Algebra and Formulae

🎩 Imagine a magician puts some sweets into a bag. You cannot see inside, but you know there are 5 sweets already in the bag and the total is 9. How many did the magician add? You probably worked it out in your head — but mathematicians write it like this: 5 + a = 9. The letter a stands for the unknown number. This clever trick is called algebra, and it is one of the most powerful tools in all of mathematics. Today you will learn to use letters to represent unknowns, write formulae, and substitute values to solve problems.

In algebra, a letter such as n, x, or a is called a variable. A variable is a symbol that holds the place of a number we do not yet know, or a number that can change. An expression is a collection of numbers and variables written together, such as 3n + 2. An equation is a statement that two expressions are equal, shown with an equals sign: 3n + 2 = 11. Solving the equation means finding the value of the variable that makes the statement true. Always read the equation like a balance scale — both sides must weigh the same.

A formula (plural: formulae) is a rule written using letters that works for many different values. For example, the perimeter of a rectangle is written as P = 2l + 2w, where l is the length and w is the width. Once you know the formula, you can substitute — that means swap in — any values you like. If l = 6 cm and w = 3 cm, then P = 2 × 6 + 2 × 3 = 12 + 6 = 18 cm. The formula does the hard work for you every time, no matter which rectangle you measure.

Algebra also helps us describe number sequences. A linear sequence goes up or down by the same amount each time — this fixed amount is called the common difference. If the first term is 3 and the common difference is 4, the sequence is 3, 7, 11, 15 ... You can write the nth term rule as: nth term = 4n − 1. Substitute n = 1: 4 × 1 − 1 = 3 ✅. Substitute n = 2: 4 × 2 − 1 = 7 ✅. The rule lets you find ANY term without listing every step — for example, the 100th term is 4 × 100 − 1 = 399.

Solving a one-step equation means doing the inverse (opposite) operation to find the variable. Think of the equation as a balance scale — whatever you do to one side, you must do to the other. For n + 5 = 12, the inverse of adding 5 is subtracting 5: n = 12 − 5 = 7. For 3n = 18, the inverse of multiplying by 3 is dividing by 3: n = 18 ÷ 3 = 6. Always check your answer by substituting it back: 3 × 6 = 18 ✅. This checking step is the mark of a confident mathematician.

🌟 You have now covered the three big ideas in algebra for Year 6. First, letters and symbols represent unknown values — this is the heart of algebra. Second, formulae are reusable rules written with letters that save time when calculating things like perimeters or sequences. Third, substituting values and using inverse operations let you solve equations and find any term in a sequence. These skills build the bridge between arithmetic and the algebra you will meet throughout secondary school and beyond. Well done — you are already thinking like a mathematician!