Algebraic manipulation broke at step 3, the age 13–14 rung, and that is the second-lowest break of all eleven strands. Everything in this guide sits above that break, so it is worth being blunt about which sections will fight back.
Likely to be hard for you:
Probably easier than you expect: E2.10 tables of values and E2.11 sketching curve shapes. Both are recognition rather than manipulation, and recognition is not where your gaps are.
Paper 2 is non-calculator: 2 hours, 100 marks, 50% of your grade. Everything here is by hand, and every line is shown.
Number sense and the four operations: solid on all seven rungs. Not one gap, from the primary rung to Extended hard.
That matters here more than anywhere. Algebra is arithmetic with letters standing in for numbers, so a student with shaky arithmetic finds every algebra topic slow and error-ridden. You are the other case: the arithmetic underneath is reliable, and what is missing is a set of rules that were never properly laid down. Rules can be laid down in weeks. Weak arithmetic takes far longer. You have the harder half already.
Every idea appears four times, with less help each time.
Sections 1 to 10 are the sub-topics of syllabus topic 2 that sit beyond your manipulation repair guide. Section 11 is a mixed set, deliberately unlabelled and out of order.
Algebraic fractions follow exactly the same four rules as ordinary fractions. The only new thing is that you must factorise before you cancel, and there is one mistake that ruins more of these than everything else combined.
| Operation | What to do |
|---|---|
| Multiply | Factorise everything, cancel across, then multiply tops and bottoms |
| Divide | Turn the second fraction upside down and multiply |
| Add / subtract | Common denominator first. Usually just the two bottoms multiplied |
| Simplify | Factorise top and bottom fully, then cancel matching brackets |
Your equations strand was secure to step 5 and broke at step 6, so linear equations and simple rearranging are already there. This section starts just below that line to make sure, then climbs through the four Extended methods: fractional equations, simultaneous equations including a non-linear pair, and the three ways of solving a quadratic.
Most equation questions on Paper 4 start as a sentence, and half the marks are for turning it into algebra. Three moves:
| move | example |
|---|---|
| name the unknown, with its unit | let the width be w cm |
| write every other quantity in terms of it | the length is 3 cm more: w + 3 |
| find the sentence that is an equation, and write it | the perimeter is 34 cm: 2(w + w + 3) = 34 |
The same idea as above: clear the fractions. When a denominator holds x, multiply every term by every denominator. The result is a linear or a quadratic equation. At the end, check that no answer makes a denominator zero.
| Method | Use it when |
|---|---|
| Factorising | It factorises. Always try this first — it is by far the quickest |
| The formula | It will not factorise, or the question says give the answer to 2 d.p. |
| Completing the square | The question says so, or asks for the turning point, or asks for the answer in exact form |
The subject is the letter on its own on one side. To change it, undo whatever has been done to the new subject, in the reverse order, doing the same to both sides. It is solving an equation with letters instead of numbers.
When the subject appears twice, there is a fixed routine: clear the fraction, collect every term with the subject on one side, factorise the subject out, then divide.
An inequality is solved in exactly the same way as an equation, with one extra rule. That one rule is where every lost mark in this sub-topic comes from.
A region question gives you two or three inequalities. Draw each boundary line as if it were an equation, decide which side you want, then shade. Cambridge's rule is to shade the side you do not want and leave the required region R clear, unless the question tells you otherwise — so read the question.
Sometimes the region is drawn and you write the inequalities. Work one boundary line at a time:
| step | what to do |
|---|---|
| 1 | find the equation of the boundary line (x = k, y = k, or y = mx + c from two points on it) |
| 2 | pick a point clearly inside R, not on any line |
| 3 | put it into the left side of the equation and see whether it is bigger or smaller than the right side: that is the direction |
| 4 | solid line: ≤ or ≥. Broken line: < or > |
Sequence questions come in three flavours and each has its own giveaway. Work out which flavour you are looking at before doing anything else, by checking the differences.
| Type | Giveaway | nth term looks like |
|---|---|---|
| Linear (arithmetic) | First differences are constant | an + b |
| Quadratic | First differences change, second differences constant | an2 + bn + c |
| Exponential (geometric) | You multiply by the same number each time | a × rn or similar |
Tn is just a name for the nth term of a sequence called T, so T3 is its 3rd term. If Tn = 2n3 − 5, then T3 = 2 × 27 − 5 = 49.
| type | giveaway | compare it with |
|---|---|---|
| cubic | the THIRD differences are constant | the cubes n3: 1, 8, 27, 64, 125 |
| a combination | none of the above fits on its own | n2, n3 or 2n, written under it term by term |
| related to another sequence | the question gives you a second sequence | that sequence, term by term |
Every proportion question on the Extended paper follows the same three steps, whatever the wording. Learn the three steps and this becomes one of the most reliable topics on the paper.
| Wording | Equation |
|---|---|
| y is directly proportional to x | y = kx |
| y is proportional to the square of x | y = kx2 |
| y is proportional to the square root of x | y = k√x |
| y is proportional to the cube of x | y = kx3 |
| y is inversely proportional to x | y = k⁄x |
| y is inversely proportional to the square of x | y = k⁄x2 |
| y is inversely proportional to the cube root of x | y = k⁄3√x |
These graphs look like a reading exercise and are really a gradient exercise. Three ideas cover every question: what the gradient means, what the area under the graph means, and what a flat section means.
| Graph | Gradient means | Area under means |
|---|---|---|
| Distance against time | Speed | Nothing useful |
| Speed against time | Acceleration | Distance travelled |
| Conversion graph | The exchange rate between the two units | Nothing useful |
Choose scales that use most of the grid, label each axis with the quantity and its unit, and plot each point with a small cross. On a travel graph join the points with ruled straight lines: a stop is a horizontal line, and coming home goes back down to distance 0.
A conversion graph is a straight line through (0, 0). To draw one, work out one point far along, say 100 euros = 115 dollars when 1 euro = 1.15 dollars, and rule a line from the origin through it.
On a curve the gradient changes from point to point. To estimate it at one point, rule a tangent there: a straight line that touches the curve at that point and does not cut across it. Rule it long, draw a big right-angled triangle under it, and read the rise and the run from the axis scales. Gradient = rise ÷ run, and it is negative if the line falls.
| graph | the gradient of a tangent means |
|---|---|
| distance–time | the speed at that instant |
| speed–time | the acceleration at that instant |
| volume–time | the rate of flow at that instant |
This sub-topic is mostly careful arithmetic rather than new theory, which is why it is rated low risk for you — your number sense scored full marks. Three things get tested: filling a table of values, plotting the curve, and reading solutions off it.
Keep to one row per term. Put negative x values in brackets. A term like 3/x2 or 12/x cannot be worked out at x = 0, so the table jumps over it, and the curve is never drawn across x = 0.
| x | −1 | 0.5 | 1 | 2 | 3 |
|---|---|---|---|---|---|
| 2x | −2 | 1 | 2 | 4 | 6 |
| 3/x² | 3 | 12 | 3 | 0.75 | 0.33 |
| y = 2x + 3/x² | 1 | 13 | 5 | 4.75 | 6.33 |
A graph of y = a × bx starts at a when x = 0 and is multiplied by b at each step: b greater than 1 is growth, b between 0 and 1 is decay. The curve gets ever closer to the x-axis on one side but never touches it. The graph lets you read values the formula cannot give by hand.
Sketching is recognition, not calculation. You are given an equation and asked for the shape, so what you need is a small gallery of shapes in your head and one rule for each about which way up it goes.
An asymptote is a line the curve gets closer and closer to without ever reaching. Draw it as a dashed line and label its equation.
| curve | vertical asymptote | horizontal asymptote | useful point |
|---|---|---|---|
| y = a/x + b | x = 0 | y = b | crosses the x-axis where a/x = −b |
| y = arx + b | none | y = b | crosses the y-axis at (0, a + b), because r⁰ = 1 |
y = 3/x + 2 is the curve y = 3/x slid up by 2, so both branches now hug the line y = 2 instead of the x-axis. It crosses the x-axis where 3/x = −2, at x = −1.5.
Written as y = a(x + p)² + q, a quadratic has its turning point at (−p, q). The squared bracket is never negative, so if a is positive the smallest y is q, when the bracket is 0: a minimum. If a is negative, q is the largest y: a maximum.
Differentiation is new notation attached to one short rule. It is rated high risk for you for one reason only: it runs entirely on index rules, and indices was your lowest break of the eleven strands. The rule itself takes a minute to learn. Getting the powers right is what will need practice.
| y | dy⁄dx | Why |
|---|---|---|
| x5 | 5x4 | Multiply by 5, drop the power to 4 |
| 3x2 | 6x | 3 × 2 = 6, and x1 is written as x |
| 7x | 7 | x is x1, so 7 × 1 × x0 = 7 |
| −4x3 | −12x2 | −4 × 3 = −12. The sign is carried through |
| 9 | 0 | A constant is a flat line, and a flat line has gradient 0 |
At the very top of a hill or the very bottom of a valley the curve is momentarily flat, so its gradient is zero. That is the whole method: set dy⁄dx = 0 and solve.
Section 6 estimated a gradient by ruling a tangent. dy/dx gives the same gradient exactly. For y = x², a tangent drawn at x = 1.5 gives a gradient of about 3, and dy/dx = 2x = 2 × 1.5 = 3 exactly. When a question says “by drawing a tangent”, it wants the drawing and an estimate; when it gives an equation and says “find”, use dy/dx.
Function questions are almost pure notation, and notation is exactly where an under-practised student loses marks quickly. Nothing here is difficult once you can read it, so read this section slowly and the marks are straightforward.
f(x) = 3x − 1 means “the rule f takes a number, triples it and subtracts 1”. f(4) means put 4 into that rule: f(4) = 12 − 1 = 11. The bracket is not multiplication.
| Notation | Means |
|---|---|
| f(3) | Substitute 3 into f |
| Domain | The set of inputs allowed |
| Range | The set of outputs produced |
| f−1(x) | The inverse — the rule that undoes f |
| fg(x) | Composite: do g first, then f on the result |
Fifteen questions drawn from all ten sections above, shuffled and unlabelled. Ordinary revision does one sub-topic at a time, which quietly does the hardest part for you — working out which method applies. Here nobody tells you. Before you write anything, name the method.