AQA GCSE Mathematics 8300: Structure and calculation – Study Guide

This study guide teaches section 3.1.1, Structure and calculation (specification references N1 to N9), of the AQA GCSE Mathematics (8300) specification, for teaching from September 2015 with exams from May/June 2017 (version 1.0). It goes deeper than the topic overview: every statement in N1 to N9 is taught with a worked example. Most of this content is assessed at both tiers. Four parts sit in the specification’s Higher content only column and are labelled Higher tier only: the product rule for counting (N5), estimating powers and roots (N6), fractional indices (N7) and surds (N8).

For the whole of Topic 1 Number, see the Number study guide. For this section, use the Structure and calculation revision notes and the Structure and calculation practice questions. The course hub is AQA GCSE Mathematics, the printable checklist lists every statement, and the free 10-minute diagnostics show where to start.

What this section covers

Spec	What you must be able to do	Tier
N1	Order positive and negative integers, decimals and fractions; use =, ≠, <, >, ≤, ≥	Both
N2	Apply the four operations, with formal written methods, to integers, decimals, fractions and mixed numbers, positive and negative; use place value; know household finance terms	Both
N3	Use inverse operations and cancellation; use priority of operations, including brackets, powers, roots and reciprocals	Both
N4	Use primes, factors, multiples, HCF, LCM and prime factorisation in index form	Both
N5	Apply systematic listing strategies	Both
N5	Use the product rule for counting	Higher tier only
N6	Use positive integer powers and real roots; recognise powers of 2, 3, 4 and 5; squares up to 15 × 15	Both
N6	Estimate powers and roots of any positive number	Higher tier only
N7	Calculate with roots and with integer indices	Both
N7	Calculate with fractional indices	Higher tier only
N8	Calculate exactly with fractions and with multiples of π	Both
N8	Calculate exactly with surds; simplify surds; rationalise denominators	Higher tier only
N9	Calculate with and interpret standard form A × 10ⁿ, with and without a calculator	Both

Paper 1 is non-calculator; Papers 2 and 3 allow a calculator. Any of this content can appear on any paper, so practise every method by hand.

Ordering and inequality symbols (N1)

To order a mix of fractions, decimals and negatives, convert everything to decimals, then place them on a number line. With negatives, the number further from zero is the smaller one: −0.8 < −0.3.

Worked example 1. Order these, smallest first: −0.6, −2/3, 5/8, 0.65, −0.06.

−2/3 = −0.666...     5/8 = 0.625
−0.666... < −0.6 < −0.06 < 0.625 < 0.65
Order: −2/3, −0.6, −0.06, 5/8, 0.65

Know all six symbols. ≠ means “is not equal to”. ≤ and ≥ include the end value, so x ≥ −2 includes −2 itself.

The four operations and place value (N2)

Formal written methods

You need a written method for each operation. For 58 × 23, split the 23:

58 × 20 = 1160
58 × 3  =  174
58 × 23 = 1334

To divide by a decimal, multiply both numbers by the same power of 10 so the divisor is a whole number: 7.56 ÷ 0.4 = 75.6 ÷ 4 = 18.9.

Place value from a given fact

If you know 58 × 23 = 1334, you can write down related results by counting decimal places.

5.8 × 2.3   = 13.34     (one d.p. + one d.p. = two d.p.)
133.4 ÷ 0.23 = 580      (133.4 is 1334 ÷ 10, and 0.23 is 23 ÷ 100,
                          so the answer is 58 × 10 = 580)

Check the size with a rough estimate: 6 × 2 = 12, so 13.34 is sensible.

Fractions, mixed numbers and negatives

Change mixed numbers to improper fractions before multiplying or dividing.

Worked example 2. Work out (a) 2 1/3 × 1 4/5 and (b) −1 5/6 + 3 1/4.

(a) 7/3 × 9/5 = 63/15 = 21/5 = 4 1/5
(b) −22/12 + 39/12 = 17/12 = 1 5/12

Sign rules: two like signs multiplied or divided give a positive; unlike signs give a negative. Subtracting a negative is adding: 4 − (−3) = 7.

Household finance terms

The specification expects you to know: cost price (what a trader pays), selling price (what the customer pays), profit (selling price minus cost price, when positive), loss (cost price minus selling price), debit (money out of an account), credit (money in), balance (what is in the account; a negative balance means overdrawn), income tax, VAT (a tax added to the price of many goods and services) and interest rate.

Worked example 3. A stall buys 40 mugs at £2.15 each. It sells 35 at £4.20 and the other 5 at £1.50. Find the profit.

Cost price:   40 × 2.15 = £86.00
Takings:      35 × 4.20 + 5 × 1.50 = 147.00 + 7.50 = £154.50
Profit:       154.50 − 86.00 = £68.50

Percentage work with VAT and interest belongs to 3.1.2 and to ratio (R9); here you only need the terms.

Inverse operations and priority of operations (N3)

Addition and subtraction undo each other, as do multiplication and division, and squaring and square-rooting. Use inverses to check: 18.9 × 0.4 = 7.56 confirms the division 7.56 ÷ 0.4 = 18.9 above.

Cancellation simplifies before you multiply:

(36 × 25) / (15 × 12)  =  (3 × 5) / (1 × 3)  ... cancel 12 into 36, 5 into 25 and 15
                       =  15 / 3 = 5

Priority of operations: brackets, then powers and roots, then multiplication and division (left to right), then addition and subtraction (left to right). The reciprocal of a number is 1 divided by it; the reciprocal of 2 1/2 = 5/2 is 2/5.

Worked example 4. Work out 18 − 3 × (4 − 6)² ÷ 2.

Brackets:   4 − 6 = −2
Power:      (−2)² = 4
× and ÷:    3 × 4 ÷ 2 = 6
Subtract:   18 − 6 = 12

Primes, factors and multiples (N4)

A prime has exactly two factors. Factors (divisors) divide a number exactly; multiples are in its times table. The unique factorisation theorem says every integer greater than 1 has exactly one prime factorisation, apart from order. Write it in index form using product notation.

Worked example 5. Two ribbons are 90 cm and 126 cm long. Both are cut into pieces of equal length, as long as possible, with nothing left over. Find the piece length and the number of pieces.

90  = 2 × 3² × 5
126 = 2 × 3² × 7
HCF = 2 × 3² = 18 cm          (lowest power of each shared prime)
Pieces: 90 ÷ 18 + 126 ÷ 18 = 5 + 7 = 12

The LCM uses the highest power of every prime: 2 × 3² × 5 × 7 = 630. Read the context: “largest equal piece” or “biggest group” needs the HCF; “next time together” or “smallest number both divide” needs the LCM.

Systematic listing and the product rule (N5)

List in a fixed order so nothing is missed or repeated. Choosing two colours from red, green, blue and yellow (order not mattering):

RG  RB  RY
    GB  GY
        BY        → 6 pairs

Tables and sample-space grids work the same way.

Higher tier only. The product rule: if one choice can be made in m ways and a second in n ways, together there are m × n ways.

Worked example 6. A code is one letter (A to Z) then two digits. How many codes are there? How many if the two digits must be different?

26 × 10 × 10 = 2600
26 × 10 × 9  = 2340    (only 9 choices remain for the second digit)

Powers and roots (N6)

Know the squares up to 15 × 15 (1, 4, 9, … 196, 225), the cubes 1, 8, 27, 64, 125 and 1000 = 10³, and 1 million = 10⁶. Recognise powers of 2 (2, 4, 8, 16, 32, 64, …), 3 (3, 9, 27, 81, 243), 4 (4, 16, 64, 256) and 5 (5, 25, 125, 625). Square roots have a positive and a negative value: x² = 49 gives x = 7 or x = −7. Higher roots work the same way: the cube root of −27 is −3, the fourth root of 81 is 3 (3⁴ = 81) and the fifth root of 32 is 2 (2⁵ = 32).

Higher tier only. Estimate a root by trapping it between known powers.

√40:  6² = 36 and 7² = 49, 40 is nearer 36  → about 6.3
∛70:  4³ = 64 and 5³ = 125, 70 is just above 64 → just over 4

Integer and fractional indices (N7)

The index laws for integer powers are in the Number study guide. Apply them to a single base:

5³ × 5⁻⁴ ÷ 5⁻² = 5^(3 − 4 + 2) = 5¹ = 5

Higher tier only. a^(1/n) is the nth root of a, and a^(m/n) = (nth root of a)^m. A negative index means take the reciprocal.

32^(−2/5)    = 1 / (fifth root of 32)² = 1 / 2² = 1/4
(125/64)^(2/3) = (5/4)² = 25/16

Calculating exactly (N8)

Exact means no rounding. Leave fractions as fractions and π as π.

Worked example 7. A quarter circle has radius 6 cm. Find its area and perimeter in terms of π.

Area      = ¼ × π × 6² = 9π cm²
Perimeter = two radii + quarter of circumference
          = 6 + 6 + ¼ × 2 × π × 6 = 12 + 3π cm

Higher tier only. A surd is a root left in exact form. Simplify by taking out the largest square factor; rationalise by multiplying top and bottom by the surd.

√98 = √49 × √2 = 7√2
√20 × √15 = √300 = √100 × √3 = 10√3
(3 + √2)(1 − √2) = 3 − 3√2 + √2 − 2 = 1 − 2√2
15/√5 = 15√5 / 5 = 3√5
2/(3√2) = 2√2 / (3 × 2) = √2/3

Standard form (N9)

A × 10ⁿ with 1 ≤ A < 10 and n an integer. For × and ÷, handle the numbers and the powers separately, then adjust A.

(3.2 × 10⁴) × (5 × 10⁻⁷) = 16 × 10⁻³ = 1.6 × 10⁻²
(4.8 × 10⁵) ÷ (1.2 × 10⁻³) = 4 × 10⁸

For + and −, match the powers first: 3.4 × 10⁵ + 6.1 × 10⁴ = 3.4 × 10⁵ + 0.61 × 10⁵ = 4.01 × 10⁵.

On a calculator, a display such as 2.5E−04 means 2.5 × 10⁻⁴ = 0.00025. When you copy it down, write it as A × 10ⁿ, never as “2.5E−04”.

Common errors

• Ordering negatives by size of digits: −0.6 is greater than −2/3.
• Losing the decimal point when using a given fact; check with an estimate.
• Working left to right instead of following priority: 18 − 3 × 4 is 6, not 60.
• Squaring a negative without brackets: (−2)² = 4, but −2² = −4.
• Using the LCM when the context asks for the largest equal group (HCF).
• Listing ordered pairs twice (RG and GR) when order does not matter.
• Treating 1 as prime, or leaving the prime factorisation out of index form when asked.
• Leaving standard form as 16 × 10⁻³ instead of adjusting to 1.6 × 10⁻².
• (Higher) Writing √20 × √15 as √35, or not simplifying √300 fully.

Next, test yourself with the revision notes self-test and then the practice questions. The Number practice questions cover the rest of Topic 1.

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).