Maths, explained clearly.
Step-by-step worked examples, common mistakes, and unlimited free practice worksheets — aligned with the UK National Curriculum and GCSE.
- 01Algebra3 min read
Exponential Growth & Decay
Exponential growth and decay describe quantities that change by multiplying by a constant factor over equal time periods. The general form y = a · bˣ represents this relationship, where a is the initial value, b is the growth factor, and x is the number of time periods. When b > 1, the quantity grows exponentially; when 0 < b < 1, it decays exponentially.
- 02Algebra3 min read
Exponents & Powers
An exponent represents how many times a base number multiplies by itself. In the expression 3⁴, the base is 3 and the exponent is 4, meaning 3 × 3 × 3 × 3 = 81. Exponents follow specific rules that make calculations with large numbers more manageable.
- 03Algebra3 min read
Inequalities
Inequalities are mathematical statements that compare two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Unlike equations which have specific solutions, inequalities describe ranges of values that satisfy the given condition. The fundamental rule when solving inequalities is that multiplying or dividing both sides by a negative number flips the inequality sign.
- 04Algebra3 min read
Linear Equations
A linear equation contains a variable raised to the first power and can be written in the form ax + b = c, where a, b, and c are constants. These equations have exactly one solution and appear as straight lines when graphed. The goal is to isolate the variable by performing inverse operations on both sides of the equation.
- 05Algebra3 min read
Logarithms
A logarithm is the inverse operation of exponentiation, answering the question: to what power must a base be raised to produce a given number? The expression log₅(25) = 2 means that 5² = 25. Logarithms appear throughout Year 12 and 13 A-level mathematics, forming the foundation for exponential equations and growth models.
- 06Algebra3 min read
Polynomials
A polynomial is an algebraic expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, where variables have non-negative integer powers. The term 'polynomial' comes from Greek, meaning 'many terms'. Common examples include 3x + 5 (linear), x² - 2x + 1 (quadratic), and 2x³ + x² - 4x + 7 (cubic).
- 07Algebra3 min read
Quadratic Equations
A quadratic equation is a polynomial equation of degree 2, written in the standard form ax² + bx + c = 0, where a ≠ 0. These equations produce curved graphs called parabolas and can have 0, 1, or 2 real solutions. The solutions represent the x-intercepts where the parabola crosses the horizontal axis.
- 08Algebra3 min read
Scientific Notation
Scientific notation expresses numbers as a coefficient between 1 and 10 multiplied by a power of 10. This standardised form appears throughout Year 8 maths and GCSE specifications as A × 10ⁿ, where A represents the coefficient and n indicates the exponent. The method transforms unwieldy numbers like 45,000 into the more manageable 4.5 × 10⁴.
- 09Algebra3 min read
Systems of Equations
A system of equations consists of two or more equations that share the same variables and must be solved together to find values that satisfy all equations simultaneously. The solution represents the point where the equations intersect when graphed. In Year 9 of the UK National Curriculum, pupils learn to solve simultaneous linear equations using substitution and elimination methods.
- 010Algebra3 min read
Two-Step Equations
A two-step equation contains one variable that requires exactly two operations to isolate. These equations follow the form ax + b = c or ax - b = c, where a represents the coefficient of the variable, b is a constant, and c is the result. The solution process involves systematically undoing operations in reverse order.
- 011Arithmetic4 min read
Addition Properties
Addition properties are fundamental mathematical rules that govern how numbers can be combined and rearranged in addition problems. The three main properties are commutative (order doesn't matter), associative (grouping doesn't matter), and identity (adding zero changes nothing). These properties form the foundation for mental maths strategies taught throughout primary and secondary education in the UK.
- 012Arithmetic3 min read
Addition
Addition combines two or more numbers to find their total sum. The process follows a systematic approach of aligning digits by place value and working from right to left, carrying over when column totals exceed 9. Addition forms the foundation for all arithmetic operations and appears throughout the UK National Curriculum from Reception through GCSE level.
- 013Arithmetic3 min read
Decimal Arithmetic
Decimal arithmetic involves performing addition, subtraction, multiplication, and division with decimal numbers. The key principle is maintaining proper alignment of decimal places during calculations. Year 4 pupils in the UK National Curriculum first encounter decimals through tenths and hundredths, progressing to three decimal places by Year 5.
- 014Arithmetic3 min read
Decimal Word Problems
Decimal word problems combine practical situations with decimal arithmetic operations. These problems typically involve money calculations, measurements, or unit prices where students must identify the correct operation and apply decimal computation rules. The key challenge lies in translating written scenarios into mathematical expressions whilst maintaining precision with decimal places.
- 015Arithmetic3 min read
Factors, GCF & LCM
Factors are whole numbers that divide evenly into another number, whilst multiples are the results of multiplying a number by whole numbers. The Greatest Common Factor (GCF) identifies the largest number that divides two or more numbers, and the Lowest Common Multiple (LCM) finds the smallest number that both original numbers divide into evenly.
- 016Arithmetic3 min read
Intro to Multiplication
Multiplication represents repeated addition of equal groups, where 4 × 3 means adding 4 three times or adding 3 four times. This fundamental arithmetic operation appears throughout the UK National Curriculum from Year 2 onwards, building from concrete examples with physical objects to abstract number work. The multiplication symbol (×) indicates how many groups and how many items per group.
- 017Arithmetic3 min read
Long Division
Long division is a written method for dividing large numbers by systematically breaking down the calculation into manageable steps. The process involves repeatedly estimating how many times the divisor fits into portions of the dividend, then multiplying, subtracting, and bringing down the next digit. This algorithm produces exact quotients and remainders for any division problem, no matter the size of the numbers involved.
- 018Arithmetic3 min read
Modular Arithmetic
Modular arithmetic is a system of arithmetic for integers where numbers wrap around when they reach a certain value called the modulus. The expression 'a mod n' represents the remainder when integer a is divided by positive integer n. For example, 17 mod 5 equals 2 because 17 divided by 5 gives quotient 3 with remainder 2.
- 019Arithmetic3 min read
Multiplication & Division in Daily Life
Multiplication and division form the foundation of everyday mathematical calculations, from sharing sweets equally amongst friends to calculating the total cost of multiple items. These operations appear constantly in real-world scenarios such as working out how many packets of biscuits to buy for a school party or determining how many weeks pocket money will last. Primary school children encounter these concepts from Year 2 onwards, building fluency through repeated practice and memorisation of multiplication tables up to 12 × 12.
- 020Arithmetic3 min read
Multiplication Properties
Multiplication properties are fundamental mathematical rules that govern how numbers behave when multiplied together. These properties include the commutative property (order doesn't matter), associative property (grouping doesn't matter), identity property (multiplying by 1), and distributive property (spreading multiplication over addition). Understanding these properties forms the foundation for mental arithmetic and algebraic manipulation in Key Stage 2 and beyond.