Year 7 Mathematics – GCSEGuide

Place Value & Ordering Numbers

Place Value

Every digit in a number has a place value depending on its position. From right to left: units, tens, hundreds, thousands, ten-thousands, hundred-thousands, millions.

Example 4,572,836 Millions=4 · Hundred-thousands=5 · Ten-thousands=7 Thousands=2 · Hundreds=8 · Tens=3 · Units=6

The value of a digit = the digit × its place value (e.g. the 7 in 4,572 = 7 × 100 = 700... wait, 7 is in the tens position here... let me fix that logic: digit × place value column)
Ordering: always compare the most significant digit first (leftmost)
Partitioning: splitting a number into its components e.g. 3,456 = 3000 + 400 + 50 + 6

Powers of 10

10¹ = 10, 10² = 100, 10³ = 1000
Multiplying by 10, 100, 1000 → digits move left by 1, 2, 3 places
Dividing by 10, 100, 1000 → digits move right by 1, 2, 3 places

Common Mistake:

People say "add a zero" when multiplying by 10. This works for whole numbers but NOT for decimals (3.5 × 10 = 35, not 3.50). Always say digits move left.

Negative Numbers

Negative numbers are less than zero. They appear on the left of zero on a number line.

Ordering Negative Numbers

The further left on a number line, the smaller the number
−10 < −3 (−10 is smaller, even though 10 is bigger than 3)

Adding and Subtracting

Adding a negative = subtracting: 5 + (−3) = 5 − 3 = 2
Subtracting a negative = adding: 5 − (−3) = 5 + 3 = 8
Use a number line — start at the first number, then move left (for subtraction) or right (for addition)

Multiplying and Dividing

Sign Rules Positive × Positive = Positive Negative × Negative = Positive Positive × Negative = Negative Negative × Positive = Negative Same rule applies for division.

Think of the sign rules like "enemies of enemies are friends": two negatives always make a positive in multiplication and division.

Factors, Multiples & Prime Numbers

Factors

A factor of a number divides into it exactly (no remainder)
Factors always come in pairs: e.g. factors of 12 are 1 & 12, 2 & 6, 3 & 4
Every number has at least two factors: 1 and itself

Multiples

Multiples are the times-table of that number: multiples of 4 are 4, 8, 12, 16, 20…
Lowest Common Multiple (LCM): the smallest number that is a multiple of both numbers e.g. LCM of 4 and 6 = 12
Highest Common Factor (HCF): the largest factor shared by both numbers e.g. HCF of 12 and 18 = 6

Prime Numbers

A prime number has exactly two factors: 1 and itself
First ten primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29
1 is NOT a prime number (it only has one factor)
2 is the only even prime number

Prime Factor Decomposition

Every whole number can be written as a product of prime factors
Use a factor tree: keep splitting until all branches are primes
e.g. 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5

Fractions

Parts of a Fraction

Numerator: the top number (how many parts you have)
Denominator: the bottom number (how many equal parts in total)

Equivalent Fractions

Multiply or divide both numerator and denominator by the same number
e.g. 1/2 = 2/4 = 3/6 = 50/100 — all equivalent

Simplifying (Cancelling)

Divide numerator and denominator by their HCF
e.g. 12/18 ÷ 6 = 2/3 (HCF of 12 and 18 is 6)

Comparing Fractions

Convert to a common denominator first, then compare numerators
e.g. 3/4 vs 5/7 → 21/28 vs 20/28 → 3/4 is larger

Adding and Subtracting Fractions

Same denominator: add/subtract numerators, keep denominator
Different denominators: find common denominator first, then add/subtract
e.g. 1/3 + 1/4 = 4/12 + 3/12 = 7/12

Multiplying and Dividing Fractions

Multiplying: multiply numerators together, multiply denominators together
e.g. 2/3 × 3/5 = 6/15 = 2/5
Dividing: flip the second fraction (reciprocal) and multiply
e.g. 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6

Mixed Numbers and Improper Fractions

Improper fraction: numerator > denominator (e.g. 7/3)
Mixed number: whole number + fraction (e.g. 2⅓)
Convert improper to mixed: divide numerator by denominator (e.g. 7÷3 = 2 rem 1 → 2⅓)

Decimals & Percentages

Decimal Place Value

After the decimal point: tenths, hundredths, thousandths
e.g. 3.472 = 3 + 4/10 + 7/100 + 2/1000

Converting Between Fractions, Decimals & Percentages

Conversion Methods Fraction → Decimal: divide numerator by denominator (1/4 = 0.25) Decimal → Percentage: multiply by 100 (0.25 → 25%) Percentage → Decimal: divide by 100 (60% → 0.6) Percentage → Fraction: put over 100, then simplify (35% = 35/100 = 7/20)

Finding Percentages

Find 10%: divide by 10. Find 1%: divide by 100
Build up: e.g. 35% = 30% + 5% = (3 × 10%) + (½ × 10%)
Using a multiplier: 20% of 80 = 0.2 × 80 = 16

Algebra – Expressions & Substitution

Writing Algebraic Expressions

A variable (like x or n) represents an unknown number
3n means 3 × n (we drop the × sign in algebra)
n/4 means n ÷ 4
n² means n × n (not n × 2)

Simplifying (Collecting Like Terms)

Like terms have the same letter(s) and power: 3x and 5x are like terms
3x + 5x = 8x; 3x + 5y cannot be simplified further
4x² + 3x − x² + 2x = 3x² + 5x

Substitution

Replace each letter with its given value, then calculate
If a = 3 and b = −2: a² + 2b = 3² + 2(−2) = 9 − 4 = 5
Always substitute into brackets and follow BIDMAS

Expanding Brackets

Multiply everything inside the bracket by the term outside
3(x + 4) = 3x + 12
−2(3x − 5) = −6x + 10 (take care with signs)

Algebra – Solving Equations

To solve an equation, find the value of the unknown that makes it true. Use inverse operations — do the same thing to both sides.

One-Step Equations

x + 7 = 12 → x = 12 − 7 = 5
3x = 18 → x = 18 ÷ 3 = 6
x/4 = 5 → x = 5 × 4 = 20

Two-Step Equations

2x + 3 = 11 → 2x = 8 → x = 4
Work in reverse BIDMAS: undo addition/subtraction first, then multiplication/division

Equations with the Unknown on Both Sides

5x − 3 = 2x + 9 → collect x terms: 3x = 12 → x = 4
Always move the smaller x term across

Checking:

Always substitute your answer back into the original equation to verify it works.

Sequences

A sequence is a list of numbers following a pattern (a rule)
Term: each number in a sequence
Common difference: the amount added or subtracted each time (arithmetic sequence)
The nth term formula lets you find any term without listing them all

Finding the nth Term of an Arithmetic Sequence

Step 1: find the common difference (d) — this is the coefficient of n
Step 2: substitute n=1 and adjust: nth term = dn + (first term − d)
e.g. sequence 5, 8, 11, 14… → d=3, nth term = 3n + 2
Check: when n=1, 3(1)+2 = 5 ✓; when n=2, 3(2)+2 = 8 ✓

Special Sequences

Square numbers: 1, 4, 9, 16, 25… (nth term = n²)
Cube numbers: 1, 8, 27, 64… (nth term = n³)
Triangular numbers: 1, 3, 6, 10, 15… (nth term = n(n+1)/2)
Fibonacci: 1, 1, 2, 3, 5, 8, 13… (each term = sum of two before)

Angles & Lines

Types of Angle

Acute: less than 90°
Right angle: exactly 90°
Obtuse: between 90° and 180°
Straight: exactly 180°
Reflex: between 180° and 360°

Angle Rules

Angles on a straight line add up to 180°
Angles around a point add up to 360°
Vertically opposite angles are equal (formed when two lines cross)
Angles in a triangle add up to 180°
Angles in a quadrilateral add up to 360°

Parallel Lines

Corresponding angles (F-angles) are equal
Alternate angles (Z-angles) are equal
Co-interior angles (C-angles) add up to 180°

2D & 3D Shapes

Properties of 2D Shapes

Triangle: 3 sides, 3 angles summing to 180°. Types: equilateral (all equal), isosceles (2 equal), scalene (all different), right-angled
Quadrilaterals: 4 sides, angles sum to 360°. Types: square, rectangle, parallelogram, rhombus, trapezium, kite
Regular polygon: all sides and angles equal. Interior angle = (n−2)×180°/n where n = number of sides

Properties of 3D Shapes

Cube: 6 square faces, 12 edges, 8 vertices
Cuboid: 6 rectangular faces, 12 edges, 8 vertices
Triangular prism: 5 faces, 9 edges, 6 vertices
Cylinder: 2 circular faces, 1 curved surface, 2 edges
Cone: 1 circular face, 1 curved surface, 1 apex vertex
Sphere: 0 flat faces, 1 curved surface, 0 edges, 0 vertices
Euler's formula: Faces + Vertices − Edges = 2 (for polyhedra)

Area & Perimeter

Key Area Formulae Rectangle: A = length × width Triangle: A = ½ × base × height Parallelogram: A = base × perpendicular height Trapezium: A = ½(a + b) × h (a and b are parallel sides) Circle: A = πr² | Circumference = 2πr or πd

Perimeter = total distance around the outside (add all sides)
Area is measured in square units: mm², cm², m², km²
Area of compound shapes: split into simpler shapes, find each area, add (or subtract) them