OxfordAQA International GCSE Mathematics: Ordering, the four operations, primes, powers and calculators (9260) – Revision Notes

These notes condense section 3.1.1 Structure and calculation (references N1 to N5 and N10) of the OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1 (for exams May/June 2018 onwards). For full explanations read the study guide, then test yourself with the practice questions. Course pages: hub and checklist. Find weak spots with a free diagnostic.

All of this unit is Core content, and all content can be assessed on the Extension tier papers. None of it is “Extension only”.

Key facts at a glance

Ref	Remember
N1	Left on a number line is smaller. For negatives, further from zero means smaller.
N2	Written methods for integers, decimals, fractions and mixed numbers, positive and negative.
N3	Inverse operations undo each other. Follow BIDMAS. A root sign or fraction line acts as a bracket.
N4	Prime = exactly two factors. HCF uses lower powers, LCM uses higher powers.
N5	Know powers of 2, 3, 4, 5. Odd roots of negatives exist; even roots of negatives do not.
N10	Use brackets, keep full display values, check angle mode, use inverse trig for angles.

N1: Ordering

Method in steps:

1. Convert everything to decimals, or to fractions over a common denominator.
2. Separate negatives from positives.
3. Order the negatives by distance from zero (largest distance is smallest number).
4. Write the final list in the original form.

Example: −5/6, −0.8, 3/5, 5/8 become −0.833…, −0.8, 0.6, 0.625. Order: −5/6, −0.8, 3/5, 5/8.

Symbols: = equal, ≠ not equal, < less than, > greater than.

N2: Four operations

Operation	Rule
Add or subtract fractions	Common denominator, then combine numerators
Multiply fractions	Cancel, then top × top, bottom × bottom
Divide fractions	Multiply by the reciprocal of the second
Mixed numbers	Convert to improper fractions first
Multiply decimals	Remove points, multiply, restore total decimal places
Divide decimals	Scale both numbers by the same power of 10
Signs	Like signs give +, unlike signs give −

Worked reminders:

5/6 + 7/9 = 15/18 + 14/18 = 29/18 = 1 11/18
7/3 × 9/14 = 3/2
23/5 − 9/2 = 46/10 − 45/10 = 1/10
(−2/3) ÷ (4/9) = −2/3 × 9/4 = −3/2
3.6 × 0.25 = 0.9
0.84 ÷ 0.12 = 84 ÷ 12 = 7

In context: decide whether the answer is a whole number of items (round down for “how many full”) or an exact amount. Always include units.

N3: Inverses and priority

BIDMAS order: brackets, indices (powers and roots), division and multiplication, then addition and subtraction. Work left to right at equal level.

Expression	Value	Note
20 − 2 × 3²	2	3² = 9, then 2 × 9 = 18
(3 + 5)² ÷ 16	4	bracket first, then square
−2²	−4	square first, then negate
(−2)²	4	brackets make the sign part of the base

Inverse pairs: + and −, × and ÷, square and square root, cube and cube root.

Reverse a chain: to undo “× 4 then − 7 = 29”, add 7 to get 36, then divide by 4 to get 9.

Reciprocal of a/b is b/a. Reciprocal of 0.25 is 4. Zero has no reciprocal.

N4: Primes, factors, multiples

Definitions:

• Prime: exactly two factors, 1 and itself. List up to 30: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
• 1 is excluded from the primes. The only even prime is 2.
• Factor (divisor): divides exactly. Multiple: in the times table.
• Prime factorisation: product of primes. Product notation: 84 = 2 × 2 × 3 × 7. Index form: 84 = 2² × 3 × 7.

Method in steps for HCF and LCM:

1. Write both numbers in index form over the same list of primes.
2. HCF: take the lower power of each prime they share (leave out primes not in both).
3. LCM: take the higher power of every prime that appears.
4. Check: HCF × LCM = product of the two numbers.

Worked reminder: 84 = 2² × 3 × 7 and 90 = 2 × 3² × 5.

HCF = 2 × 3 = 6
LCM = 2² × 3² × 5 × 7 = 1260
Check: 6 × 1260 = 7560 = 84 × 90

Primality test: try the primes up to √n. 87 = 3 × 29, so 87 is not prime. 1000 = 2³ × 5³.

Question wording: “at the same time again”, “next together” mean LCM. “Largest size”, “greatest number of identical groups” mean HCF.

N5: Powers and roots

Base	Powers to remember
2	2, 4, 8, 16, 32, 64, 128, 256, 512, 1024
3	3, 9, 27, 81, 243, 729
4	4, 16, 64, 256, 1024
5	5, 25, 125, 625, 3125

Square numbers: 1² to 15² (the last is 225). Cubes: 1³ to 10³ (the last is 1000).

Must-know distinctions:

• (−5)³ = −125 but −5² = −25.
• ∛(−125) = −5, but √(−25) is not a real number.
• √49 = 7 (positive root only). Solving x² = 49 gives x = 7 or x = −7.
• 3⁴ = 81 means 3 × 3 × 3 × 3, not 3 × 4.

N10: Calculator use

Task	Advice
Long fraction	Bracket the numerator and the denominator separately
Rounding	Keep full values until the last line, then round once
Trig side	Rearrange first: opposite = hypotenuse × sin, or adjacent × tan
Trig angle	Use the inverse function, for example tan⁻¹(opposite ÷ adjacent)
π	Use the calculator value; otherwise 3.142
Angle unit	Check it is set to degrees before any trig

Examples: 6.3 × tan 41° = 5.48 (3 s.f.). sin⁻¹(5.2 ÷ 8.4) = 38.2° (1 d.p.). cos⁻¹(4.5 ÷ 7.5) = 53.1° (1 d.p.).

Mixed worked reminders

A lamp flashes every 14 seconds and a buzzer sounds every 20 seconds. Both start together. When do they next coincide?

14 = 2 × 7
20 = 2² × 5
LCM = 2² × 5 × 7 = 140

They coincide again after 140 seconds, which is 2 minutes 20 seconds.

A tray holds 2 1/2 litres. Each cup takes 0.3 litres. How many full cups, and how much is left?

2.5 ÷ 0.3 = 25 ÷ 3 = 8.33...
8 full cups use 8 × 0.3 = 2.4 litres
Left: 2.5 − 2.4 = 0.1 litres

Evaluate (√(49 − 13) − 2³) ÷ (−1/2).

49 − 13 = 36, so √36 = 6
6 − 8 = −2
−2 ÷ (−1/2) = −2 × (−2) = 4

Quick checks before you move on

• Can you recite 2¹⁰, 3⁶, 4⁵ and 5⁵ without working?
• Can you say why 1 is not prime?
• Can you convert any mixed number to an improper fraction and back?
• Do you know where to put brackets so a calculator divides by a whole denominator?

Quick self-test

1. Put in order, smallest first: 0.3, −0.35, 1/4, −1/3.
2. Work out 3 1/3 − 1 3/4.
3. Work out 1.5 × 0.06.
4. Work out 5 − 2 × (3 − 7)².
5. A number is multiplied by 6, then 4 is subtracted, giving 38. Find the number.
6. Write 252 as a product of prime factors in index form.
7. Find the HCF and LCM of 30 and 45.
8. Write down the value of 2⁷ and 5⁴.
9. Evaluate ∛(−343) and (−2)⁶.
10. Find the angle x if tan x = 11/9, to 1 d.p.

Answers

1. −0.35, −1/3, 1/4, 0.3. (−0.35 < −0.333… and 1/4 = 0.25 < 0.3.)
2. 10/3 − 7/4 = 40/12 − 21/12 = 19/12, which is 1 7/12.
3. 15 × 6 = 90 and there are 3 decimal places, so 0.09.
4. (3 − 7)² = 16, then 2 × 16 = 32, then 5 − 32 = −27.
5. 38 + 4 = 42, and 42 ÷ 6 = 7.
6. 252 = 2² × 3² × 7.
7. 30 = 2 × 3 × 5 and 45 = 3² × 5. HCF = 3 × 5 = 15. LCM = 2 × 3² × 5 = 90.
8. 2⁷ = 128 and 5⁴ = 625.
9. ∛(−343) = −7 because (−7)³ = −343. (−2)⁶ = 64.
10. x = tan⁻¹(11/9) = 50.7° (1 d.p.).

Where marks are usually lost

• Putting negative decimals or fractions in the wrong order by reading the digits and ignoring the sign.
• Adding the denominators when adding fractions.
• Multiplying or dividing mixed numbers without changing them to improper fractions first.
• Losing a decimal place in multiplication, giving 0.9 instead of 0.09 style answers.
• Reading −2² as 4.
• Choosing the wrong power in HCF and LCM, or including a prime that is not shared when finding the HCF.
• Listing 1 as a prime, or leaving a prime factorisation as a product that still contains a composite number.
• Rounding in the middle of a calculator calculation, so the final digits drift.
• Calculating a trig angle with the calculator in the wrong angle unit.
• Giving a final answer with no units in a context question, or not rounding down when asked how many whole items.

Official syllabus

OxfordAQA International GCSE Mathematics (9260), Version 5.1, for exams May/June 2018 onwards, published by OxfordAQA: section 3.1.1 Structure and calculation, references N1 to N5 and N10.