HookTen extra balls made the Lottery three times harder to win
In October 2015, Camelot added ten balls to Lotto, the National Lottery's main game — 49 numbers became 59. That sounds like a change of about 20%. It was not. The chance of hitting the jackpot collapsed from 1 in 13,983,816 to 1 in 45,057,474 — the game became roughly 3.2 times harder to win — and within months a record run of rollovers pushed the jackpot to £66 million in January 2016, the biggest in Lotto history, forcing Camelot to rewrite the rollover rules. A small change to the numbers; an enormous change to the outcome.
Why did ten balls do that much damage? Because the number of possible six-ball tickets is built by repeated multiplication — 59 choices, then 58, then 57, and so on — and multiplication grows things violently. Everything in this section is that machinery: how numbers are built from place value and primes, how the four operations combine them, why the written methods you learned at primary school actually work, and how powers and standard form tame numbers too big or too small to write out. Foundation papers spend more marks here than on any other section, and almost every later topic quietly depends on it.
ModelOrdering numbers — the number line settles every argument
Every comparison question is a number-line question. Negative numbers get bigger as they move towards zero: −2 is greater than −7 because it sits further right. The UK's coldest recorded temperature is −27.2°C, logged at Braemar in 1895 and again in 1982, and matched at Altnaharra in December 1995 — three record-holders sharing one value, because nothing in Britain has ever sat further left on the thermometer.
Decimals are ordered by place value, column by column: 0.72 beats 0.702 because in the hundredths column it shows 2 against 0 — the extra digit on the end of 0.702 is worth nothing until it is compared in its own column. The classic Foundation error is 'longer means larger'. It does not: 0.5 is bigger than 0.4999. Fractions are ordered by converting to a common denominator or, faster on a calculator paper, to decimals.
The six symbols do precise jobs: = equal, ≠ not equal, < less than, > greater than, ≤ less than or equal, ≥ greater than or equal. AQA will ask you to write an inequality as well as read one, and the difference between ≤ and < is exactly the distinction that error intervals (section N1.3) are marked on later.
Order −4, 0.45, 2/5, −4.5 and 0.405, smallest first. Convert what you can: 2/5 = 0.4. Negatives first: −4.5 sits further left than −4, so it is smaller. Then compare the positives column by column: 0.4, 0.405 and 0.45 all show 4 tenths; the hundredths digits are 0, 0 and 5, so 0.45 is largest; the thousandths split the rest — 0 against 5 — so 0.4 < 0.405. Final chain: −4.5 < −4 < 2/5 < 0.405 < 0.45. Writing the chain with symbols, not just a list, is what secures the final mark.
MechanismThe four operations — why the written methods work
Formal written methods are non-negotiable on Paper 1, the non-calculator paper. Column addition and subtraction work because place value lets you trade ten of one column for one of the next — that is all 'carrying' and 'borrowing' are. Long multiplication splits a number by place value and multiplies each part separately; short 'bus stop' division shares out each column in turn and carries the remainder down. These are not rituals: each one is place value made mechanical.
Decimals bend to the same methods. To compute 3.24 × 2.6, ignore the points: 324 × 26 = 8,424. Then count the decimal places in the question — two plus one makes three — and place the point: 8.424. It works because you multiplied one number by 100 and the other by 10, so your whole-number answer is exactly 1,000 times too big, and moving the point three places undoes it.
Fractions: adding and subtracting need a common denominator, because you can only count pieces that are the same size. Multiplying goes straight across the top and the bottom. Dividing flips the second fraction and multiplies — because dividing by a half asks how many halves fit, and there are twice as many halves as wholes. Mixed numbers go improper first, every time; carrying '2 and a bit' through the middle of a calculation is where the marks die.
Work out 2⅓ + 1¾. Improper first: 2⅓ = 7/3 and 1¾ = 7/4. Common denominator 12: 7/3 = 28/12 and 7/4 = 21/12. Add the numerators: 28 + 21 = 49, so the total is 49/12 = 4 1/12. Now a division: 3½ ÷ 1¼ = 7/2 ÷ 5/4 = 7/2 × 4/5 = 28/10 = 2 4/5. Every line of that working is evidence for a method mark on Paper 1 — an unsupported answer risks scoring one mark where the full chain scores three.
ModelPriority of operations — the sum that broke Twitter
In August 2019, the expression 8 ÷ 2(2+2) went viral: millions of people argued for days over whether the answer was 16 or 1. The honest verdict is that the notation is deliberately ambiguous — but the GCSE convention is not. Brackets first, then indices (powers and roots), then division and multiplication with equal priority worked left to right, then addition and subtraction the same way. Under that convention the viral sum reads 8 ÷ 2 × 4, worked left to right: 16.
The structure underneath is inverse operations: addition undoes subtraction, multiplication undoes division, squaring undoes square-rooting. Inverses are how you check answers — 8,424 ÷ 26 should hand back 324 — and later they are how you solve every equation you will ever meet. The reciprocal of a number is 1 divided by it: the reciprocal of 4 is 1/4, the reciprocal of 2/3 is 3/2, and any number multiplied by its reciprocal gives exactly 1. Zero has no reciprocal, because nothing multiplies by 0 to make 1.
Your calculator applies the priority rules automatically, which cuts both ways: type 12 ÷ 3 + 1 hoping for twelve-over-four and it gives you 5, because it divides before it adds. Bracket what belongs together: 12 ÷ (3 + 1) = 3.
MechanismPrimes, HCF & LCM — the factor-tree machine
A prime has exactly two factors, itself and 1 — so 1 is not prime (it has only one factor) and 2 is the only even prime. Primes matter because every whole number is built from them in exactly one way. Nature noticed before mathematicians did: North America's periodical cicadas surface every 13 or 17 years, and biologists argue those prime-numbered cycles keep the insects out of sync with predators whose populations peak every 2, 4 or 6 years.
A factor tree breaks a number down: split it into any factor pair, keep splitting until every branch ends in a prime, then collect the leaves. For 360: 360 = 2 × 180 = 2 × 2 × 90, and so on down to 2 × 2 × 2 × 3 × 3 × 5, written in index notation as 2³ × 3² × 5. AQA insists on the index form for full marks — the bare list of primes is working, not an answer.
HCF and LCM come straight off the prime factorisations. The highest common factor takes each shared prime at its lowest power; the lowest common multiple takes every prime that appears anywhere, each at its highest power. In the wild, LCM questions dress up as buses leaving a stop together (every 12 and 15 minutes — side by side again after 60), gym classes realigning, or hot-dogs sold in packs of different sizes; HCF questions dress up as 'largest equal groups' or the biggest square tile that fits a floor.
Find the HCF and LCM of 84 and 360. Factor trees give 84 = 2² × 3 × 7 and 360 = 2³ × 3² × 5. HCF: the shared primes at their lowest powers are 2² and 3, so HCF = 4 × 3 = 12. LCM: every prime at its highest power — 2³ × 3² × 5 × 7 = 8 × 9 × 5 × 7 = 2,520. Self-check the pair: 84 × 360 = 30,240 and 12 × 2,520 = 30,240. HCF × LCM always equals the product of the two original numbers — a free verification almost nobody uses.
CaseSystematic listing — count everything exactly once
Foundation papers regularly hand out three or four marks for pure organisation: how many two-course meals, how many three-digit codes, how many outfit combinations. The method is fixed — hold the first choice constant, cycle through everything else in order, then move the first choice on. Alphabetical or numerical order is not fussiness; it is the visible proof that you have missed nothing and repeated nothing, and it is what the mark scheme is looking for.
Take digit cards 3, 5 and 8, each used once, making two-digit numbers: 35, 38, 53, 58, 83, 85 — six of them, and the ordered list demonstrates it. A café offering 3 starters and 4 mains serves 3 × 4 = 12 different two-course meals, and a table with starters down the side and mains across the top makes all 12 visible at once. (The formal 'product rule for counting' belongs to the Higher tier, but Foundation examiners still expect the multiplication instinct on two-step problems — backed by a list or table that shows the cases.)
The Lottery numbers from the introduction are the same idea at industrial scale: 59 choices for the first ball, 58 for the second, 57 for the third and onwards. The odds got 3.2 times worse precisely because each extra ball multiplies through every choice that follows it.
ModelPowers and roots — repeated multiplication, and its undo button
A power is repeated multiplication: 2³ = 2 × 2 × 2 = 8 — emphatically not 2 × 3. Foundation expects the square numbers up to 15² = 225, the cubes 1, 8, 27, 64, 125 and 1,000, and recognition of powers of 2, 3, 4 and 5 — know that 64 is simultaneously 2⁶, 4³ and 8², that 81 = 3⁴, and that 625 = 5⁴. Roots run the machine backwards: √49 = 7 because 7² = 49, and the cube root of 64 is 4 because 4³ = 64.
Powers grow absurdly fast, which is why they keep starring in real stories. In January 2002, Californian school student Britney Gallivan demolished the myth that paper cannot be folded more than seven times, folding a roll roughly 1.2 km long twelve times. The reason each fold gets so brutal is doubling: if you could somehow fold a 0.1 mm sheet 42 times, the stack — 0.1 mm × 2⁴² — would stand roughly 440,000 km tall, comfortably past the Moon at about 384,000 km.
The index laws make calculation quick: multiplying powers of the same base adds the indices, dividing subtracts them, and a power of a power multiplies them. Negative indices are on the Foundation paper: a negative index means reciprocal, so 2⁻³ = 1/2³ = 1/8 — a small positive number, not a negative one. (Fractional indices are Higher-tier only.)
Write 3⁴ × 3² ÷ 3⁵ as a single power, then evaluate it. Multiplying adds the indices: 3⁴ × 3² = 3⁶. Dividing subtracts: 3⁶ ÷ 3⁵ = 3¹ = 3. Two lines — no need to grind out 81 × 9 = 729 and divide by 243, though the long way confirms it: 729 ÷ 243 = 3. One with a negative index: 5⁻² = 1/5² = 1/25 = 0.04.
DataExact answers and standard form — π stays put, powers of ten do the lifting
'Give your answer exactly' or 'in terms of π' means stop before the decimal: a circle of radius 5 cm has area π × 5² = 25π cm², exactly. Push it through a calculator and you get 78.539..., which rounds, which is no longer exact. The same discipline applies to fractions: 1/3 is exact; 0.33 is an approximation that poisons every later step it touches. Foundation does not do surds — that is Higher-tier — but exact fraction and π answers are firmly on your papers.
Standard form writes any number as A × 10ⁿ, where 1 ≤ A < 10 and n is an integer. Real numbers demand it: the SARS-CoV-2 virus measures roughly 1 × 10⁻⁷ m across; the ONS estimated the UK's mid-2023 population at about 6.83 × 10⁷ people; the 45,057,474 Lottery combinations from the introduction are about 4.5 × 10⁷. The power counts how many places the point moves — positive for big numbers, negative for small ones — and the 1 ≤ A < 10 condition is marked: 45 × 10⁶ is the right size but the wrong form.
To multiply in standard form, multiply the front numbers and add the powers; to divide, divide and subtract. To add or subtract, either match the powers first or drop back into ordinary numbers — the powers do not combine just because you are.
(3 × 10⁴) × (2 × 10⁵): multiply 3 × 2 = 6, add the powers 4 + 5 = 9, giving 6 × 10⁹. (8 × 10⁵) ÷ (4 × 10²): divide 8 ÷ 4 = 2, subtract 5 − 2 = 3, giving 2 × 10³. Now the trap: (3 × 10⁴) + (2 × 10³) is not 5 × 10⁷ — convert instead: 30,000 + 2,000 = 32,000 = 3.2 × 10⁴. Finally, is 45 × 10³ in standard form? No: 45 is not between 1 and 10, so rewrite it as 4.5 × 10⁴.
VocabularyKey terms the mark scheme pays for
TrapsMisconceptions that cost marks
ExamWhat examiners want
Paper 1 is the non-calculator paper, and this section is its backbone: expect a long multiplication or division, fraction arithmetic and an HCF/LCM problem, all marked method-first. AQA mark schemes award M marks for a correct method even when the arithmetic slips — a complete long-multiplication layout with one wrong digit typically keeps most of its marks, while a bare wrong answer keeps nothing. So write every stage: improper fractions before adding, the factor trees before an HCF, the index-notation line before a final answer.
On Papers 2 and 3 the calculator moves the difficulty into interpretation. Standard-form questions expect you to use the ×10ˣ key rather than typing out '× 10 ^' with rogue brackets, and to write the full display down before rounding. If a question says 'give your answer in terms of π' or 'exactly', a decimal loses the final mark even on a calculator paper — exactness is a form of answer, not a level of precision.
Two habits worth a grade: check divisions by multiplying back (inverse operations sit on the spec precisely because they are the checking tool), and when a question mixes fractions, decimals and percentages, convert everything into one form before comparing — mark schemes routinely award a mark for the conversions alone.