数论

• Order and calculate with positives, negatives, decimals and fractions; the four operations with formal written methods.
• Primes, factors, HCF/LCM by prime decomposition; indices, roots, surd 无理数s (H) and standard form 标准形式.
• Fractions ↔ decimals ↔ percentages as operators; rounding, estimation and bounds (H).

Ordering and symbols: order positive and negative integers, decimals and fractions on a number line; use =, ≠, <, >, ≤, ≥.

The four operations (N2): formal written methods for integers, decimals, proper and improper fractions and mixed numbers, all signs — set in context including household finance: profit, loss, cost price, selling price, debit, credit, balance, income tax, VAT, interest rate.

Inverse operations and priority (N3): cancellation to simplify; BIDMAS 运算顺序 — brackets, indices/powers, roots, division/multiplication, addition/subtraction.

Primes and structure (N4): prime numbers, factors, multiples, common factors, common multiples, HCF, LCM; prime factorisation 质因数分解 in index form (product notation; unique factorisation). Worked pattern:

• 200 = 2³ × 5²
• HCF = product of the lowest powers of common primes; LCM = product of the highest powers of all primes.

(N5) systematic listing and the product rule for counting (H).

Indices (N7): laws — a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m-n), (a^m)^n = a^(mn), a^(-n) = 1/a^n, a^(1/2) = √a; (H) fractional indices a^(m/n) = the n-th root of a^m.

Surds (N8, H): √12 = √(4 × 3) = 2√3 — simplify; rationalise denominators (multiply by the conjugate: 1/(3+√2) × (3−√2)/(3−√2) = (3−√2)/7).

Standard form (N9): A × 10ⁿ, 1 ≤ A < 10, n an integer — add/subtract (match powers), multiply/divide (multiply the As, add the ns); interpret calculator displays.

Estimation (N14): round each value to 1 significant figure, compute, decide over- or under-estimate; check calculations by approximation.

Rounding and error intervals 误差区间 (N15): to decimal places and significant figures (know not to round mid-calculation); (H) inequality notation for error intervals — 3.6 kg to 1 d.p. means 3.55 ≤ m < 3.65; truncation gives 3.6 ≤ m < 3.7.

Bounds (N16, H): upper bound 上界 of a length 9 m to the nearest metre is 9.5; in calculations, UB of × and + uses UBs, ÷ uses UB ÷ LB; give the answer to an appropriate degree of accuracy (often the bound width decides).

Conversions (N10): 3.5 = 7/2, 0.375 = 3/8; (H) recurring decimal 循环小数s ↔ fractions: 0.6̄ = 6/9 = 2/3; 0.4̄5̄ (pair) → 45/99 = 5/11. Method: x = 0.6̄, 10x = 6.6̄, 9x = 6.

Fractions as operators (N12): "3/5 of 40" = 40 × 3/5 = 24; percentage as multiplier 乘数 — increase by 15% = × 1.15; decrease by 20% = × 0.8; reverse percentage: find the original before the change by dividing by the multiplier.

Units (N13): metric conversions for length, area, volume, capacity; compound measures (speed, density, pressure) link to topic 3.