What you'll be able to do by the end
- Number: use negative numbers in all four operations; use BIDMAS; find factors, multiples, HCF and LCM; use prime factorisation; calculate with powers and roots
- Algebra: use algebraic notation; collect like terms; expand and factorise; substitute into expressions and formulae; solve linear equations (including brackets and unknowns on both sides); rearrange simple formulae
- Proportional reasoning: simplify and use ratios; convert between fractions, decimals and percentages; calculate percentage change with multipliers; solve direct and inverse proportion
Part 1 — Negative numbers
Moving right = bigger. Moving left = smaller. So −7 is smaller than −2 — the bigger the digit after the minus sign, the smaller the number. Adding moves right, subtracting moves left: −3 + 5 = 2; 4 − 9 = −5; −2 − 6 = −8.
Two signs together
| You see | Becomes | Example |
|---|---|---|
| + + | + | 5 + (+3) = 8 |
| + − | − | 5 + (−3) = 2 |
| − + | − | 5 − (+3) = 2 |
| − − | + | 5 − (−3) = 8 |
Two signs the same → plus. Two signs different → minus. "Minus a minus is a plus" is the one everybody remembers — and the one that trips people up if they apply it to multiplication without thinking.
Multiplying and dividing
Same signs → positive. Different signs → negative. −3 × 4 = −12; 3 × −4 = −12; −3 × −4 = 12; −12 ÷ −3 = 4.
✅ Practice — negative numbers
- −5 + 8 =
- 3 − 11 =
- −4 − 7 =
- 6 − (−4) =
- −2 + (−5) =
- −6 × 3 =
- −7 × −2 =
- 20 ÷ −4 =
- −18 ÷ −6 =
- Order smallest first: −3, 2, −7, 0, 5
Part 2 — Order of operations (BIDMAS)
Brackets · Indices (powers) · Division and Multiplication (left to right) · Addition and Subtraction (left to right).
3 + 4 × 2 → multiply first (4 × 2 = 8), then add → 11 (not 14). (3 + 4) × 2 → brackets first → 14. 20 − 3² → indices first (9) → 11. 12 ÷ 4 + 2 × 3 → 3 + 6 → 9.
BIDMAS makes it look as though division comes before multiplication. It doesn't — they're the same level, done left to right. 20 ÷ 5 × 2 = 4 × 2 = 8 (not 20 ÷ 10 = 2).
✅ Practice — order of operations
- 5 + 2 × 3 =
- (5 + 2) × 3 =
- 10 − 6 ÷ 2 =
- 4² + 5 =
- 3 × (8 − 5) =
- 24 ÷ 6 × 2 =
- 2 + 3 × 4 − 1 =
- (10 − 4)² ÷ 4 =
Part 3 — Factors, multiples & primes
| Term | Meaning | Example |
|---|---|---|
| Factor | Divides into it exactly | Factors of 12: 1, 2, 3, 4, 6, 12 |
| Multiple | The times table of a number | Multiples of 4: 4, 8, 12, 16… |
| Prime | Exactly two factors — 1 and itself | 2, 3, 5, 7, 11, 13… |
1 is not prime — it has only one factor. 2 is prime — and the only even prime, because every other even number divides by 2.
Prime factorisation — every number is a product of primes
60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.
HCF and LCM using prime factors — 24 and 36
24 = 2³ × 3, and 36 = 2² × 3². HCF — take the lowest power of each shared prime: 2² × 3 = 12. LCM — take the highest power of each prime: 2³ × 3² = 72.
The HCF must divide into both (12 goes into 24 and 36 ✓); the LCM must be divisible by both (72 ÷ 24 = 3, 72 ÷ 36 = 2 ✓). Also: HCF is never bigger than the smaller number, and LCM never smaller than the bigger one.
Powers and roots
5² = 25 ("five squared"); 2³ = 8 ("two cubed"); √49 = 7 (7 × 7 = 49); ∛27 = 3. Square numbers to know: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.
✅ Practice — factors, multiples, primes
- List all the factors of 18
- First five multiples of 7
- Is 21 prime? Explain.
- Write 48 as a product of prime factors
- HCF of 18 and 24
- LCM of 6 and 8
- Work out 6²
- Work out √81
Part 4 — Algebra: the notation
Algebra uses letters for numbers we don't know yet, and the rules of writing it are strict: 3a means "three lots of a" (not a3); ab means a × b; a² means a × a; a/3 means a ÷ 3.
3a means "three lots of a" — if a = 4, then 3a = 12. a³ means a × a × a — if a = 4, then a³ = 64. Students confuse these constantly.
Key words
| Word | Meaning |
|---|---|
| Term | A single part — in 3a + 5b − 2, the terms are 3a, 5b and −2 |
| Coefficient | The number in front of a letter — in 3a, it's 3 |
| Expression | Letters and numbers, no equals sign — 3a + 5 |
| Equation | Has an equals sign, can be solved — 3a + 5 = 20 |
| Formula | A rule connecting quantities — A = l × w |
Collecting like terms
Like terms have exactly the same letters and powers, and only they can be added or subtracted. 3a + 5a = 8a; 7b − 2b = 5b; 4a + 3b − a + 2b = 3a + 5b. But 3a and 3b are different letters, and 2a and 2a² are different powers — 3a + 3b cannot be simplified.
Expanding brackets — multiply everything inside by what's outside
3(a + 4) = 3a + 12; 5(2x − 3) = 10x − 15; a(a + 3) = a² + 3a.
−2(a + 5) — the −2 multiplies both terms: −2 × a = −2a, −2 × 5 = −10. Answer: −2a − 10, not −2a + 10.
Factorising — the reverse
Find the highest common factor and take it outside: 6a + 9 = 3(2a + 3); 4x − 12 = 4(x − 3); a² + 5a = a(a + 5). Check by expanding.
Substitution — replace the letters with numbers
If a = 4 and b = 3, then 2a + 3b = 8 + 9 = 17. If x = 5, then x² − 2x = 25 − 10 = 15. If a = −2, then 3a + 4 = −6 + 4 = −2.
✅ Practice — algebra basics
- Simplify 5a + 3a
- Simplify 7x − 2x + 4x
- Simplify 3a + 2b + 5a − b
- Expand 4(a + 3)
- Expand 6(2x − 5)
- Expand −3(a + 2)
- Factorise 8a + 12
- Factorise x² + 4x
- If a = 3, find 5a − 2
- If x = 4 and y = 2, find 3x + 2y
- If a = −3, find a² + 1
- If n = 5, find 2n²
Part 5 — Solving equations
An equation is a balance — whatever you do to one side, you must do to the other. One step: x + 7 = 12 → subtract 7 → x = 5; 3x = 21 → divide by 3 → x = 7. Two steps: 2x + 5 = 17 → subtract 5 (2x = 12) → divide by 2 → x = 6.
To build 2x + 5 you multiply by 2, then add 5. To undo it, subtract 5, then divide by 2. Last thing done, first thing undone.
With brackets, and unknowns on both sides
3(x + 4) = 27 → expand (3x + 12 = 27) or divide first (x + 4 = 9) → x = 5. For unknowns on both sides — 5x + 3 = 2x + 18 → subtract 2x (3x + 3 = 18) → subtract 3 (3x = 15) → x = 5.
Move the smaller letter term, so you're left with a positive. In 5x + 3 = 2x + 18, move the 2x — moving the 5x works but is fiddlier.
Rearranging formulae — same skill, the answer is a letter
Make r the subject of C = 2πr → divide by 2π → r = C / 2π. Make b the subject of A = ½bh → ×2 (2A = bh) → ÷h → b = 2A / h.
✅ Practice — equations
- x + 9 = 15
- x − 4 = 11
- 5x = 35
- x/3 = 6
- 2x + 7 = 19
- 4x − 5 = 23
- 3(x + 2) = 21
- 5(x − 3) = 20
- 4x + 2 = 2x + 10
- 7x − 3 = 3x + 17
- Make h the subject of A = bh
- Make x the subject of y = 3x + 2
Part 6 — Sequences
A sequence is a list of numbers following a rule. The term-to-term rule is what you do to get from one term to the next: 3, 7, 11, 15… → add 4; 100, 50, 25… → divide by 2; 2, 6, 18, 54… → multiply by 3.
Nth term — the position rule
For a linear sequence 5, 8, 11, 14, 17…: it goes up by 3, so the rule starts with 3n. When n = 1, 3n = 3, but the first term is 5 — a difference of +2. So nth term = 3n + 2. Check: n = 4 → 3(4) + 2 = 14 ✓.
| Sequence | Difference | nth term |
|---|---|---|
| 4, 7, 10, 13 | +3 | 3n + 1 |
| 6, 11, 16, 21 | +5 | 5n + 1 |
| 2, 4, 6, 8 | +2 | 2n |
| 10, 8, 6, 4 | −2 | −2n + 12 |
The coefficient of n is always the difference. Then substitute n = 1 and adjust to get the first term.
✅ Practice — sequences
- Next two terms: 4, 9, 14, 19, __, __
- Term-to-term rule: 80, 40, 20, 10
- nth term: 5, 9, 13, 17
- nth term: 7, 9, 11, 13
- nth term is 4n − 1 — first three terms
- nth term is 2n + 5 — find the 20th term
Part 7 — Proportional reasoning
Ratio
Simplify: divide all parts by the HCF — 15 : 25 → both ÷ 5 → 3 : 5. Share in a ratio: £240 in 3 : 5 → total parts 8 → one part 240 ÷ 8 = 30 → £90 and £150.
Percentage change with multipliers
Increase 10% → ×1.1; increase 25% → ×1.25; decrease 20% → ×0.8. Increase £80 by 15% → 80 × 1.15 = £92; decrease £80 by 15% → 80 × 0.85 = £68. Percentage change = (change ÷ original) × 100: 40 → 50 is (10 ÷ 40) × 100 = 25% increase.
Direct and inverse proportion
Direct (one doubles, the other doubles): 6 items cost £15 → one item £2.50 → ten items £25. Inverse (one doubles, the other halves): 4 people take 9 hours → total work 4 × 9 = 36 person-hours → 36 ÷ 6 = 6 hours for 6 people.
More people means less time. If your answer went up when the number of workers went up, you've treated it as direct.
✅ Practice — proportion
- Simplify 24 : 36
- Share £350 in the ratio 2 : 5
- Write ⅜ as a decimal and a percentage
- Increase £60 by 20%
- Decrease 250 by 30%
- A price rises from £40 to £46. Percentage increase?
- 8 items cost £30 — find the cost of 12
- 5 workers take 6 days — how long for 3 workers?
Answers
Negative numbers
1. 3 · 2. −8 · 3. −11 · 4. 10 · 5. −7 · 6. −18 · 7. 14 · 8. −5 · 9. 3 · 10. −7, −3, 0, 2, 5
Order of operations
1. 11 · 2. 21 · 3. 7 · 4. 21 · 5. 9 · 6. 8 · 7. 13 · 8. 9
Factors, multiples, primes
1. 1, 2, 3, 6, 9, 18 · 2. 7, 14, 21, 28, 35 · 3. No — it has factors 1, 3, 7, 21 (four, not two) · 4. 2⁴ × 3 · 5. 6 · 6. 24 · 7. 36 · 8. 9
Algebra basics
1. 8a · 2. 9x · 3. 8a + b · 4. 4a + 12 · 5. 12x − 30 · 6. −3a − 6 · 7. 4(2a + 3) · 8. x(x + 4) · 9. 13 · 10. 16 · 11. 10 · 12. 50
Equations
1. x = 6 · 2. x = 15 · 3. x = 7 · 4. x = 18 · 5. x = 6 · 6. x = 7 · 7. x = 5 · 8. x = 7 · 9. x = 4 · 10. x = 5 · 11. h = A/b · 12. x = (y − 2)/3
Sequences
1. 24, 29 · 2. Divide by 2 · 3. 4n + 1 · 4. 2n + 5 · 5. 3, 7, 11 · 6. 45
Proportion
1. 2 : 3 · 2. £100 and £250 · 3. 0.375 and 37.5% · 4. £72 · 5. 175 · 6. 15% · 7. £45 · 8. 10 days
The mistakes that cost the most
Left to right instead of BIDMAS. 3 + 4 × 2 is 11, not 14.
Thinking −7 is bigger than −2. On a number line, −7 is further left.
Applying "two minuses make a plus" to addition. −3 − 5 is −8, not 2. The rule is about two signs next to each other, not two negative numbers.
Adding unlike terms. 3a + 2b stays as it is.
Forgetting the second term when expanding. −2(a + 5) is −2a − 10, not −2a + 10.
Confusing 3a and a³. With a = 4, one is 12 and the other is 64.
Doing the same thing to only one side of an equation. It's a balance.
Inverse proportion scaled directly. More workers, less time. Check the direction.
Teaching notes
Algebra is arithmetic with a gap in it. Students who panic at letters often cope fine with "what number goes in the box?" Start there, then replace the box with x, and much of the fear disappears.
Substitution before manipulation. A student who can work out 3a + 2 when a = 5 understands what the letters mean. Simplifying and solving are much easier after that, and much harder before.
Solving equations should always be checked. Substitute the answer back in — ten seconds, catches most errors, and teaches that the answer is a claim that can be tested.
The nth term is where sequences click or don't. Emphasise the shortcut — the coefficient is always the difference — and drill it on ten sequences in a row. Most students get it within twenty minutes once the pattern is explicit.
BIDMAS needs the caveat taught with it. D and M are equal, and so are A and S. Students who learn the acronym without this get 20 ÷ 5 × 2 wrong and can't see why.
This unit is the bridge to GCSE. The proportional reasoning here is a lighter version of the GCSE ratio and proportion topic; the algebra becomes quadratics and simultaneous equations. A student secure here has a very different GCSE experience.