AQA GCSE Mathematics 8300: Structure and calculation – Revision Notes

For full explanations and worked examples, use the Structure and calculation study guide. These notes condense section 3.1.1, Structure and calculation (N1 to N9), of the AQA GCSE Mathematics (8300) specification, for teaching from September 2015 with exams from May/June 2017 (version 1.0). Content marked (Higher) is in the specification’s Higher content only column: the product rule (N5), estimating powers and roots (N6), fractional indices (N7) and surds (N8). Everything else is assessed at both tiers.

The course hub is AQA GCSE Mathematics and the printable checklist lists every statement. For the rest of Topic 1, see the Number revision notes. The free 10-minute diagnostics show which areas to revise first.

Key definitions

Term	Meaning
Integer	A whole number: positive, negative or zero
Prime	A number with exactly two factors (1 is not prime; 2 is the only even prime)
Factor (divisor)	Divides the number exactly
Multiple	In the number’s times table
HCF	Highest common factor: the largest number that divides all of them
LCM	Lowest common multiple: the smallest number they all divide into
Reciprocal	1 ÷ the number; the reciprocal of a/b is b/a
Inverse operation	Undoes another: + and −, × and ÷, squaring and square-rooting
Standard form	A × 10ⁿ with 1 ≤ A < 10 and n an integer
Surd (Higher)	A root left in exact form, such as √3
Cost price / selling price	What the trader pays / what the customer pays
Profit / loss	Selling price − cost price / cost price − selling price
Debit / credit	Money out of / into an account
Balance	Money in the account; negative means overdrawn

Also know: income tax, VAT and interest rate as household finance terms (N2 notes).

Facts to know by heart (N6, N9)

Set	Values
Squares to 15 × 15	1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Cubes	1, 8, 27, 64, 125; 1000 = 10³
Powers of 2	2, 4, 8, 16, 32, 64, 128, 256
Powers of 3	3, 9, 27, 81, 243
Powers of 4	4, 16, 64, 256
Powers of 5	5, 25, 125, 625
Large numbers	1 thousand = 10³, 1 million = 10⁶

Ordering and symbols (N1)

Method in steps

1. Convert every value to a decimal (same number of d.p.)
2. Place negatives first: the larger the digits, the smaller the number
3. Write the original values, not your decimals, in the final order

Symbols: = (equals), ≠ (not equal), < (less than), > (greater than), ≤ and ≥ (include the end value).

Calculating (N2, N3)

Priority of operations

Brackets → Powers and roots → × and ÷ (left to right) → + and − (left to right)

A reciprocal counts as a division. A square root sign acts like a bracket over everything under it.

Place value from a given fact. Count decimal places for ×. For ÷, adjust each number to the given fact and track how many times you scaled by 10. Always finish with a rough estimate.

Fractions. Mixed numbers to improper fractions first. × : multiply across. ÷ : multiply by the reciprocal. + and − : common denominator.

Negatives. Like signs give +; unlike signs give −. Subtracting a negative means adding.

Cancellation (N3). Cancel common factors between any numerator and any denominator before multiplying out.

Primes, HCF and LCM (N4)

Method in steps

1. Build a factor tree; stop at primes
2. Write as a product in index form, primes in ascending order
3. HCF: shared primes only, lowest power of each
4. LCM: every prime, highest power of each

Worked reminder: 84 = 2² × 3 × 7 because 84 = 4 × 21 = 2 × 2 × 3 × 7.

Check: HCF × LCM = the product of two numbers. This works for two numbers only.

Listing and counting (N5)

Systematic listing: fix the first item, run through all options for the second, then move the first item on. Decide at the start whether order matters (AB and BA the same or different?) and whether repeats are allowed.

(Higher) Product rule: m ways then n ways gives m × n ways. If choices cannot repeat, the number of options drops by one each time.

Worked reminder: three-digit numbers using only the digits 1, 2 and 3, repeats allowed: 3 × 3 × 3 = 27.

Indices and roots (N6, N7)

Rule	Example
aᵐ × aⁿ = aᵐ⁺ⁿ	3⁴ × 3² = 3⁶
aᵐ ÷ aⁿ = aᵐ⁻ⁿ	2⁵ ÷ 2⁸ = 2⁻³
(aᵐ)ⁿ = aᵐⁿ	(5²)³ = 5⁶
a⁰ = 1	9⁰ = 1
a⁻ⁿ = 1/aⁿ	10⁻² = 1/100
(Higher) a^(1/n) = nth root of a	81^(1/4) = 3
(Higher) a^(m/n) = (nth root of a)ᵐ	4^(5/2) = 2⁵ = 32

Roots and integer indices are Foundation (N7). (Higher) For a fractional index, root first, then power. For a negative fractional index, take the reciprocal as well.

(Higher) Estimating a root: find the two consecutive whole numbers whose powers trap it, then judge which end it is nearer. √55 lies between 7 (49) and 8 (64), a little under 7.5.

Exact calculation (N8)

Multiples of π (Foundation): keep π as a symbol. Area of a circle of radius r = πr²; circumference = 2πr. A semicircle of radius 3 has area 4.5π and curved edge 3π.

(Higher) Surd rules

√a × √b = √(ab)            √a ÷ √b = √(a/b)
√a × √a = a                k√a + m√a = (k + m)√a

(Higher) Method in steps: simplify √n

1. Find the largest square number that divides n
2. √n = √(square) × √(rest)
3. Take the square root of the square part

Worked reminder: √180 = √36 × √5 = 6√5.

(Higher) Rationalise a single-surd denominator: multiply top and bottom by that surd, then simplify. 9/√3 = 9√3/3 = 3√3.

Standard form (N9)

Method in steps

× or ÷ : numbers together, powers together (add for ×, subtract for ÷), then adjust A
+ or − : rewrite both with the same power of 10, combine, then adjust A
Adjust: if A ≥ 10, divide A by 10 and add 1 to n; if A < 1, multiply A by 10 and subtract 1 from n

Converting: 0.000 81 = 8.1 × 10⁻⁴ (the point moves 4 places right). 7.06 × 10⁵ = 706 000.

Calculator displays: read E or ×10 in the display as a power of 10, then write it properly as A × 10ⁿ.

Must-know distinctions

• Factor vs multiple: factors of 12 are 1, 2, 3, 4, 6, 12; multiples of 12 are 12, 24, 36, …
• HCF vs LCM: “largest equal pieces” means HCF; “next time together” means LCM.
• −3² vs (−3)²: −9 vs 9. The power acts before the minus unless brackets say otherwise.
• Negative index vs negative number: 2⁻³ = 1/8, a positive number.
• Basic vs additional Foundation: both are on Foundation papers; only the Higher content only column is restricted to Higher.
• Exact vs rounded: “in terms of π” or “exact” means no decimals.

Quick self-test

1. Write < or > between −0.09 and −0.1.
2. Given 27 × 34 = 918, write down 2.7 × 3.4.
3. Work out 4 − 2 × (−3)².
4. Write 600 as a product of prime factors in index form.
5. Find the HCF and LCM of 48 and 180.
6. (Higher) A shop offers 6 shirts, 4 pairs of trousers and 3 pairs of shoes. How many different outfits of one of each are there?
7. Work out 6⁻².
8. (Higher) Work out 64^(−5/6).
9. A circle has radius 9 cm. Give its area in terms of π.
10. (Higher) Simplify √200.
11. (Higher) Rationalise 12/√6.
12. Work out (8 × 10⁻³) × (7 × 10⁶), giving your answer in standard form.

Answers

1. −0.09 > −0.1 (−0.09 is closer to zero).
2. 9.18
3. 4 − 2 × 9 = 4 − 18 = −14
4. 2³ × 3 × 5²
5. 48 = 2⁴ × 3 and 180 = 2² × 3² × 5, so HCF = 2² × 3 = 12 and LCM = 2⁴ × 3² × 5 = 720.
6. 6 × 4 × 3 = 72
7. 1/36
8. sixth root of 64 = 2; 2⁵ = 32; so 1/32.
9. π × 9² = 81π cm²
10. √100 × √2 = 10√2
11. 12√6 / 6 = 2√6
12. 56 × 10³ = 5.6 × 10⁴

Where marks are usually lost

• Squaring a negative inside a calculation without brackets, so 4 − 2 × (−3)² becomes 4 + 18.
• Writing the decimal versions in an ordering answer when the question gave fractions.
• Using a given fact but placing the decimal point by guesswork instead of counting places.
• Leaving a prime factorisation as 2 × 2 × 2 × 3 × 5 × 5 when index form was asked for.
• Finding the LCM when the context needs the HCF, or the reverse.
• Missing or repeating items in a list because no fixed order was used.
• (Higher) Adding choices instead of multiplying them with the product rule.
• Giving 6⁻² as −36 or −12.
• Leaving standard form as 56 × 10³ or 0.56 × 10⁵.
• (Higher) Rationalising but not cancelling, leaving 12√6/6.

Now try the Structure and calculation practice questions, then the Number practice questions.

Official syllabus

AQA GCSE Mathematics (8300) specification, for teaching from September 2015, exams from May/June 2017, version 1.0, published by AQA – section 3.1.1 Structure and calculation (N1 to N9).