In arithmetic, you work with specific numbers. For example, $3 + 5 = 8$. This is always true. Algebra takes a step back. Instead of just using numbers, we use symbols, usually letters, to represent numbers. This lets us describe relationships and patterns in a more general way.

Think about the cost of a few cinema tickets. If one ticket is £12, two tickets are $2 \times 12$, and three are $3 \times 12$. We can generalise this. If the cost of one ticket is $c$ and you buy $n$ tickets, the total cost is $n \times c$. In algebra, we usually drop the multiplication sign to keep things tidy. We write it as $nc$. This is called an algebraic expression.

There are a few conventions to remember. We write the number before the letter (e.g., $4x$ not $x4$). When multiplying a variable by itself, like $y \times y$, we use index notation: $y^2$. These small rules make expressions easier for everyone to read and understand. An expression is a collection of terms with no equals sign, whereas an equation sets two expressions equal to each other.

Often, we build expressions from worded problems or patterns. Imagine you're buying snacks. A drink costs $d$ pounds and a bag of popcorn costs $p$ pounds. If you buy 3 drinks and 2 bags of popcorn, the total cost is an expression: $3d + 2p$.

What if you go back later and buy another drink and 3 more bags of popcorn? The cost for this trip is $d + 3p$. The total for both trips is $(3d + 2p) + (d + 3p)$. This looks a bit messy. We can tidy it up by collecting 'like terms'. These are terms that contain the same variable raised to the same power. Here, $3d$ and $d$ are like terms, and $2p$ and $3p$ are like terms.

By adding the number parts, called coefficients, we get $4d + 5p$. This simplified expression is much neater and represents the same total cost.

The same laws of indices you know from arithmetic apply to algebra. When variables are involved, these rules help simplify complex expressions.

Brackets are used to group terms. To get rid of them, we use a process called expanding. For a single bracket, we use the distributive law and multiply the term outside by everything inside.

When we have two brackets multiplied together, like $(x+3)(x+4)$, we need to make sure every term in the first bracket multiplies every term in the second one. A common way to organise this is the grid method.

By adding the terms from the grid ($x^2 + 4x + 3x + 12$) and collecting like terms, we get the expanded expression: $x^2 + 7x + 12$.

Factorising is the reverse of expanding. Instead of removing brackets, we put them back in. The simplest method is to find the highest common factor (HCF) of all the terms in the expression.

We place the HCF outside a set of brackets. Then, we figure out what needs to go inside the brackets by dividing each original term by the HCF. This process is called 'factorisation'.

So, the factorised form is $3(2x + 3)$. You can always check your answer by expanding the brackets back out. If you get the expression you started with, you've done it correctly.

Ready to test your skills? Let's see how well you can build, simplify, and manipulate these new expressions.

Being able to move smoothly between words, symbols, and different forms of expressions is the foundation of algebra. It's a powerful tool for describing the world and solving problems.