On the Foundations Check your algebra strand was secure to step 2 and broke at step 3 — the age 13–14 rung. That is the second-lowest break of the eleven strands, and it is a long way below where the Extended paper will ask you to work.
It maps onto three syllabus sections: E2.1 Introduction to algebra, E2.2 Algebraic manipulation and E2.4 Indices II. E2.2 and E2.4 are both rated high non-calculator risk, and Paper 2 is non-calculator: 2 hours, 100 marks, 50% of your grade. So everything below is done by hand, every line written out.
Algebra is not one topic sitting beside the others — it is the spine that runs through them. It turns up inside solving equations, inside straight-line graphs, inside rearranging a volume formula, inside trigonometry. A gap here leaks into topics that look completely unrelated, which is why this guide is worth the hours.
One more thing the check picked up. Across several different strands, the error that kept coming back was a sign error — a minus that got lost, or a minus that appeared from nowhere. Five of your wrong answers were ones you were confident about, which is the signature of a rule being applied smoothly and wrongly rather than a topic being unknown. Algebra is where sign errors do the most damage, because one dropped minus in line 2 poisons every line after it. So signs are the through-line of this guide: they come back in every single section.
Your number sense scored solid on all seven rungs — primary level right the way up to Extended hard. Not one break. The four operations, place value, negatives in arithmetic, order of operations: all working.
That matters, because it rules out the explanation people reach for first. The arithmetic engine underneath is fine. What is missing is the layer above it: the habit of working with a letter the same way you already work with a number. That is a much smaller, much more fixable thing than “bad at maths”.
Each idea appears four times, and each time you get less help: worked in full, then last step is yours, then last two are yours, then all yours. Read the full one properly — do not skim to the questions. The reveal buttons let you check one step at a time instead of seeing the whole answer at once.
Work with a pen and paper next to the laptop. Write every line. The single biggest cause of sign errors is doing two steps in your head and typing the result.
Almost every algebra mistake that is not a sign mistake is a sign mistake wearing a disguise. So this comes first, and it is not a table to memorise. It is one picture — the number line — and everything else falls out of it.
On the number line, plus means move right and minus means move left. A negative number is itself a “turn round” instruction. So when you meet two signs together, you carry out both instructions, one after the other.
Read it as two turns. The − outside says “go the opposite way”. The
−2 inside is already pointing left. Opposite of left is right. Two turns bring you
back to facing right, so you move right 2.
+ + and − − both give +. + − and
− + both give −. Same signs make plus, different signs make minus.−5 − 3 as if it were
−5 − (−3) and answering −2. There is only one sign on the 3
here, so it is a plain subtraction: start at −5, move left 3, land on −8. The
double-negative rule only fires when you can actually see two signs stacked, usually with a bracket
between them.You do not need to memorise the four-line table either. Follow the pattern down a times table and the answer forces itself.
The left column drops by 1 each row, so the answers must go up by 3 each row. By the time the second number is negative, the answers have crossed zero and gone positive. Nothing was decided by a rule — the pattern had no choice.
−2 × −3 × −4
has three minuses — odd — so the answer is negative: −24. Division behaves exactly the
same way.−3 − 3 = −6 but −3 × −3 = +9. Two negatives
added take you further left; two negatives multiplied bring you back positive. Check which operation
you are actually doing before you touch the signs.This is one of the most examined traps at this level, and it is worth thirty seconds of care every time you see it. The difference is entirely about what the power is attached to.
A power binds more tightly than a minus sign. In −3² there is no bracket, so
the little 2 is glued to the 3 only, and the minus sign is left outside waiting. Read it as
“the negative of three squared”. In (−3)² the bracket sweeps the
minus sign inside, so the whole of −3 gets squared.
Fourteen of them, quick. Do them in order and do not skip the ones that look easy — the pairs are deliberately placed next to each other so you have to notice the difference rather than settle into a rhythm.
A letter is a number you have not been told yet. That is all it is. It obeys every rule that numbers obey — you can add it, multiply it, square it — and the only thing you cannot do is work out what it equals until someone tells you.
Two pieces of shorthand to have straight, because a lot of confusion at this level is really just notation:
5a means 5 × a. The multiplication sign is left out to avoid it being
mistaken for the letter x.a² means a × a. It does not mean 2 × a. Those are wildly
different: if a = 5, then a² = 25 but 2a = 10.a⁄3 or a⁄3 means a ÷ 3.“Collecting like terms” sounds technical. It is counting objects that are the same kind of object. Five somethings plus three more of the same something is eight of them, and you do not need to know what the something is worth.
The middle row is the one people fight. 4x + 3y looks unfinished, so there is
a pull to write 7xy or 7. But x and y are two different unknowns — four of one thing and three of
another. Four apples plus three oranges is not seven applesoranges. 4x + 3y is the final answer;
it is allowed to be two terms long.
3a²b and 7a²b are like
terms. 3a²b and 7ab² are not.x² + 3x = 4x²
or 4x is one of the most common errors in the whole subject. x² is x multiplied by
itself; x is a single x. If x = 10 then x² + 3x = 100 + 30 = 130, whereas 4x² would be 400.
Substituting a number is how you catch yourself.7a − 3b − 2a the terms are +7a,
−3b and −2a. If you rearrange to put the a terms together you must
carry the − with the 2a: 7a − 2a − 3b. Leaving the minus behind is where the sign
error creeps in.Substitution is where the check caught you, and the shape it caught you on was 3a² with a negative a. It is worth being precise about why that particular question is harder than it looks: two rules collide in it, and they have to be applied in the right order.
3a² with a = −2 becomes 3(−2)². The
bracket is what stops the minus sign getting separated from the 2, and once it is a reflex you stop
losing marks to it entirely.With the bracket in place the order of operations does the rest. Work inside brackets first, then powers, then multiply and divide, then add and subtract.
3 × −2² without the bracket.
Without it the square attaches to the 2 alone, giving 3 × −4 = −12 — wrong sign,
and it looks plausible enough to leave alone. The bracket is the whole defence.3a² = 3 × 4 = 12, but (3a)² = (−6)² = 36.
The power in 3a² is attached to the a only — the 3 is just sitting there waiting to multiply
at the end.A bracket means “treat everything inside me as one lump”. So
3(2n + 5) is three lots of the whole lump, which means three lots of the 2n and
three lots of the 5. The number outside reaches every term inside, not just the first one it meets.
3(2n + 5) = 6n + 5 is the single most common expanding error, and it happens because your
eye moves on after the first product. The fix is mechanical: draw the two arrows before you write
anything. If a bracket has two terms in it, your answer has two terms in it.This is where the sign errors live. A negative outside the bracket multiplies every term inside, and because multiplying by a negative flips signs, every sign inside the bracket changes.
Look at the second product carefully, because it is the one that catches people. −2 × −5 is two minus signs multiplied, so the answer is +10, not −10. The −5 inside the bracket came out positive.
6 − (n − 4) has no number outside the
bracket, only a minus sign. That minus is really −1. So it becomes
6 − n + 4 = 10 − n. Writing the 1 in while you are learning this is worth doing —
“subtract the whole bracket” is where a lot of quietly wrong lines come from.Two brackets multiplied means every term in the first must meet every term in the second. Two terms times two terms gives four products, always. If you have written three, you have missed one; if you have written five, you have doubled one up.
The grid is the method to use. It is slower to draw than the acronym everyone learns, and it is worth it, because a grid physically cannot leave a product out — an empty cell is visible. It also keeps working when the brackets get longer, which the acronym does not.
Write the terms with their signs along the top and down the side. The −5 goes in as −5, not as 5 with the minus remembered separately. Then fill each cell with the product of its row and column, and add the four cells up.
The two middle cells are the only ones that can be collected — they are both x terms. +3x − 5x = −2x. The x² and the −15 have nothing to pair with, so they stay.
(x − 4)² means (x − 4)(x − 4). It does not
mean x² − 16, and it does not mean x² + 16 either. The power applies to the whole
bracket, so you have to write the bracket out twice and grid it like any other pair.
Multiply two of the brackets first and simplify. Then multiply that answer by the third bracket with the same grid: three terms down the side and two along the top make six cells. Six products, then collect.
| × | +2x | +1 |
|---|---|---|
| +x2 | +2x3 | +x2 |
| +x | +2x2 | +x |
| −6 | −12x | −6 |
Factorising is expanding played backwards. You are given the answer and asked to reconstruct the bracket. That means you already have a perfect checking method built in: expand your answer. If it does not come back to the question, it is wrong, and you can see so in ten seconds without asking anyone.
Do the numbers and the letters separately, then put them together.
12a² − 8a = 2(6a² − 4a) is true but it is not finished, and it earns
nothing. Once you have written the bracket, look inside it: if the terms still share a factor, you did
not take enough out.5x + 5 = 5(x + 1), not
5(x). When a term is entirely swallowed by the factor, what is left behind is 1, not
nothing. Expanding your answer catches this instantly.12a² − 8a the second term is
−8a, so what stays behind after dividing by 4a is −2. The minus does not
evaporate and it does not come outside; it sits inside the bracket on its term.When four terms have no factor common to all of them, pair them up. Take a common factor out of each pair. If it is going to work, the same bracket appears twice, and that bracket is itself a common factor.
You already know what the answer looks like, because section 5 built it. Expanding
(x + a)(x + b) gives x² + (a + b)x + ab. So the two numbers in the
brackets have to multiply to give the constant and add to give the number in front of x.
Factorising is hunting for that pair.
Write out the pairs. Do not try to spot it — four lines on paper is faster than two minutes of staring, and it does not go wrong under exam pressure.
This is the part that repays being learnt properly, because it cuts the number of pairs you have to test roughly in half. Look at the constant first, then the x coefficient.
| Constant | x term | The two numbers | Example |
|---|---|---|---|
| positive | positive | both positive | x² + 9x + 20 = (x + 4)(x + 5) |
| positive | negative | both negative | x² − 7x + 12 = (x − 3)(x − 4) |
| negative | positive | one of each; the bigger one is positive | x² + 2x − 15 = (x + 5)(x − 3) |
| negative | negative | one of each; the bigger one is negative | x² − 5x − 24 = (x − 8)(x + 3) |
The reasoning behind every row is the same. Two numbers multiply to a positive only if their signs match, and then their sum carries that shared sign. They multiply to a negative only if their signs differ, and then the sum takes the sign of whichever number is larger in size.
You met this in section 5 without being told its name: (n + 6)(n − 6) = n² − 36,
because the middle terms cancelled. Run it backwards and you get a factorisation you can write down on
sight.
Two conditions have to hold, and both matter. There must be no x term, and it must be a subtraction of two square numbers. So x² − 49 factorises to (x + 7)(x − 7), because 49 = 7².
x² + 49 does not
factorise — there is no pair of numbers that multiplies to +49 and adds to 0. Only the
difference works, which is why the name says difference.The test: are the first and last terms both squares, and is the middle term 2 × their square roots? In x² − 10x + 25 the roots are x and 5, and 2 × x × 5 = 10x, so x² − 10x + 25 = (x − 5)². The sign of the middle term goes in the bracket.
The pair-hunting of this section needs one change. Multiply a × c. Find two numbers that multiply to ac and add to b. Use them to split the middle term into two, then factorise the four terms by grouping, as in section 6.
Every term has an x, and often a number, in common. Take the highest common factor out first, then factorise the quadratic left in the bracket. “Factorise completely” means both steps.
An index is a counter. n⁵ means n × n × n × n × n
— the little number tells you how many n values are being multiplied together. Every index rule
below comes straight out of that, and if you ever forget one you can rebuild it by writing the letters
out, which takes fifteen seconds and never lies.
Five n values and three more n values make eight n values. So multiplying means adding the powers. Not multiplying them: n⁵ × n³ is n⁸, not n¹⁵.
| Rule | What you do | Example |
|---|---|---|
| am × an | add the powers | n⁵ × n³ = n⁸ |
| am ÷ an | subtract the powers | n⁷ ÷ n³ = n⁴ |
| (am)n | multiply the powers | (n³)⁴ = n¹² |
| a⁰ | always 1 | 5t⁰ = 5 × 1 = 5 |
| a−n | one over it | n−3 = 1 ÷ n³ |
n⁵ × n³ = n¹⁵ is wrong. Powers get multiplied only when one power
sits on top of another, as in (n³)⁵. Writing the letters out settles it every time.This is the one the syllabus keeps testing, and the one that is worth slowing down for.
| power | means | example |
|---|---|---|
| x−n | 1 ÷ xn | x−3 = 1/x3 |
| x1/n | the nth root of x | x1/2 = √x, 81/3 = 2 |
| xm/n | the nth root, then to the power m | 82/3 = 22 = 4 |
The three rules above still apply: add the powers when multiplying, subtract when dividing, multiply when one power sits on another. And numbers and letters are still separate jobs. A fractional power on a bracket applies to the number too.
Write both sides as powers of the same base. Then the powers must be equal, and you solve an ordinary equation. Logarithms are not needed. Know the powers of 2 (2, 4, 8, 16, 32, 64), of 3 (3, 9, 27, 81) and of 5 (5, 25, 125).
Fifteen questions with nothing labelled, in no particular order. This is deliberately harder than doing fifteen factorising questions in a row, and it is the version that matters — in the exam nobody tells you which method the question wants. Half the skill is recognising the shape.
Before you start each one, say to yourself what kind of question it is. Expand, factorise, substitute, collect, or index rules. Then do it on paper, and check the signs before you type.
Work down the sections in order rather than jumping to the ones that look hardest. Each section genuinely uses the one before it: expanding two brackets needs the sign rules, factorising a quadratic needs expanding, and the index trap is the sign rules again in disguise.
A realistic plan is three passes. First pass: read the worked-in-full example in each section and write it out yourself on paper, copying it. Second pass: do the faded examples, revealing one step at a time and only after you have written your own attempt. Third pass: the mixed set, cold, a few days later.
One habit to carry into every other topic: when a line of working contains a minus sign, slow down for that line. Not for the question — for that line. Your diagnostic says the errors are not spread evenly; they cluster where a sign has to be carried, flipped or distributed. That is a small enough target to actually hit.