➕ Year 8 Mathematics

Expanding & Factorising

Expanding Double Brackets (FOIL)

• Multiply each term in the first bracket by each term in the second bracket
• First, Outer, Inner, Last
• (x + 3)(x + 5) = x² + 5x + 3x + 15 = x² + 8x + 15
• (x + 4)(x − 2) = x² − 2x + 4x − 8 = x² + 2x − 8
• (x − 3)² = (x − 3)(x − 3) = x² − 6x + 9
• Difference of two squares: (x + a)(x − a) = x² − a². e.g. (x + 5)(x − 5) = x² − 25

Factorising Quadratics (x² + bx + c)

• Find two numbers that multiply to give c and add to give b
• x² + 7x + 12: two numbers × to 12, + to 7 → 3 and 4 → (x + 3)(x + 4)
• x² + x − 6: two numbers × to −6, + to 1 → 3 and −2 → (x + 3)(x − 2)
• x² − 9 = (x + 3)(x − 3) — difference of two squares

Factorising by Extracting a Common Factor

• Find the highest common factor of all terms and place it outside the bracket
• 6x² + 10x = 2x(3x + 5)
• 12xy − 8y² = 4y(3x − 2y)

Linear Graphs (y = mx + c)

• m = gradient (how steep the line is = rise ÷ run)
• c = y-intercept (where the line crosses the y-axis)
• Positive gradient: line goes up from left to right. Negative gradient: goes down.
• Gradient = (y₂ − y₁) ÷ (x₂ − x₁) between any two points on the line

Drawing a Straight Line

• Method 1: Use the gradient and y-intercept. Plot (0, c), then use the gradient to find another point
• Method 2: Make a table of x and y values, plot points, draw a straight line through them
• e.g. y = 2x − 1: gradient = 2, y-intercept = −1. When x=0, y=−1; when x=2, y=3

Finding the Equation of a Line

• Step 1: Find the gradient (m) between two given points
• Step 2: Substitute one point and m into y = mx + c; solve for c
• e.g. points (1, 5) and (3, 11): m = (11−5)/(3−1) = 6/2 = 3. Using (1,5): 5 = 3(1) + c → c = 2. Equation: y = 3x + 2

Parallel and Perpendicular Lines

• Parallel lines have the same gradient (m): y = 3x + 1 and y = 3x − 4 are parallel
• Perpendicular lines: their gradients multiply to give −1 (negative reciprocal)
• If a line has gradient 2, a perpendicular line has gradient −½

Inequalities

• < means "less than" | > means "greater than" | ≤ means "less than or equal to" | ≥ means "greater than or equal to"
• Solve like an equation, but if you multiply or divide by a negative number, flip the inequality sign
• e.g. 3x + 5 < 17 → 3x < 12 → x < 4
• e.g. −2x ≥ 8 → x ≤ −4 (sign flips when dividing by −2)

Representing on a Number Line

• Open circle ○ for strict inequalities (< or >): the value itself is NOT included
• Closed circle ● for ≤ or ≥: the value IS included
• Arrow pointing in the direction of the solution

Double (Combined) Inequalities

• −3 < 2x + 1 ≤ 7: subtract 1 from all parts → −4 < 2x ≤ 6 → −2 < x ≤ 3
• Integer values satisfying −2 < x ≤ 3: −1, 0, 1, 2, 3

Simultaneous Equations

Two equations with two unknowns; solve to find values of both unknowns that satisfy both equations simultaneously.

Elimination Method

• Add or subtract the equations to eliminate one variable
• If the coefficients don't match, multiply one (or both) equations first

Substitution Method

• Rearrange one equation to get y = … (or x = …)
• Substitute into the other equation
• e.g. y = 2x − 1 and 3x + y = 9 → 3x + (2x − 1) = 9 → 5x = 10 → x = 2, y = 3

Percentage Change

Percentage Increase and Decrease

• Percentage change = (change ÷ original) × 100
• Multiplier method: increase by 15% → multiply by 1.15; decrease by 20% → multiply by 0.80
• e.g. A coat costs £80. After a 30% discount: 80 × 0.70 = £56

Reverse Percentages (Finding the Original)

• A price after 25% increase is £150. Find the original: 150 ÷ 1.25 = £120
• Always divide by the multiplier to reverse it
• A common error: taking 25% OFF the sale price (incorrect) instead of dividing by 1.25

Compound Percentage Change

• Applied repeatedly over multiple periods
• Compound interest formula: A = P(1 + r/100)ⁿ where P = principal, r = rate, n = years
• e.g. £500 at 4% compound interest for 3 years: 500 × (1.04)³ = 500 × 1.1249 = £562.43
• Depreciation (e.g. car value): A = P(1 − r/100)ⁿ

Standard Form

• A way of writing very large or very small numbers: A × 10ⁿ where 1 ≤ A < 10 and n is an integer
• 4,600,000 = 4.6 × 10⁶ (move decimal point 6 places left)
• 0.000082 = 8.2 × 10⁻⁵ (move decimal point 5 places right)

Calculating in Standard Form

• Multiply: multiply the A values; add the powers of 10
• (3 × 10⁴) × (2 × 10³) = 6 × 10⁷
• Divide: divide the A values; subtract the powers
• (8 × 10⁶) ÷ (4 × 10²) = 2 × 10⁴
• Adding/subtracting: convert to the same power first
Check:

If the A value after calculation is not between 1 and 10, adjust: 25 × 10³ = 2.5 × 10⁴

Pythagoras' Theorem

• Applies to right-angled triangles only
• The hypotenuse is the longest side, opposite the right angle

• e.g. a = 3, b = 4 → c = √(9 + 16) = √25 = 5
• Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17
• Checking if a triangle is right-angled: if a² + b² = c², it is right-angled

Trigonometry (SOH-CAH-TOA)

• Used to find missing sides and angles in right-angled triangles
• Label sides relative to the angle θ: Opposite (O), Adjacent (A), Hypotenuse (H)

Finding a Missing Side

• Identify the angle and the two sides involved
• Choose the correct ratio (SOH, CAH, or TOA)
• Rearrange and calculate: if sin 30° = O/H → O = H × sin 30°
• e.g. Hypotenuse = 10cm, angle = 40°. Find adjacent: cos 40° = A/10 → A = 10 × cos 40° = 7.66 cm

Finding a Missing Angle

• Use the inverse function: θ = sin⁻¹(O/H), θ = cos⁻¹(A/H), θ = tan⁻¹(O/A)
• e.g. O = 5, H = 8 → θ = sin⁻¹(5/8) = sin⁻¹(0.625) = 38.7°

Volumes of 3D Shapes

• Volume is measured in cubic units: mm³, cm³, m³
• For a prism, find the area of the cross-section first, then multiply by the length
• Surface area of a cylinder: 2πr² + 2πrh (two circles + curved surface)

Probability & Tree Diagrams

Basic Probability Rules

• P(event) = number of favourable outcomes ÷ total number of outcomes
• Probability scale: 0 (impossible) to 1 (certain)
• P(A) + P(not A) = 1 (complementary events)
• P(A or B) = P(A) + P(B) — only when A and B are mutually exclusive
• P(A and B) = P(A) × P(B) — only when A and B are independent

Tree Diagrams

• Used for two or more events in sequence; show all possible outcomes
• Multiply along branches to find the probability of a combined outcome
• Add probabilities of different branches that give the same result
• All probabilities on branches from one point must sum to 1

Relative Frequency

• Estimated probability based on experimental results
• Relative frequency = number of times event occurs ÷ total number of trials
• As the number of trials increases, relative frequency approaches the theoretical probability

Statistical Diagrams

Scatter Graphs

• Plot two variables to look for a relationship (correlation)
• Positive correlation: as x increases, y increases (points slope up)
• Negative correlation: as x increases, y decreases (points slope down)
• No correlation: points scattered randomly
• Line of best fit: a straight line drawn through the middle of the data; used to make predictions
• Correlation does NOT imply causation — two things may correlate without one causing the other

Stem-and-Leaf Diagrams

• Organise data while retaining original values; easy to find median, mode, and range
• Leaves are written in order; a key is essential
• Back-to-back stem-and-leaf: compare two data sets with a shared stem

Cumulative Frequency & Box Plots

• Cumulative frequency: running total of frequencies. Plot at upper class boundaries.
• From the CF graph: find median (at n/2), lower quartile (n/4), upper quartile (3n/4)
• Interquartile range (IQR) = Upper Quartile − Lower Quartile (middle 50% of data)
• Box plot: shows minimum, LQ, median, UQ, maximum on a number line