Imagine you have a pizza cut into 8 equal slices. If you eat 3 of those slices, you haven't eaten the whole pizza, just a part of it. A fraction is simply a way to describe a part of a whole. In this case, you ate $3/8$ of the pizza.

So, in our $3/8$ example, 3 is the numerator, and 8 is the denominator. It's a simple way of saying "three out of eight equal pieces."

Not all fractions look the same. They fall into a few different categories based on the relationship between the numerator and the denominator.

Think back to the pizza. If you eat 1, 2, or even 7 slices, you’re eating a proper fraction of the whole pie ($1/8$, $2/8$, $7/8$). You haven't eaten the whole thing yet.

What if you were really hungry and ate a whole 8-slice pizza and 3 slices from another one? You've eaten 11 slices in total, and each pizza was cut into 8. You could represent this as the improper fraction $11/8$. Any time the top number is bigger than the bottom number, like $5/4$ or $10/3$, it's an improper fraction.

This brings us to our last type: mixed numbers.

A mixed number is just a more intuitive way to write an improper fraction. Instead of saying you ate $11/8$ pizzas, you could say you ate 1 whole pizza and $3/8$ of another. We write this as $1\ 3/8$. It's the same amount, just written differently.

Fractions aren't just for pizzas and pies. They are numbers, just like 1, 2, and 3. They live on the number line, filling in the gaps between the whole numbers.

As you can see, proper fractions are the values between 0 and 1. Once you hit 1 or go beyond it, you're in the territory of improper fractions and mixed numbers. For example, $9/4$ is the same as two wholes and one quarter more, or $2\ 1/4$.

Now, let's test your understanding of these fundamental concepts.

Grasping what fractions are and the different forms they take is the first and most important step. Once you're comfortable with these ideas, you'll be ready to see what else they can do.