3.1 · Number

OxfordAQA International GCSE Mathematics (9260) unit 3.1 (specification for first assessment 2018). Core rows N1–N20 run from ordering and the four operations through factors and indices, standard form 标准形式, sets, rounding, FDP operators, percentage change and ratio 比; the Extension additions add index-form factorisation, fractional powers, exact surd work, bounds calculations, recurring decimals, reverse percentages, the compound interest formula and exponential growth and decay. The 2016 specimen pair (1E/2E, Extension tier) samples this unit at HCF, surds, fraction chains, bounds and exponential models; lines the specimens never sample are taught here as specification originals and labelled so.

Must-know for unit 3.1 (NO formulae sheet in 9260 — the recall list is the specification itself):

Order positive and negative integers 整数, decimals and fractions on the number line 数轴 with $=$, $\ne$, $<$, $>$ — convert to one form first when comparing ($\tfrac35 = 0.6$). The four operations with formal written methods cover integers, decimals, simple fractions proper and improper, and mixed numbers 带分数: $2\tfrac13 + 1\tfrac14 = \tfrac{28}{12} + \tfrac{15}{12} = \tfrac{43}{12} = 3\tfrac7{12}$. Inverse operations cancel and simplify: multiplication undoes division, subtraction undoes addition — use cancellation early ($\tfrac{15}{28} \times \tfrac{7}{9} = \tfrac{15}{4} \times \tfrac{1}{9} = \tfrac{5}{12}$ after cancelling the 7 and 3).

Even, odd, prime 质数, factor, multiple 倍数, common factor, common multiple — with the highest common factor 最大公因数 and lowest common multiple 最小公倍数 found from prime factorisations 质因数分解. The Extension addition writes every factor in index form 指数形式: $504 = 2^3 \times 3^2 \times 7$ and $588 = 2^2 \times 3 \times 7^2$, so the HCF is $2^2 \times 3 \times 7 = 84$ (the 1E specimen's answer) and the LCM is $2^3 \times 3^2 \times 7^2 = 3528$.

Positive integer powers with square and cube roots ($\sqrt{144} = 12$, $\sqrt[3]{125} = 5$); the index laws multiply and divide by adding and subtracting indices. The Extension addition brings fractional powers 分数指数 ($27^{1/3} = 3$, $8^{2/3} = 4$: cube-root then square) and exact calculation with surds 根式 — simplify by extracting squares and rationalise denominators with conjugates:

The 1E specimen's surd answer $4 + 4\sqrt2$ comes from this same machinery (family-level citation).

Worked example: a negative fractional power. Evaluate $\left(\tfrac{16}{81}\right)^{-3/4}$. Deal with the three parts of the power one at a time. The minus sign flips the fraction; the denominator 分母 $4$ takes a fourth root; the numerator 分子 $3$ cubes:

Root first, then power: $\sqrt[4]{81} = 3$ is easy, while $81^3$ is not. A negative power never makes the answer negative.

Standard form $A \times 10^n$ with $1 \le A < 10$ covers very large and very small measurements; the calculator's $\times 10^x$ key keeps arithmetic exact. Use the calculator "effectively and efficiently": know the trigonometric and square-root functions, keep full precision through the working, and round only at the end.

Worked example: standard form without a calculator. Work out $(6 \times 10^5) \times (4 \times 10^{-2})$. Multiply the numbers and add the powers: $24 \times 10^{3}$. That is not standard form, because $24 \ge 10$, so write $24 = 2.4 \times 10^1$ and the answer is $2.4 \times 10^4$.

Sets 集合: the language and notation — $n(A)$ for the count, $A'$ the complement 补集, $A \cup B$ the union 并集, $A \cap B$ the intersection 交集 — with Venn diagrams 韦恩图 to organise counts (the same toolkit as the probability unit).

Round to an appropriate degree of accuracy — decimal places 小数位 and significant figures 有效数字 — and know each rounded value carries an interval: to the nearest unit $u$, the value lies within $\tfrac u2$ either side. The Extension addition CALCULATES with upper and lower bounds — the upper bound 上界 and lower bound 下界: choose the extreme the question needs.

Worked example: 2E specimen family, adapted. A floor of 96 m² is measured to 2 s.f.; a safe occupancy needs 5.1 m² per person to 1 d.p. The greatest number of people that can be GUARANTEED room: divide the smallest floor by the largest space — LB $= 95.5$, UB per person $= 5.15$, so $\tfrac{95.5}{5.15} = 18.5\ldots$; round DOWN to 18, because 19 people could need $19 \times 5.15 = 97.85$ m², more than the floor may be. (The specimen's own numbers give $19.8\ldots$, also rounded down, to 19.) State which bound each number needs BEFORE dividing.

Equivalent fractions 等值分数 cancel; fractions, decimals and percentages all act as OPERATORS: $35\%$ of 240 is $0.35 \times 240 = 84$; $\tfrac35$ of a quantity divides it by 5 and multiplies by 3. Express one quantity as a fraction or percentage of another: 7 as a percentage of 28 is $\tfrac{7}{28} \times 100 = 25\%$. The Extension addition converts between fractions and recurring decimals 循环小数 in both directions — $0.\overline{45} = \tfrac{45}{99} = \tfrac5{11}$ by the algebraic shift-and-subtract method (specification line, not sampled by the specimens).

Worked example: a recurring decimal with a delay. Write $0.2\overline{7}$ (that is $0.2777\ldots$) as a fraction in its simplest form. Let $x = 0.2\overline{7}$. Move the point until the repeating part starts, then one more cycle:

Subtract the two lines whose decimal parts match exactly ($100x$ and $10x$), not $x$ itself, whose decimal part starts with the 2.

Percentage change uses multipliers 乘数: increase by $12\%$ is $\times 1.12$; decrease by $17\%$ is $\times 0.83$. The Extension additions cover reverse percentage 反向百分比 (percentages) problems and the compound interest formula 复利公式 $A = P\left(1 + \tfrac{r}{100}\right)^n$: £2000 at 3% for 4 years gives $2000 \times 1.03^4 = £2251.02$; reverse: a £92 price after a 15% increase means an original of $92 \div 1.15 = £80$ (specification lines, not sampled by the specimens).

Worked example: reverse percentage. In a sale, a coat costs £68 after a $15\%$ reduction. Find the original price. The sale price is $85\%$ of the original, so original $\times\, 0.85 = 68$ and the original is $68 \div 0.85 = £80$. The usual mistake is to add $15\%$ of £68 ($68 \times 1.15 = £78.20$): the $15\%$ was of the ORIGINAL price, not the sale price.

Ratio notation simplifies by dividing through and links to fractions (share £350 in $2:5$: parts of $\tfrac27$ and $\tfrac57$ — £100 and £250); apply ratio to problems (recipes, maps, mixtures) and use common rates 比率 including rates of pay (£9.20 per hour for 7.5 hours = £69) and best-buy comparisons (unit pricing — specification lines, not sampled). Direct proportion 正比例 scales both quantities ($y = kx$); inverse proportion 反比例 scales them oppositely ($y = \tfrac kx$).

Worked example: a ratio from a difference. Amir and Bea share money in the ratio $3 : 5$. Bea gets £24 more than Amir. The difference is $5 - 3 = 2$ parts, so 2 parts are £24 and 1 part is £12. Amir gets $3 \times 12 = £36$ and Bea gets $5 \times 12 = £60$. Check: $60 - 36 = 24$. ✓

Worked example: inverse square proportion. $y$ is inversely proportional to $x^2$, and $y = 5$ when $x = 2$. Find $y$ when $x = 4$. Write $y = \dfrac{k}{x^2}$; then $5 = \dfrac{k}{4}$, so $k = 20$. When $x = 4$, $y = \dfrac{20}{16} = 1.25$. Doubling $x$ divides $y$ by $2^2 = 4$, a quick check.

The final Extension addition models exponential 指数 growth and decay: $y = a b^t$ with $b > 1$ growing and $0 < b < 1$ decaying. The 2E specimen fits a decay model from data — successive values $240, 192, 153.6$ have ratio $0.8$, so $y = 240 \times 0.8^t$ (the specimen's fitted $a$-family) — and SOLVES with it: £12000 losing 20% per year, $V = 12000 \times 0.8^t$, first falls below £2500 when $0.8^t < 0.2083$, i.e. $t > \ln 0.2083/\ln 0.8 \approx 7.0$: during the 8th year (the specimen's "how many years" answer family). Growth works identically: £1200 at 7% passes £2000 after $\ln(5/3)/\ln 1.07 \approx 7.6$, i.e. 8 full years.

• Order integers, decimals and fractions with $=$, $\ne$, $<$, $>$ on the number line (N1).
• Four operations with formal methods on integers, decimals, fractions, mixed numbers; inverse-operation cancellation (N2, N3).
• Even/odd/prime/factors/multiples vocabulary; HCF and LCM from index-form prime factorisations (N4, Extension; 1E specimen HCF 84).
• Integer powers and roots; index laws; fractional powers; exact surd arithmetic incl. rationalising (N5–N7, Extension).
• Standard form; effective calculator use (N8, N10).
• Sets: $n(A)$, $A'$, $\cup$, $\cap$ with Venn diagrams (N9).
• Rounding to appropriate accuracy; upper/lower bound calculations choosing the right extremes (N11, Extension; 2E specimen 19).
• Equivalent fractions; FDP as operators; fraction/percentage of; recurring decimal conversions (N12–N14, Extension).
• Percentage change with multipliers; reverse percentage; compound interest formula (N15, Extension).
• Ratio notation and sharing; ratio problems; rates of pay and unit pricing (N16–N19).
• Direct and inverse proportion; exponential growth and decay models fitted and solved (N20, Extension; 2E specimen family).

Boundary notes: NO formulae sheet is provided in 9260 — the recall list above is examinable as memory. Lines marked "not sampled" (fractional powers, recurring decimals, reverse percentage, rates of pay and best buys) are taught from the specification text as originals; the specimen pair is Extension-tier only, so Core-tier demand (papers 1–2 Core, 1 h 30, 80 marks) has NO local evidence and holds to spec-text teaching only. Worked examples are adapted from the 2016 specimen pair or original.