Before diving into algebra, let's refresh some key ideas about numbers. You're already familiar with integers (whole numbers and their opposites), fractions, decimals, and percentages. The key is to be comfortable moving between them and performing calculations fluently.

For example, calculating a 25% discount on a $60 item is the same as finding $60 \times 0.25$ or $60 \times \frac{1}{4}$. All three methods give you the same answer: $15.

When you're dealing with rates and ratios, you're just comparing numbers. A ratio compares quantities with the same unit, while a rate compares quantities with different units.

A common ratio problem involves scaling quantities up or down. If you want to double the recipe, you just multiply both parts of the ratio by 2. You'd use 4 cups of flour and 2 cups of sugar, maintaining the 4:2 ratio, which simplifies back to 2:1.

Rates, on the other hand, involve different units. Speed is a classic example.

Algebra uses letters, called variables, to represent unknown numbers. An algebraic expression is a combination of numbers, variables, and operation symbols. For example, $5x + 3$ is an expression.

One of the most common tasks in algebra is simplifying expressions. This involves combining 'like terms'. Like terms are terms that have the exact same variable part. For instance, $3x$ and $7x$ are like terms, but $3x$ and $7y$ are not. Neither are $3x$ and $7x^2$.

Another key skill is factorisation, which is like the reverse of expanding brackets. You look for the highest common factor (HCF) of all the terms in the expression and 'pull it out'.

Consider the expression $6x + 9$. The terms are $6x$ and $9$. The highest number that divides into both 6 and 9 is 3. So, 3 is the HCF.

An expression becomes an equation when you set it equal to something. For example, $3a + 7b = 20$ is an equation. Our goal is often to find the value of the variable that makes the equation true.

First, let's look at evaluating expressions. This is when you're given the values for the variables and just need to substitute them in to find the result. This is also called substitution.

Solving a linear equation means finding the value of the variable. The main strategy is to perform the same operation on both sides of the equation to isolate the variable.

Let's solve for $x$ in the equation $4x - 5 = 11$.

Linear equations can be graphed on a Cartesian plane, which has a horizontal x-axis and a vertical y-axis. The graph of a linear equation is always a straight line.

To graph an equation like $y = 2x + 1$, you need to find some coordinate pairs $(x, y)$ that make the equation true. The easiest way is to pick a few simple values for $x$ and calculate the corresponding values for $y$.

Once you have two or more points, you can draw a straight line through them. Every point on that line is a solution to the equation $y = 2x + 1$.

Mastering these number and algebra skills provides a strong foundation for tackling more complex problems. Practice is key to building speed and confidence.