A1: Algebraic Notation
Use and interpret algebraic notation: ab, 3y, a², a³, a/b, brackets
📋 Key Concepts
Definition: Algebraic notation uses letters (variables) to represent numbers. Different formats show different operations.
Standard Notation Forms
| Notation | Meaning | Example |
|---|---|---|
| ab | a multiplied by b | If a = 3, b = 4: ab = 12 |
| 3y | 3 multiplied by y | If y = 5: 3y = 15 |
| a² | a multiplied by itself (a × a) | If a = 4: a² = 16 |
| a³ | a × a × a | If a = 2: a³ = 8 |
| a/b | a divided by b | If a = 12, b = 3: a/b = 4 |
| (a + b) | Brackets show calculation first | If a = 2, b = 3: (a + b) = 5 |
Important: In algebra, we don't write multiplication signs. Instead of a × b, we write ab. Instead of 3 × y, we write 3y.
📝 Variables and Coefficients
Variable: A letter that represents an unknown number (e.g., x, y, n, a).
Coefficient: The number multiplied by a variable (e.g., in 5x, the coefficient is 5).
Coefficient: The number multiplied by a variable (e.g., in 5x, the coefficient is 5).
Example 1
Identify the coefficient in each term:
a) 7x → coefficient is 7
b) y → coefficient is 1 (understood)
c) -3a → coefficient is -3
d) -n → coefficient is -1
📝 Powers and Indices
Rule: Powers show how many times to multiply a number by itself.
Example 2
Write without using powers:
a) x² = x × x
b) 2a³ = 2 × a × a × a
c) 4m²n = 4 × m × m × n
Example 3
Calculate when a = 3:
a) a² = 3 × 3 = 9
b) 2a³ = 2 × 3 × 3 × 3 = 2 × 27 = 54
c) a² + a = 9 + 3 = 12
📝 Brackets
Rule: Brackets indicate that the calculation inside must be done first.
Example 4
Calculate when x = 4:
a) 3(x + 2) = 3 × (4 + 2) = 3 × 6 = 18
b) 2(x² - 1) = 2 × (16 - 1) = 2 × 15 = 30
c) (x + 1)² = (4 + 1)² = 5² = 25
Example 5
Write using algebraic notation:
a) "Multiply x by 3 then add 5" → 3x + 5
b) "Add 5 to x then multiply by 3" → 3(x + 5)
c) "Square x then subtract 2" → x² - 2
📝 Division Notation
Forms: Division can be written as fractions: a/b or a ÷ b or a⁄b
Example 6
Calculate when a = 12 and b = 4:
a) a/b = 12/4 = 3
b) a⁄2 = 12⁄2 = 6
c) a + b⁄4 = 16⁄4 = 4
🧮 Worked Examples — Exam Style
Algebraic notation questions are about translation: words ↔ symbols. The order of operations matters — '3 lots of (x + 2)' is 3(x + 2), not 3x + 2.
Write 'add 3 to x, then multiply the result by 4' in symbols.
The brackets control the order: 4(x + 3)
4x + 3 would mean ‘multiply x by 4, THEN add 3’ — different instruction.
B1 for 4(x + 3). Word-order questions test whether you can see which operation happens first.
Mia is n years old. Write an expression for her age in 5 years, and for her brother's age, who is half Mia's age now.
In 5 years: n + 5
Brother now: n/2 (or ½n)
B1 each. Context notation questions (ages, money, counters) are the standard Foundation format — define what YOUR letter means.
Exam wording to expect: "Write an expression for …", "in terms of n", or forming expressions from a context.
❓ Practice Questions
Q1: Write 3 × x × x using powers.
Q2: Find the value of 4y when y = 7.
Q3: Find the value of a² when a = 5.
Q4: Calculate 2(x + 3) when x = 4.
Q5: Write "multiply x by 2 then add 1" in algebraic notation.
Q6: Find the value of x + y⁄3 when x = 8 and y = 4.
Q7: Write 'subtract x from 5' in symbols
Q8: Write an expression for the cost of 3 pens at £p each plus a £2 bag
✅ Answers
- 3x²
- 28
- 25
- 14
- 2x + 1
- 4
🧮 Worked Examples — Harder Exam Style
Pens cost p pence each. Rulers cost 10p more than pens. Write an expression for the total cost of 4 pens and 2 rulers, and simplify it.
4 pens: 4p. Ruler: p + 10, so 2 rulers: 2p + 20
Total: 4p + 2p + 20 = 6p + 20 pence
M1 for 4p + 2(p + 10), A1 for 6p + 20. Forming expressions from a context is the opening step of most Foundation algebra word problems.
📋 Model Answers — Full Working
Every mark comes from the working, not just the answer. These model answers show the lines a mark scheme expects to see.
Q1. x × x = x², so 3 × x² = 3x²
Q2. 4y means 4 × y = 4 × 7 = 28
Q3. a² = 5² = 25
Q4. 2(x + 3) = 2(4 + 3) = 2 × 7 = 14. Brackets first.
Q5. 2x + 1 (multiply first, then add — the words give the order)
Q6. (x + y)/3 means the SUM is divided by 3: (8 + 4) ÷ 3 = 4. The fraction line brackets the top.
Q7. 5 − x — ‘subtract x FROM 5’ means x is taken away from 5, so 5 comes first
Q8. 3p + 2
📌 Key Facts to Memorise
- 4y means 4 × y; ab means a × b; 3(x + 2) means 3 × (x + 2)
- x² means x × x (not 2x)
- Writing rules: number before letter (5a), letters alphabetically (3ab), no × or ÷ signs
- "Subtract x from 5" = 5 − x; "subtract 5 from x" = x − 5 — order flips!
- A fraction bar brackets its whole numerator: (x + y)/3 = (x + y) ÷ 3
🗺️ In the Exam
Notation underpins all of algebra: 1–2 marks directly (translation questions), and indirectly everywhere else. Paper 1 favourite because it needs no calculator.
Links to: substitution (A2), simplifying (A4), modelling with algebra (A21).
🧭 Board Focus — AQA
Spec 8300. Three papers, each 80 marks and 1h30: Paper 1 non-calculator, Papers 2 and 3 calculator. Foundation/Higher tiers. Roughly a third of the marks sit under AO2 (reasoning) and AO3 (problem-solving), so show every line of working — method marks survive arithmetic slips.
Always check: specification details change — confirm paper structure and set texts against the current specification for your exam year.
🎯 Exam Tips
- Remember: no multiplication sign needed between numbers and letters (3y means 3 × y)
- The coefficient of x in "x" is 1, not 0
- Brackets change the meaning: 3x + 5 ≠ 3(x + 5)
- Always substitute carefully when evaluating expressions
- Check if you need to square before or after other operations
🧠 Problem-Solving Strategies
Problem-Solving
When translating words into algebra: identify the variable first, then build the expression step-by-step following the order of operations. Always check whether brackets are needed — they change the meaning entirely.
Multi-Step Problem
A rectangle has length 3x cm and width (x + 2) cm. The perimeter is 40 cm. Find the area of the rectangle.
Solution:
Step 1: Use perimeter formula: 2(3x) + 2(x + 2) = 40
6x + 2x + 4 = 40
8x = 36, so x = 4.5
Step 2: Find dimensions: length = 3(4.5) = 13.5 cm, width = 4.5 + 2 = 6.5 cm
Step 3: Area = 13.5 × 6.5 = 87.75 cm²
⚠️ Common Errors
Watch Out!
1. Wrong: 3x means 3 + x Correct: 3x means 3 × x
2. Wrong: 2x² means (2x)² = 4x² Correct: 2x² means 2 × x² (square first, then multiply)
3. Wrong: 3x + 5 and 3(x + 5) are the same Correct: 3x + 5 = 3×x + 5, but 3(x + 5) = 3x + 15
✍️ 6-Mark Exam Question
Extended Answer
6 marks: The formula for the cost of a party is C = 15n + 50, where C is the total cost in pounds and n is the number of guests. (a) Explain what each part of the formula represents. (b) Find the cost when n = 20. (c) If the budget is £200, what is the maximum number of guests?
(a) 15n means £15 per guest (cost of food per person). 50 is the fixed cost (venue hire). C is the total cost.
(b) C = 15(20) + 50 = 300 + 50 = £350
(c) 15n + 50 ≤ 200 → 15n ≤ 150 → n ≤ 10. Maximum 10 guests.
Mark scheme: (a) 2 marks — 1 for explaining 15n, 1 for explaining 50. (b) 1 mark for substitution, 1 for answer. (c) 1 mark for inequality, 1 for correct answer.
📊 AO3: Reason & Interpret
Reasoning and Interpretation
Two phone contracts are available. Contract A: C = 10n + 25. Contract B: C = 15n + 10. C is total cost (£), n is months.
(a) Which contract is cheaper for 6 months?
(b) After how many months do the contracts cost the same?
(c) A customer says "Contract A is always better value." Is this correct? Explain.
Answers: (a) A: £85, B: £100 — Contract A is cheaper. (b) 10n + 25 = 15n + 10 → 15 = 5n → n = 3 months. (c) No — Contract B is cheaper for fewer than 3 months.
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