Algebra for Beginners: The Basics of Algebra for the 11+ Exam

Why Algebra Belongs in 11+ Preparation

For many children, algebra is the first time maths stops being purely about numbers and starts being about ideas. The word itself, loosely translated from its Arabic roots, means balancing and restoring — but a friendlier way to think about it is as the mystery of the unknown.

Up to this point, children have added, subtracted, multiplied, and divided. There is no real mystery there — it is arithmetic. Algebra introduces something new: variables, also called unknowns, represented by letters. Suddenly maths becomes a puzzle to solve, and that shift is exactly what grammar school entrance exams want to test.

Did you know? Most 11+ maths papers include some basic algebra, usually disguised as “find the missing number” questions. A child who understands what a variable is will find these far less intimidating than one who has only memorised arithmetic.

What Is a Variable?

A variable is simply a letter that stands in the place of a number. Take this equation:

x + 1 = 5

We are on the hunt for the number that belongs where the x sits. If we try 4, we get 4 + 1 = 5, which is true. So we say x = 4. That is a complete algebraic solution: we started with an unknown and worked out what it had to be.

The letter does not have to be x. Consider:

a + 10 = 12

Here, a = 2, because 2 + 10 = 12. The key ideas to share with your child are simple:

• A variable stands in the place of a number.
• Variables are usually written as letters of the alphabet.
• “Solving” an equation just means finding the number the letter represents.

This foundation underpins much of the trickier reasoning later, so it is well worth practising until it feels natural. Our 11+ Maths Test Questions and Answers guide has more worked examples in this style.

The Different Types of Numbers

Before children manipulate variables confidently, they benefit from knowing the families of numbers they are working with. Here they are, from simplest to most advanced.

A helpful point for children: every integer is also a rational number, because 5 is the same as 5/1. And the square root of 2 is real but not rational — it cannot be written as a neat fraction.

Tip: Use the real number line to make this concrete. Draw a line, mark the integers, then show that there is no gap between them — every point is a number. If you drew a line with only fractions, there would be tiny “holes” where numbers like √2 belong.

For the 11+ itself, children will spend almost all their time with natural numbers, integers, and simple fractions — so master those first.

The Properties That Make Maths Work

Three quiet rules govern arithmetic. Children use them instinctively, but naming them makes mental maths faster and more accurate.

1. The Commutative Property (order does not matter)

If you have 5 blue marbles and 2 red marbles, it does not matter whether you count 5 + 2 or 2 + 5 — you always get 7. The same holds for multiplication: five groups of three (5 × 3) and three groups of five (3 × 5) both give 15.

• Addition: a + b = b + a
• Multiplication: a × b = b × a

2. The Associative Property (grouping does not matter)

This one is about where you put the brackets:

• (9 + 3) + 5 = 9 + (3 + 5) — both equal 17
• (5 × 3) × 2 = 5 × (3 × 2) — both equal 30

Because the grouping makes no difference, we usually just drop the brackets entirely.

3. Subtraction Is the Awkward One

Subtraction breaks both rules, which is why it causes so many lost marks:

• It is not commutative: 6 - 2 = 4, but 2 - 6 = -4.
• It is not associative: (9 - 4) - 1 = 4, but 9 - (4 - 1) = 6.
• It clashes with the order of operations: in 10 - 6 + 2, you must work left to right, giving 6 — not addition first, which wrongly gives 2.

Tip: Teach your child to rewrite subtraction as adding a negative number. So 6 - 2 becomes 6 + (-2). Because addition is commutative and associative, all the helpful properties come back — and the negative sign stays attached to the right number. This single habit prevents a surprising number of careless errors.

A Simple Way to Practise at Home

Algebra rewards little-and-often practice far more than occasional cramming. Here is a gentle weekly routine you can adapt.

Keep sessions short and praise the method, not just the answer. A child who can explain why x = 4 understands far more than one who simply guessed correctly. For broader strategy, our How to Improve Your Child’s 11+ Maths Score article pairs well with this routine, and the Practical 11+ Guide puts maths in the context of the whole exam.

Practise Algebra With Our 11+ Maths Apps

Understanding the basics of algebra is one thing; building speed and confidence is another — and that comes from structured, repeated practice. Two apps in our suite are designed for exactly this.

The 11+ Maths Learn & Test app is the ideal starting point. It covers 1,280 questions across 32 topics, with detailed study notes spanning shapes, algebra, fractions, data handling, time, money, and more. The “learn then test” structure means your child first reads a clear explanation, then immediately applies it — perfect for new ideas like variables and number properties.

Once the fundamentals are secure, the 11+ Maths Practice Papers app builds exam stamina with 1,200 maths questions across 24 full practice papers, covering every 11+ maths topic with detailed explanations. Working through full papers helps children handle algebra questions under realistic time pressure, exactly as they will meet them on exam day.

Together, these maths apps are part of our wider collection of 8,100+ questions across 7 specialist 11+ apps, covering maths, English, verbal reasoning, non-verbal reasoning, and vocabulary — everything your child needs to prepare with confidence for grammar school entrance exams.

Tip: Start with Learn & Test to build understanding, then switch to Practice Papers for timed revision. The combination of knowing the method and practising under exam conditions is what turns shaky algebra into reliable marks.