➕ Year 10 Mathematics

GCSE-level algebra, trigonometry, graphs, ratio, statistics, and probability.

Surds & Indices

Rules of Indices

  • aᵐ × aⁿ = aᵐ⁺ⁿ (multiply → add powers)
  • aᵐ ÷ aⁿ = aᵐ⁻ⁿ (divide → subtract powers)
  • (aᵐ)ⁿ = aᵐⁿ (power of a power → multiply)
  • a⁰ = 1 (anything to the power 0 is 1)
  • a⁻ⁿ = 1/aⁿ (negative power → reciprocal)
  • a^(1/n) = ⁿ√a (fractional power → root)
  • a^(m/n) = (ⁿ√a)ᵐ (fractional power → root then power)

Surds

  • A surd is an irrational root that cannot be simplified to a whole number or fraction: √2, √3, √5 are surds
  • Simplifying surds: √12 = √(4×3) = 2√3. Find the largest perfect square factor.
  • Adding/subtracting: only like surds can be combined: 3√2 + 5√2 = 8√2, but √2 + √3 cannot simplify further
  • Multiplying: √a × √b = √(ab). √3 × √3 = 3 (a surd multiplied by itself gives a rational number)
  • Expanding brackets with surds: (3 + √2)(3 − √2) = 9 − 2 = 7 (difference of two squares)

Rationalising the Denominator

  • We never leave a surd in the denominator of a fraction. Multiply numerator and denominator by the surd (or its conjugate).
  • Simple: 1/√3 × (√3/√3) = √3/3
  • With conjugate: 1/(3+√2) × (3−√2)/(3−√2) = (3−√2)/(9−2) = (3−√2)/7
Simplify: (2+√5)² = 4 + 4√5 + 5 = 9 + 4√5 Rationalise: 6/(√3+1) Multiply by (√3−1)/(√3−1): = 6(√3−1)/(3−1) = 6(√3−1)/2 = 3(√3−1) = 3√3 − 3

Advanced Algebra

Simultaneous Equations

  • Elimination: make coefficients of one variable equal by multiplying, then add or subtract equations to eliminate it
  • Substitution: rearrange one equation, substitute into the other
  • One linear + one quadratic: substitute the linear equation into the quadratic → solve the resulting quadratic → substitute back
Solve: 3x + 2y = 17 and x − y = 1 From second equation: x = y + 1 Substitute: 3(y+1) + 2y = 17 → 5y = 14 → y = 2.8, x = 3.8

Algebraic Fractions

  • Simplify by factorising numerator and denominator first, then cancel common factors
  • (x²−9)/(x+3) = (x+3)(x−3)/(x+3) = x−3 (cancel (x+3))
  • Adding: get a common denominator (same process as numerical fractions)
  • 3/(x+1) + 2/(x−1) = [3(x−1) + 2(x+1)] / [(x+1)(x−1)] = (5x−1)/(x²−1)
  • Solving equations with algebraic fractions: multiply through by the LCD to clear fractions, then solve

Rearranging Formulae

  • When the subject appears more than once: factorise and divide
  • Make x the subject of: 3x + ax = b → x(3 + a) = b → x = b/(3+a)
  • Make x the subject of: y = (x+2)/(x−1) → y(x−1) = x+2 → yx − y = x + 2 → yx − x = y + 2 → x(y−1) = y+2 → x = (y+2)/(y−1)

Ratio & Proportion

Percentage Change

  • Percentage increase: new = original × (1 + r/100). e.g. 20% increase → multiply by 1.2
  • Percentage decrease: new = original × (1 − r/100). e.g. 15% decrease → multiply by 0.85
  • Reverse percentage: if a price after 20% increase is £120, original = 120/1.2 = £100
  • Percentage change = (change ÷ original) × 100
  • Compound: applies the percentage to the new amount each time. After n years at r%: final = P × (1 + r/100)ⁿ

Direct and Inverse Proportion

  • Direct proportion: y ∝ x means y = kx. The graph is a straight line through the origin. Double x → double y.
  • Inverse proportion: y ∝ 1/x means y = k/x. The graph is a reciprocal curve. Double x → halve y.
  • Direct as a square: y ∝ x² means y = kx². e.g. distance ∝ time² under uniform acceleration
  • Finding k: use a given pair of values to calculate k, then write the equation
y ∝ x². When x = 3, y = 36. Find y when x = 5. 36 = k(3²) → k = 4. Equation: y = 4x² When x = 5: y = 4(25) = 100

Bounds and Accuracy

  • Measurements are always given to a degree of accuracy. A length of 6.4 cm (1 d.p.) has lower bound 6.35 and upper bound 6.45.
  • General rule: LB = measurement − (half the unit of accuracy); UB = measurement + (half the unit of accuracy)
  • Upper bound of a + b = UB(a) + UB(b); Lower bound = LB(a) + LB(b)
  • Upper bound of a × b = UB(a) × UB(b)
  • Upper bound of a/b = UB(a) / LB(b) (divide by the smallest to get the biggest)

Trigonometry

Right-Angled Trigonometry (SOH-CAH-TOA)

  • sin θ = Opposite/Hypotenuse, cos θ = Adjacent/Hypotenuse, tan θ = Opposite/Adjacent
  • To find a side: choose the ratio that connects the known angle with the known side and the unknown side
  • To find an angle: use the inverse function (sin⁻¹, cos⁻¹, tan⁻¹)
  • Exact values to memorise: sin 30° = 1/2, sin 45° = 1/√2, sin 60° = √3/2; cos 30° = √3/2, cos 45° = 1/√2, cos 60° = 1/2; tan 30° = 1/√3, tan 45° = 1, tan 60° = √3

The Sine Rule

a/sin A = b/sin B = c/sin C Use when: you know two angles and a side (AAS), or two sides and a non-included angle (SSA). To find a side: a = b × sin A / sin B To find an angle: sin A = a × sin B / b → A = sin⁻¹(...)

The Cosine Rule

Finding a side: a² = b² + c² − 2bc cos A Finding an angle: cos A = (b² + c² − a²) / 2bc Use when: you know two sides and the included angle (SAS), or all three sides (SSS).

Area of a Non-Right-Angled Triangle

Area = ½ab sin C where a and b are two sides and C is the angle between them.

Graphs

Quadratic, Cubic, and Reciprocal Graphs

  • Quadratic (y = ax² + bx + c): U-shaped (a > 0) or n-shaped (a < 0) parabola. Roots = where it crosses the x-axis. Vertex = turning point.
  • Cubic (y = ax³ + ...): S-shaped curve. One or three real roots. If a > 0, goes up from bottom-left to top-right.
  • Reciprocal (y = k/x): two separate curves in opposite quadrants (if k > 0: Q1 and Q3). Asymptotes at x = 0 and y = 0.
  • Exponential (y = aˣ): always positive, always crosses the y-axis at (0,1) if a > 0. Rapid growth (a > 1) or decay (0 < a < 1).

Transformations of Graphs

  • y = f(x) + a: translate up by a (or down if a < 0)
  • y = f(x + a): translate left by a (note: +a moves LEFT — counter-intuitive)
  • y = af(x): stretch vertically by scale factor a (from the x-axis)
  • y = f(ax): stretch horizontally by scale factor 1/a (from the y-axis)
  • y = −f(x): reflection in the x-axis
  • y = f(−x): reflection in the y-axis

Solving Equations Graphically

  • The solutions to f(x) = g(x) are the x-coordinates of the intersections of y = f(x) and y = g(x)
  • Example: to solve x² = 2x + 3 graphically, draw y = x² and y = 2x + 3 on the same axes; the x-values of intersections are the solutions
  • Alternatively: rearrange to f(x) − g(x) = 0 and find where the resulting curve crosses the x-axis

Proof

Types of Mathematical Proof

  • Algebraic proof: use algebra to show a statement is always true. Use general representations: consecutive integers n and n+1; even numbers 2n; odd numbers 2n+1; consecutive even numbers 2n and 2n+2.
  • Proof by counter-example: to disprove a statement, find just ONE example where it is false

Key Proof Techniques

  • To show something is always even: show it equals 2 × (some integer)
  • To show something is always odd: show it equals 2k + 1 for some integer k
  • To show a product is always divisible by n: factorise so n appears as a factor
  • Geometric proof: circle theorem proofs use properties (isoceles triangles from radii, angle at centre = 2 × angle at circumference)
Prove that the sum of three consecutive integers is always divisible by 3. Let the integers be n, n+1, n+2. Sum = n + (n+1) + (n+2) = 3n + 3 = 3(n+1) This is always a multiple of 3. □

Statistics

Sampling

  • Random sampling: every member of the population has an equal chance of being selected. Uses a random number generator or lottery method.
  • Stratified sampling: the population is divided into subgroups (strata) and a proportional number is selected from each. Number from a stratum = (stratum size / total population) × sample size. Ensures the sample reflects the population's composition.
  • Systematic sampling: every nth member of the list is selected. Simple but can introduce bias if there is a pattern in the list.
  • Convenience sampling: choosing whoever is easiest to reach. Highly biased — not a reliable method.

Histograms (revisited)

  • Frequency density = frequency ÷ class width. The y-axis shows frequency density.
  • Area of bar = frequency. Total area = total frequency.
  • To read off a frequency from a histogram: frequency = frequency density × class width

Interpreting Data

  • Correlation: positive (as x increases, y increases), negative (as x increases, y decreases), no correlation
  • Correlation ≠ causation: two things can be correlated without one causing the other (both may be caused by a third variable)
  • Line of best fit: drawn through the mean point (x̄, ȳ). Used for interpolation (predicting within the data range) — reliable. Extrapolation (predicting outside the range) — unreliable.

Probability

Set Notation and Venn Diagrams

  • ξ (universal set): all elements under consideration
  • A ∩ B: elements in both A AND B (intersection)
  • A ∪ B: elements in A OR B or both (union)
  • A' (complement): elements NOT in A
  • P(A ∪ B) = P(A) + P(B) − P(A ∩ B) — inclusion-exclusion principle

Conditional Probability

  • P(B|A) = P(A ∩ B) / P(A) — the probability of B given A has occurred
  • Events are independent if P(B|A) = P(B) — knowledge of A doesn't affect probability of B
  • For dependent events (without replacement): probabilities change after each selection
  • Tree diagrams: multiply along branches (AND), add between branches (OR). Final probabilities must sum to 1.

Expected Outcomes

  • Expected frequency = probability × number of trials
  • If a die is rolled 600 times, expected frequency of a 6 = (1/6) × 600 = 100
  • Actual frequencies will deviate from expected — this is normal. The more trials, the closer actual frequencies approach expected.