Mathematics (UK GCSE)
Mathematics for UK GCSE (Years 10-11, ages 14-16). Covers Number, Algebra, Ratio/Proportion, Geometry, Probability, Statistics and Graphs, aligned with AQA/Edexcel/OCR/WJEC foundation and higher tier specifications.
Ämne: Matematik · Nivå: Högstadium (13–15) · 431 kort
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- Place value: each digit's value depends on its position. In 3,452.67 the 4 represents 400 (hundreds) and the 6 represents 0.6 (tenths).
- Ordering decimals: line up the decimal points and compare digit by digit from left to right. 0.42 < 0.5 because 4 tenths < 5 tenths, even though 0.42 has more digits.
- A prime number has exactly two distinct factors: 1 and itself. 1 is not prime (only one factor). The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
- Prime factorisation expresses a number as a product of primes. 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5. Every integer > 1 has a unique prime factorisation (Fundamental Theorem of Arithmetic).
- HCF (Highest Common Factor) is the largest number that divides two values exactly. LCM (Lowest Common Multiple) is the smallest positive number both values divide into. For 12 and 18: HCF = 6, LCM = 36.
- Useful identity: HCF(a,b) × LCM(a,b) = a × b. So if you know HCF you can find LCM by dividing the product by HCF. Example: 12 × 18 = 216, divided by HCF 6 gives LCM 36.
- Index laws (positive integer indices): a^m × a^n = a^(m+n); a^m ÷ a^n = a^(m-n); (a^m)^n = a^(mn); (ab)^n = a^n b^n; a^0 = 1 for any non-zero a.
- Negative index: a^(-n) = 1/a^n. So 2^(-3) = 1/2³ = 1/8. A negative index turns the value into its reciprocal raised to the positive power.
- Fractional index: a^(1/n) = the n-th root of a. So 8^(1/3) = ³√8 = 2. More generally a^(m/n) = (ⁿ√a)^m, e.g. 27^(2/3) = (³√27)² = 3² = 9.
- Standard form (scientific notation): a × 10^n where 1 ≤ a < 10 and n is an integer. Example: 47,000 = 4.7 × 10⁴; 0.00023 = 2.3 × 10⁻⁴. Used for very large or very small numbers.
- A surd is an irrational root, e.g. √2, √3, √5. Surd rules: √(ab) = √a × √b; √(a/b) = √a/√b; (√a)² = a. Example: √50 = √(25×2) = 5√2.
- Rationalising the denominator removes surds from the bottom of a fraction. For 1/√a multiply top and bottom by √a: 1/√3 = √3/3. For 1/(a+√b) multiply by the conjugate (a−√b).
- Percentage of an amount: multiply by the decimal equivalent. 15% of 80 = 0.15 × 80 = 12. To increase by 15% multiply by 1.15; to decrease by 15% multiply by 0.85.
- Reverse percentages: to find an original value after a percentage change, divide by the multiplier. If a price after 20% VAT is £72, then original = 72 ÷ 1.20 = £60.
- Compound interest formula: Final amount = P × (1 + r/100)^n, where P is the principal, r is the percentage rate per period and n is the number of periods. Interest is added to the running total each period.
- Depreciation (compound decrease): Final value = P × (1 − r/100)^n. A £20,000 car losing 15% of value each year is worth 20000 × 0.85³ ≈ £12,283 after 3 years.
- Converting between fractions, decimals and percentages: ½ = 0.5 = 50%; ¼ = 0.25 = 25%; ⅛ = 0.125 = 12.5%; ⅓ = 0.333… = 33.3%; ⅖ = 0.4 = 40%; ⅘ = 0.8 = 80%.
- Adding/subtracting fractions: find a common denominator first. 2/3 + 1/4 = 8/12 + 3/12 = 11/12. Multiplying fractions: multiply numerators and denominators directly: 2/3 × 5/7 = 10/21.
- Dividing fractions: multiply by the reciprocal (flip the second fraction). 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8. "Keep, change, flip."
- Recurring decimals to fractions: let x = 0.444…, then 10x = 4.444…, so 10x − x = 4 giving x = 4/9. For two-digit recurrence (e.g. 0.272727…) multiply by 100 instead.
- Upper and lower bounds: a measurement rounded to a given accuracy lies within ± half a unit. 12 cm to nearest cm has lower bound 11.5 cm and upper bound 12.5 cm (often written < for the upper bound).
- Combining bounds: for sums use UB+UB, LB+LB. For products use UB×UB, LB×LB. For quotients UB of a/b is UB(a)/LB(b); LB of a/b is LB(a)/UB(b). For subtraction UB(a−b) = UB(a) − LB(b).
- Rounding to significant figures (s.f.): count from the first non-zero digit. 0.004789 to 2 s.f. = 0.0048. The number of s.f. tells you how many meaningful digits to keep.
- Estimation using rounding: round each value to 1 s.f. and compute. For (38 × 21)/9.4 estimate as (40 × 20)/10 = 80. Useful for checking calculator answers.
- Order of operations (BIDMAS/BODMAS): Brackets, Indices (or Orders), Division/Multiplication (left to right), Addition/Subtraction (left to right). Same precedence operations evaluate left to right.
- Negative number rules: + × + = +, − × − = +, + × − = −, − × + = −. Same applies for division. Two negatives "cancel out". Example: −6 ÷ −2 = 3; −5 × 4 = −20.
- Square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. A square number is n² for some integer n. Cube numbers: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.
- Algebraic conventions: 3a means 3 × a; ab means a × b; a² means a × a; a/b means a ÷ b. No × sign needed between a number and a letter or between two letters.
- Collecting like terms: combine terms with the same letter and power. 4a + 3b − 2a + 5b = 2a + 8b. You cannot combine 3a + 2b because they are unlike terms.
- Expanding single brackets: multiply every term inside by the term outside. 3(2x + 5) = 6x + 15. Watch the sign: −2(x − 4) = −2x + 8.
- Expanding double brackets (FOIL): (x+3)(x+5) = x² + 5x + 3x + 15 = x² + 8x + 15. Multiply First, Outside, Inside, Last and then collect like terms.
- Difference of two squares: (a + b)(a − b) = a² − b². Useful for factorising: x² − 49 = (x + 7)(x − 7). Also 9x² − 25 = (3x + 5)(3x − 5).
- Factorising a quadratic x² + bx + c: find two numbers that multiply to c and add to b. For x² + 7x + 12 use 3 and 4: (x + 3)(x + 4).
- Factorising ax² + bx + c with a ≠ 1: find two numbers that multiply to a×c and add to b, then split bx and factorise by grouping. 2x² + 7x + 3 → split 7x as 6x+x → (2x+1)(x+3).
- The quadratic formula solves ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) / (2a). Use when factorising is hard. The expression b² − 4ac is called the discriminant.
- Discriminant b² − 4ac tells you about roots of ax² + bx + c = 0: > 0 → two distinct real roots; = 0 → one repeated root; < 0 → no real roots (curve does not touch x-axis).
- Completing the square for x² + bx + c: rewrite as (x + b/2)² − (b/2)² + c. Example: x² + 6x + 1 = (x + 3)² − 9 + 1 = (x + 3)² − 8. Useful for finding turning points.
- Turning point of a parabola from completed-square form (x − p)² + q: vertex is at (p, q). For y = x² − 6x + 5 = (x − 3)² − 4, the minimum is at (3, −4).
- Solving a linear equation: do the same operation to both sides to isolate the variable. 3x − 7 = 14 → add 7 → 3x = 21 → divide by 3 → x = 7. Always check by substituting back.
- Solving simultaneous linear equations by elimination: scale equations so one variable has matching coefficients, then add or subtract to eliminate. Substitute back to find the other variable.
- Linear-quadratic simultaneous equations: rearrange the linear equation to make one variable the subject, substitute into the quadratic, then solve. Each solution gives a pair (x, y).
- Solving inequalities works like equations, with one twist: multiplying or dividing both sides by a negative number reverses the inequality sign. E.g. −2x > 6 → x < −3.
- Quadratic inequalities: solve the corresponding equation, sketch the parabola, then read off where it is positive/negative. For x² − 4 > 0: roots at ±2, parabola opens upward, so x < −2 or x > 2.
- Arithmetic sequence: each term differs from the previous by a constant d (common difference). nth term = a + (n − 1)d, where a is the first term. Example: 5, 8, 11, 14: nth term = 3n + 2.
- Geometric sequence: each term is the previous multiplied by a constant r (common ratio). nth term = ar^(n−1). Example: 3, 6, 12, 24 has r = 2 so nth term = 3 × 2^(n−1).
- Quadratic sequence: second differences are constant. The nth term has the form an² + bn + c, where 2a equals the second difference. Example: differences 5, 7, 9 give second difference 2 → a = 1.
- Fibonacci-style sequence: each term is the sum of the previous two: a, b, a+b, a+2b, 2a+3b, 3a+5b… The classical Fibonacci sequence starts 1, 1, 2, 3, 5, 8, 13, 21, 34.
- Function notation: f(x) names a rule. f(x) = 3x + 2 means "apply rule 3x+2 to x". f(4) means substitute 4: f(4) = 3(4) + 2 = 14. f(−1) = 3(−1) + 2 = −1.
- Composite function fg(x) means "first apply g, then apply f to the result". If f(x) = 2x+1 and g(x) = x², then fg(x) = f(x²) = 2x² + 1, whereas gf(x) = (2x+1)².
- Inverse function f⁻¹(x) reverses f. To find it: write y = f(x), swap x and y, then solve for y. If f(x) = 3x − 5, then x = 3y − 5 → y = (x + 5)/3, so f⁻¹(x) = (x + 5)/3.