Accuracy, notation and setting out

A correct method can still lose marks through the way the answer is written: a value rounded too early, a final answer to the wrong accuracy, a missing $+C$, or a "show that" argument that jumps to the result. The rules behind these marks are consistent across Papers 1, 2 and 3, and once known they are easy to follow. Most of them also make the working clearer and easier to check.

The accuracy rule

Unless a question says otherwise, every numerical answer should be given exactly or correct to three significant figures (3 s.f.). This instruction is printed on the cover of each paper, and it applies to every part of every question.

Exact answers

An exact answer is one written without rounding: $\frac{5}{3}$, $2\sqrt{7}$, $\ln 4$, $\frac{\pi }{6}$, $e^{-2}$. On Paper 1 almost every answer is exact, because the numbers are designed for it. On Paper 2, an exact answer is needed only when the question asks for it ("find the exact value", "give your answer in the form $a\sqrt{b}$"); otherwise 3 s.f. is the usual form.

An exact answer should still be finished. Arithmetic that can be completed should be completed: $\sqrt{\frac{25}{4}}$ should become $\frac{5}{2}$, and $\frac{3}{0.1}$ should become $30$. A fraction that is not in its lowest terms, such as $\frac{10}{4}$, is generally accepted as long as the numerator and denominator are integers, but simplifying it costs nothing.

Three significant figures

Significant figures count from the first non-zero digit. Three significant figures of $0.0066807$ are $0.00668$; of $8342.26$ they are $8340$; of $180.252$ they are $180$.

An answer given to more than 3 s.f. that is correct is usually accepted unless the question specifies the accuracy, but an answer given to fewer, such as $0.067$, is not accurate enough and loses the accuracy mark.

Angles, money and other contexts

The 3 s.f. rule applies to angles too, so $36.8698\dots \mathrm{°}$ is $36.9\mathrm{°}$. Many students give angles in degrees to one decimal place, which for angles of $10\mathrm{°}$ or more is at least as accurate as 3 s.f. and so is also accepted. Money is usually given to the accuracy the question asks for, often the nearest cent or the nearest whole unit. When nothing is specified, 3 s.f. still applies, although two decimal places for money is also correct and more natural.

Probabilities are given as fractions or decimals, exact or to 3 s.f. A probability of $0.000234$ is written to 3 s.f., not as $0.000$.

Intermediate values and premature rounding

Rounding an intermediate value and then using the rounded value in later steps is called premature rounding. The error it introduces grows through the calculation and can change the final answer in the third significant figure, which loses the accuracy mark.

The safe habit has two parts. Keep every intermediate value at full accuracy (stored on the GDC, or written to at least five or six significant figures), and round only the final answer. In the working, write intermediate values with an ellipsis or extra figures, such as $k=0.186538\dots$, to show that the full value was used.

Units

In context questions, give the units with every final answer: $11.2$ m, $3.35$ m s${}^{-1}$, $180$ cm, $0.421$ units${}^2$ for an area with no units given. Read the units in the question carefully: a time given in minutes, a model with $t$ in years, a length in centimetres with a final answer asked for in metres. A missing unit is not always penalised, but a wrong unit (or a conversion error) usually is.

Notation that loses marks

Mathematical notation is part of the answer. Several habits look harmless but cost marks or make working unreadable.

Equals signs

An equals sign states that two things are equal. A chain such as

is false at almost every step, even though the student's thinking is correct. Write each equation on its own line, or connect steps with words or with $⟹$.

Similarly, do not write $=$ between an expression and its derivative ("$x^3=3x^2$"). Write $\frac{d}{dx}(x^3)=3x^2$ or $f^{\mathrm{\prime }}(x)=3x^2$.

Calculus notation

• An indefinite integral needs $+C$: $\int 2xdx=x^2+C$. Without it the final mark is usually lost.

• Every integral needs its $dx$ (or $dt$, $dy$): ${\int }_0^2{x}^{2}dx$, not ${\int }_0^2{x}^2$.

• Use $f^{\mathrm{\prime }}(x)$ or $\frac{dy}{dx}$ consistently, and write ${\lim }_{h\to 0}$ in front of every line of a limit until the limit is taken.

Calculator notation

Calculator syntax is not mathematical notation. Avoid `2.5E7` (write $2.5\times {10}^7$), `x^2` (write $x^2$), an asterisk for multiplication (write $\times$ or nothing), `sqrt(5)` (write $\sqrt{5}$), and function names such as `normalcdf` and `binompdf` (write the probability statement, such as $P(X\le 4)$ with $X\sim B(12,0.3)$).

Other common slips

How marks are awarded

IB markschemes use a small set of mark types. Knowing them explains why working matters so much.

A mark written in brackets, such as (M1) or (A1), can be implied by later correct work. If a correct final answer appears without the step, the bracketed mark is still given. Marks without brackets, such as M1, are awarded only when the method is seen, so a correct answer with no working may not earn them.

A marked solution

The following short question shows how marks attach to lines of working.

The last case is the strongest argument for showing working. A short line of method protects most of the marks against a single slip.

Follow-through

If an answer to one part is wrong and the wrong value is then used correctly in a later part, the later part can earn its marks, marked FT. The error is penalised once, where it happens.

For example, a question asks (a) find $f^{\mathrm{\prime }}(x)$ for $f(x)=x^3-6x$, and (b) find the gradient of the curve at $x=2$. A student who writes $f^{\mathrm{\prime }}(x)=3x^2-6x$ in part (a) loses marks there. In part (b), the student substitutes correctly: $f^{\mathrm{\prime }}(2)=3(4)-12=0$. The correct gradient is $6$, but part (b) still earns full follow-through marks, because the method in (b) is correct for the derivative the student found.

Follow-through has limits. It is usually not awarded when the earlier error makes a later part much simpler (for example, a wrong derivative that turns a quadratic equation into a linear one), or when the follow-through answer is clearly impossible, such as a negative length or a probability greater than $1$, and no comment is made. Follow-through can only be awarded when the working is visible.

A related rule covers misreading. If a value is copied from the question incorrectly and then used correctly, a small penalty is usually applied once and the rest of the work can be marked as if the copied value were correct, provided the question has not become easier.

Setting out a "show that"

A "show that" question prints the answer, so every mark is for the steps. Examiners look for a clear logical route from the given information to the printed result.

1. Start from what is given, or from one side of the result.

2. Write every algebraic step on its own line, including steps that seem obvious.

3. Do not work backwards from the result, and do not manipulate both sides of an identity at once.

4. End with the printed result, written exactly as it appears, and a short statement that it has been reached.

The final line matters

In a "show that", the printed result must actually appear at the end of the working. A student who reaches $24+4k-4k^2\ge 0$ and stops has not finished, even though the result follows. Write the final step and the result.

Presentation on the page

• Number every part clearly: (a), (b)(i), (b)(ii). In Section B and Paper 3, start each question on a new page.

• Cross out errors with a single line instead of erasing them. Crossed-out work is not marked if a replacement is given, so do not leave two different answers to one part: the examiner may mark the first one, or the worse one.

• Write the final answer clearly, often as a sentence or underlined, so it is not mistaken for an intermediate value.

• Draw diagrams in pencil, label everything, and keep them large enough to read.

• Keep working near the question it belongs to. Work for one question written in the space for another may be missed.

Checklist

• To do:

  I give every answer exactly or to 3 s.f., unless the question asks for something else.

• To do:

  I keep intermediate values unrounded and round only the final answer.

• To do:

  I give units with context answers and answer in the words of the question.

• To do:

  I write $+C$, $dx$, coordinates in brackets and probability statements correctly.

• To do:

  I never write calculator syntax as working.

• To do:

  I understand M, A, R, AG and FT marks and show the working they need.

• To do:

  I set out "show that" proofs forwards, step by step, ending with the given result.