Paper 1: no technology

Paper 1 is the paper sat without any technology: no graphic display calculator (GDC), no scientific calculator, only a pen, pencil, ruler and a clean copy of the formula booklet. It tests the same course content as Paper 2, but every question is designed so that it can be answered by hand. That changes the kind of mathematics that appears: exact values, algebraic manipulation, calculus done by hand, proofs and sketches are at the centre of the paper, and the numbers are chosen so that the arithmetic works out.

Both SL and HL students sit a Paper 1. The HL paper is longer and includes the additional higher level content, but its structure and the skills it rewards are the same. Every question on the paper is compulsory: there is no choice of questions.

How the paper is built

The paper opens with five minutes of reading time, during which writing is not allowed. It then has two sections.

Section A: short-response questions

Section A is a run of short questions, typically worth 4 to 8 marks each. Each question is answered in the space printed under it in the question paper, with the working written in the same space. The questions usually test one or two ideas each: solving a log equation, finding an exact trigonometric value, differentiating with the chain rule, finding a term of a binomial expansion, solving a probability problem with fractions.

Section A questions are roughly ordered from shorter and more routine to longer and less familiar, but that ordering is loose. A question late in Section A may be quicker than one early on.

Section B: extended-response questions

Section B contains a small number of long, multi-part questions, each typically worth 13 to 20 marks. They are answered in a separate answer booklet, and each question should start on a new page. A Section B question develops one situation through several parts: a function is analysed and then sketched, a sequence is set up and then summed, a probability model is built and then used.

The parts of a Section B question are linked. Later parts often say "hence", use a result printed in an earlier "show that" part, or build on a diagram drawn earlier. The first parts are usually routine, so every Section B question has accessible marks at the start, even for a topic that feels weak.

Planning the time

Paper 1 gives a little more than one minute per mark at both SL and HL. A 6-mark question is therefore worth about 6 to 7 minutes, and a 16-mark Section B question about 18 minutes. Use the marks printed beside each part as a budget, both for time and for how much to write: a 1-mark part needs a single result, a 7-mark part needs a full method.

Using the reading time

Five minutes is enough to read every question once. Useful things to do with it:

• Read Section B first, since those questions carry the most marks and are the longest to understand. Decide which one looks most familiar.

• Spot the "show that" parts in Section B. Their answers are printed, so they can be used in later parts even if the proof itself is not finished.

• Note any Section A question that looks long or unfamiliar, to leave until later.

• Plan the first two or three lines of the first question, so writing can start straight away.

Moving on when stuck

The marks are not in order of difficulty, so getting stuck on one question costs marks elsewhere. A good rule is to leave a question after spending about one and a half times its mark allocation in minutes without progress, mark it in the margin, and return to it at the end. Leave space to finish it.

In Section B, a stuck part rarely ends the question. Later parts often give a result to use ("Show that $f^{\mathrm{\prime }}(x)=\dots$"; "Given that $k=3$, find…"), so read on and collect the marks that follow.

Arithmetic without a calculator

Paper 1 questions are written so that the arithmetic is manageable by hand. Fluency still matters: slow or careless arithmetic is a common reason for running out of time and for losing accuracy marks.

Exact values

Answers on Paper 1 are usually exact values: fractions, surds, multiples of $\pi$, logarithms and powers of $e$. Leave answers such as $\frac{2\sqrt{3}}{3}$, $\ln 5$ or $\frac{\pi }{6}$ in that form unless the question asks for something else. A decimal approximation of an exact answer is not what is asked for and may not earn the final mark.

The exact values of the trigonometric ratios of the standard angles are not in the formula booklet and must be known by heart.

Together with the symmetries of the unit circle (see 3.5), these give the exact values in every quadrant.

The sign decision in part (a) is where most marks are lost. A Paper 1 question gives the interval for a reason: it decides the sign, and an answer of $\pm \frac{4}{5}$ does not earn the final mark.

Surds, fractions and powers

Expect to simplify surds, rationalise denominators, work with negative and fractional indices, and combine algebraic fractions. These skills are taught in their own subtopics, but on Paper 1 they appear inside larger questions, so they must be quick and reliable.

Logarithms and exponentials

Log equations on Paper 1 are built to give exact answers. The usual method is to combine the logarithms into one using the laws of logarithms, convert to exponential form, solve, and then check every solution against the domain of the original logarithms.

Calculus by hand

Paper 1 is where differentiation and integration are tested as techniques. Expect to differentiate with the chain, product and quotient rules, find equations of tangents and normals, locate and classify stationary points, integrate standard functions and evaluate definite integrals exactly.

Exact definite integrals

A definite integral on Paper 1 is evaluated with an antiderivative, and the limits are usually chosen so that the result simplifies.

The step $e^{2\ln 3}=9$ is the kind of simplification Paper 1 expects you to know: $e^{\ln a}=a$ and $k\ln a=\ln a^k$. An answer left as $\frac{1}{2}{e}^{2\ln 3}-\frac{1}{2}$ is correct but unfinished.

Stationary points and sketches

Without a GDC, a curve's shape comes from its algebra: intercepts from setting $x=0$ and $y=0$, stationary points from $f^{\mathrm{\prime }}(x)=0$, their nature from the second derivative or a sign table, and asymptotes from the form of the function. "Sketch" questions on Paper 1 expect these features to be found and labelled, not guessed.

A Section B question in full

The following question has the typical shape of a Section B question: routine differentiation, then a stationary point problem, then a "hence" part that uses the results, then a sketch.

Several features of this question are typical. Part (a) is worth 2 marks and needs one line. Part (c) says "justifying", so a correct classification without the second derivative values (or a sign test) loses the reasoning mark. Part (d) says "hence", so the answer must come from the stationary values just found. In part (e), the sketch only earns its marks if the labelled points match the work in (b) and (c).

Checking work without a calculator

Without a GDC, every check has to be done by hand, so build cheap checks into the working.

• Substitute back. A solution to an equation can be checked in the original equation in a few seconds. Do it for log, trigonometric and quadratic equations.

• Check the size. A probability must lie between 0 and 1; a length must be positive; an area must be positive; an angle in a triangle must be less than $\pi$.

• Differentiate an answer to an integral. Differentiating the result should give back the integrand.

• Use the "niceness" of the numbers. Paper 1 numbers are chosen to work out. If a calculation produces $\frac{173}{29}\sqrt{7}$ in a question that seemed simple, there is probably an arithmetic error earlier.

Writing answers that earn the method marks

Paper 1 markschemes give most of the marks for method. A correct final answer with no working may not earn full marks, and a wrong final answer with a clear correct method can earn most of them (see Accuracy, notation and setting out). Write each step on its own line, connected by words or by $=$ signs that are true.

Some typical places where method marks are earned on Paper 1:

• writing the formula used before substituting (for example $S_n=\frac{n}{2}(2u_1+(n-1)d)$ with values in place)

• setting a derivative equal to zero to find a stationary point

• writing the antiderivative in square brackets before substituting the limits

• drawing a tree diagram or listing a sample space in probability

• stating the identity used to rewrite a trigonometric equation.

Answers in the correct form

Many Paper 1 questions ask for an answer in a particular form: "in the form $a+b\sqrt{3}$ where $a,b\in \mathbb{Z}$", "in the form $\ln k$", "as a single fraction". The final mark is for matching that form exactly. Read the question again before writing the final line.

The final minutes

Keep the last five to ten minutes for the questions skipped earlier and for checks. A partial answer to a skipped question earns method marks; a blank space earns nothing. If time is short, write the method in outline (the equation to solve, the formula to use) rather than leave the part empty.

Checklist

• To do:

  I know the standard exact trigonometric values and can find them in every quadrant.

• To do:

  I can simplify surds, logs and indices quickly and accurately by hand.

• To do:

  I can differentiate and integrate by hand and evaluate definite integrals exactly.

• To do:

  I check every solution of a log or trigonometric equation against the domain or interval.

• To do:

  I can sketch a curve from its algebra, with intercepts, stationary points and asymptotes labelled.

• To do:

  I budget about one minute per mark and move on when stuck.

• To do:

  I use printed "show that" results in later parts, even if I could not prove them.