Accuracy, notation and setting out

Introduction

Many marks in this course are lost after the mathematics has been done correctly. An answer rounded to 2 significant figures instead of 3, a value carried forward after early rounding, a probability written as a calculator command, a missing unit or a missing line of working can each cost a mark on an answer that was otherwise right. These losses are entirely avoidable, because the rules behind them are fixed and the same on every paper.

The rules fall into three groups. Accuracy covers how many figures to give, when to give an exact value and how to keep values accurate through a long question. Notation covers writing answers in correct mathematical form rather than in the calculator's form. Setting out covers how working is shown so that the examiner can award the marks it earns. All three make sense once the way a markscheme awards marks is clear, so that comes first.

How a markscheme awards marks

Every question is marked against a markscheme, which lists each mark, what earns it and any notes on what is or is not accepted. Markschemes for this course use a small set of codes.

The codes describe what each mark is for. A method mark is for a correct method, an accuracy mark for a correct answer or value, and a reasoning mark for a valid reason.

An accuracy mark usually depends on the method mark before it: a correct-looking answer reached by a wrong method normally earns nothing. On one line, "M1A1" typically means M1 for the method (for example, substituting into a formula) and A1 for the correct values or answer. Several A marks on one line, such as A1A1A1 for three coefficients, are awarded independently.

Reasoning marks appear in parts that ask for an explanation, a justification or a conclusion from a test. Markschemes pair them with the conclusion's accuracy mark and add the note "do not award R0A1": a correct conclusion with no valid reason scores nothing.

In a "show that" question the answer is printed in the question, so writing it earns nothing. The markscheme labels the final line AG.

Implied marks, follow-through and alternatives

A mark printed in brackets, such as (M1) or (A1), is an implied mark: it is awarded if the work is seen, or if it is implied by later correct working or a correct final answer. So a correct answer with no working earns all the bracketed marks for that part. Unbracketed marks (M1, R1, A1) need the working to be seen. In the markschemes for this course since 2021, implied method marks are more common than unbracketed ones, but many method marks still need to be seen.

Follow-through (FT) lets correct working from a wrong earlier answer earn its marks, so one error is penalised once rather than in every later part. It needs working to be shown, and markschemes limit it when the error makes a later part much easier or produces an impossible value, such as a probability greater than 1. When a question can be answered in more than one way, the markscheme lists the alternatives (METHOD 1, METHOD 2, or EITHER … OR … THEN), and any correct method is marked in line with them unless the question names a method. Older markschemes in past-paper collections also show N, G and C marks (marks for answers with no working); these belong to earlier courses and are not used in this one.

Accuracy

Three significant figures by default

Unless a question says otherwise, every final numerical answer is given exactly or correct to three significant figures (3 s.f.). Markschemes print the 3 s.f. answer, often followed by the unrounded value in brackets: $0.177(0.176953\dots )$. An answer given to fewer figures is not accurate enough; one given to more figures than 3, if correctly rounded, is usually accepted, but the safe choice is to write the unrounded value and then the 3 s.f. value.

Zeros need care. To 3 s.f., 98.96 is 99.0, not 99, because the zero after the decimal point is the third significant figure. Leading zeros in a small number are not significant, so 0.0315926 to 3 s.f. is 0.0316. Trailing zeros in a large whole number are placeholders, so 61,334.3 to 3 s.f. is 61,300.

Accuracy set by the question or the context

When a question states an accuracy, it replaces the 3 s.f. rule, and a mark depends on it. "Give your answer to the nearest dollar", "correct to two decimal places" and "to one decimal place" all appear regularly, and markschemes say, for example, "award A0 if not rounded to a whole number". Money is usually given to the nearest cent or the nearest whole unit, as the question asks; when it gives no instruction, 2 decimal places (the nearest cent) or 3 s.f. are usually accepted.

The context can also fix the accuracy. Numbers of people, objects, payments or months must be whole numbers, and the direction of rounding depends on what the question asks.

Exact answers

When a question asks for an exact value, a decimal approximation loses the mark. Exact values keep fractions, $\pi$, roots and logarithms unevaluated: the volume of a cylinder of radius 3 cm and height 5 cm is exactly $45\pi$ cm³, and the hypotenuse of a right-angled triangle with shorter sides 5 and 7 is exactly $\sqrt{74}$. Writing 141 cm³ or 8.60 is correct to 3 s.f. but is not exact. Examiners' reports note that most candidates in one session lost a mark by giving a length to 3 s.f. when the exact length was asked for.

Early rounding

Rounding in the middle of a calculation is the most frequently reported cause of lost final marks in this course. A value rounded to 3 s.f. and then reused can move the final answer enough to change its third significant figure. The rule is to keep full accuracy in the GDC until the final answer of each part and to write the rounded values on the page only for the reader.

Some markschemes accept answers that come from 3 s.f. intermediate values, with notes such as "accept 37.1 and 37.4 from use of 2 sf and/or 3 sf values". This is a concession for particular questions, not a rule, and cannot be relied on.

Show that with a rounded answer

When a "show that" part gives a rounded value, such as "show that the angle is 44.0°, correct to three significant figures", the working must include a value to more figures than the given one, such as 44.0486…, before the rounded value. Without it, the answer looks copied from the question. One Paper 3 markscheme, for example, required a correct value to 4 significant figures or more before the final mark could be awarded.

Notation

No calculator notation

Answers must be written in correct mathematical notation, not the calculator's. The two commonest problems are standard form and function commands.

• A display such as 5.2E30 or 1.14E10 means $5.2\times {10}^{30}$ or $1.14\times {10}^{10}$, and must be written that way. In one session, answers written in the calculator's E form scored at most one mark out of two.

• A command such as binomcdf(20, 0.3, 7) or poissoncdf(5.4, 2) is not mathematical notation. Define the random variable and write the probability: $X\sim \mathrm{B}(20,0.3)$, $\mathrm{P}(X\le 7)=0.772$.

The question's variables and symbols

Use the letters the question defines. If the time is $t$ and the height is $h$, a regression equation is $h=-0.452t+3.17$, not $y=-0.452x+3.17$; markschemes can withhold an accuracy mark for an otherwise correct equation written with $x$ and $y$. If a new variable is needed, define it in words before using it ("let $n$ be the number of complete months").

Standard notation is expected everywhere else too. A few forms come up often.

The equals sign means the two sides are equal. A chain of working in which each "=" joins things that are not equal (an expression equal to its value at a point, then equal to a later answer) is wrong notation and can make correct working hard to follow.

Units

Give units with every answer that has them: m, cm², cm³, s, m s⁻¹, °C, dollars. Most markschemes show units in brackets, which means they are not required for the mark, but some parts require them ("units must be given for the final A1"), and wrong units, such as cm² for a volume, are listed in examiners' reports as a common loss. Since giving the right unit never loses marks, the safe habit is to give it every time.

Setting out working

The markscheme can only award marks it can see. When the GDC does the calculation, the working is shorter than in a hand calculation, but every part except a "write down" or "state" should still have a method line. A good method line shows what was calculated and how: a formula with values substituted, the equation entered into the GDC, the distribution with its parameters, the TVM entries, or the test with its data.

Setting out also means making the order and the final answer clear. Write each part's label, keep its working together, and finish with a clear final answer, with units and the stated accuracy. If two different final answers are left for the same part, normally only the first is marked, and the mark can be lost. Cross out any attempt you are replacing and leave a single answer.

The cup question brings together the main accuracy rules: an exact answer kept as a root, a volume found from the unrounded height, a unit conversion stated, a count rounded in the direction the context requires, and a 3 s.f. answer that keeps its trailing zero.

Common mistakes

Rounding a context answer to the nearest whole number by habit is a close second. Before rounding a number of people, cups, buses, years or payments, decide whether the context needs rounding up, rounding down or a multiple of some period, and write a few words that show the decision.

Summary

• A markscheme awards method (M), accuracy (A) and reasoning (R) marks; AG marks a given answer; FT allows follow-through with working shown.

• Bracketed marks are implied by a correct answer; unbracketed marks need the working seen; "do not award R0A1" means a conclusion needs its reason.

• Give final answers exactly or to 3 significant figures unless the question says otherwise; keep trailing zeros that are significant (99.0).

• Follow any stated accuracy exactly, and round counts in the direction the context needs.

• Give exact answers (fractions, $\pi$, roots) when "exact" is asked for.

• Never round in the middle of a calculation; keep full values in the GDC.

• Write standard form and probabilities in mathematical notation, never in calculator notation; use the question's variables.

• Give units with every answer that has them, and give one clear final answer per part.

Self-check

• To do:

  I can explain what M1, (M1), A1, R1, AG and FT mean in a markscheme.

• To do:

  I can say which marks a correct answer with no working earns.

• To do:

  I can round any value to 3 significant figures, including small numbers and values ending in zero.

• To do:

  I can tell 3 significant figures from 3 decimal places.

• To do:

  I can decide whether a context answer should be rounded up, down or to a multiple.

• To do:

  I can give an exact answer in terms of $\pi$, a fraction or a root.

• To do:

  I can keep unrounded values in my GDC through a multi-part question.

• To do:

  I can convert calculator output such as 3.7E−5 into standard form.

• To do:

  I can write a probability, an integral and a hypothesis in correct notation.

• To do:

  I can give units, including squared and cubed units, with every answer.