and both are tested by formulas containing P(A ∩ B), so students substitute whichever formula they recall first. Conditional probability is also read left-to-right as 'A and B together', which drops the restriction to the given event.
What to do instead
Separate the two ideas with a concrete numerical example you can reconstruct in the exam: mutually exclusive means the sets do not overlap so P(A ∩ B) = 0; independent means P(A ∩ B) = P(A)P(B). Draw a Venn diagram or two-way table for every probability question before touching a formula, and read P(A|B) aloud as 'within B, how much is A', shading B first. Check independence numerically by comparing P(A ∩ B) with P(A)P(B) rather than asserting it.
Misusing the regression line - using the y-on-x line to predict x from a given y, extrapolating well outside the data range, or claiming that a high correlation coefficient proves causation.
The calculator returns a single line labelled y = ax + b, which invites the assumption that it works in both directions; the distinction between the two least-squares lines is a small part of the syllabus and is easily forgotten. Interpretation questions feel like common sense, so students answer in loose everyday language.
What to do instead
When a value of x is given, substitute into the y-on-x equation; when a value of y is given and x is required, either use the x-on-y line or, more safely, solve the y-on-x equation and state the assumption. Always quote r (and r² where asked) with its sign and strength, and add an explicit caveat sentence: correlation does not imply causation, and prediction outside the observed range is unreliable. Note the data range in your working before extrapolating.
Omitting the constant of integration on indefinite integrals, and differentiating composite trigonometric, exponential and logarithmic functions without the chain rule - writing the derivative of sin(2x) as 2cos(2x)'s inner factor dropped, or of ln(3x) as 1/x.
The '+C' carries no marks in a student's mental model because it never changes a numerical answer, so it is treated as decoration and dropped under time pressure. Chain-rule omissions happen because the outer derivative is produced quickly and correctly, creating a false sense of a finished step.
What to do instead
Make '+ C' part of the written answer line itself rather than something appended afterwards - never write an indefinite integral without it, even in rough working that you intend to copy up. For derivatives, use a two-pass method: identify the inner function, differentiate the outer, then multiply by the inner derivative, and only then simplify. Practise a mixed set of sin(kx), cos(kx), e^(kx) and ln(kx) until the factor k is produced automatically.
Treating a definite integral as an area without regard to sign - reporting a negative area for a curve below the axis, or finding the area between two curves without first locating their intersection points and splitting the integral at any crossing.
The integral is learned as an 'area machine', so the possibility of signed contributions and of regions above and below the axis within one interval is not considered; intersection-finding is seen as a separate topic rather than a necessary step of the area calculation.
What to do instead
For any area question, sketch the curve or curves first, mark which parts lie below the axis, and split the integral at each zero or intersection. Take absolute values of each piece or subtract the lower function from the upper one over each sub-interval, then add the magnitudes. Distinguish explicitly between 'evaluate the integral' and 'find the area', and state in words which quantity the question wants before calculating.