first half

Calculating with Density

Rule: Density = Mass ÷ Volume (D = M ÷ V). Example: Mass 50g, Volume 10cm³ = 5 g/cm³.

Finding the Volume of Pyramids

Rule: Volume = (1/3) × Base Area × Height. Example: Base area 12, height 9 = (1/3) × 12 × 9 = 36.

Finding the Volume of Spheres

Rule: Volume = (4/3) × π × r³. Example: Radius 3 cm = (4/3) × π × 27 = 36π.

Identifying Parts of Circles

Rule: Radius = centre to edge. Diameter = edge to edge. Chord = line touching two points. Sector = "pizza slice".

Enlargement by a Positive Scale Factor

Rule: Multiply distance from centre by scale factor. Example: Side length 3 with scale factor 2 becomes 6.

Enlargement by a Negative Scale Factor

Rule: Multiply distance by scale factor and flip across centre. Example: Scale factor -2 doubles distance on the opposite side.

Constructing Direct Proportion Equations

Rule: y = kx. Find k first. Example: If y=12 when x=4, k = 12÷4 = 3, so y = 3x.

Constructing Inverse Proportion Equations

Rule: y = k / x. Find k first. Example: If y=2 when x=3, k = 2×3 = 6, so y = 6/x.

Fractional Indices (x^(a/b))

Rule: Bottom = root, top = power. Example: 8^(2/3) = (∛8)² = 2² = 4.

Upper and Lower Bounds (Error Intervals)

Rule: Find limits of rounded numbers. For division/subtraction: Max = Top Max ÷ Bottom Min. Min = Top Min ÷ Bottom Max.

Completing the Square

Rule: Write x² + bx + c as (x + p)² + q, where p is half of b. Example: x² + 6x + 5 becomes (x + 3)² - 4.

The Quadratic Formula

Rule: Used for ax² + bx + c = 0 when it doesn't factorise. Formula: x = [-b ± √(b² - 4ac)] / 2a.