ordering fractions

Welcome! I am excited to help you master ordering fractions—it is a super useful skill that makes dealing with measurements, recipes, and logic puzzles a breeze.

Before we dive into our first visual, I'd love to know a little bit about your starting point.

To help me tailor this to your level, could you let me know how comfortable you currently are with basic fractions, and why you're interested in learning how to order them?

I know what fractions are, but I get stuck comparing them.

It is completely normal to feel stuck when comparing fractions! They can look a bit like a secret code, but once you know how to read them, they tell you exactly how big or small they are. Let's pull back the curtain and look at the foundation first.

What Exactly is a Fraction?

A fraction is simply a way of showing parts of a whole, like slices of a pizza. Every fraction has two numbers that work together as a team: the on the top, and the on the bottom.

Side-by-side fraction models representing three-quarters and one-quarter.

To make sure this feels solid before we start comparing them, let's look at how the slices are sliced. To have a true fraction, every single piece must be exactly the same size. If you cut a cake unevenly, you can't use basic fractions to describe the pieces accurately!

Can you show me an example of unequal parts?

You've got it! Seeing unequal parts side-by-side makes it immediately obvious why we can't use basic fractions for them. If the pieces are different sizes, saying you have "one out of four" slices doesn't tell us how much you actually have.

Two pizzas side-by-side: the left pizza is divided into two equal halves, while the right pizza is sliced unevenly into one massive piece and one tiny sliver.

In the visual, you can see that the unequal slice on the right is totally different from the clean, mathematical halves on the left. Once we ensure our slices are perfectly equal, we can start comparing them. Let's test this out with a simple scenario: what happens when our fractions have the exact same ? For example, if you have a pizza cut into 8 equal slices, and you are comparing

3/8

of the pizza to

5/8

of the pizza, which amount is larger?

Let's continue to comparing fractions with like denominators

Let's dive right into comparing fractions that share the exact same denominator. When the bottom numbers are identical, it means our wholes are sliced into pieces of the exact same size, making the comparison wonderfully straightforward.

Same Denominator? Just Count the Slices

Think of the as telling us the size of the slices, and the as telling us how many slices we actually have. If you and a friend both have pizzas cut into 8 equal slices, your slices are identical in size. If you have

3/8

of a pizza (3 slices) and your friend has

5/8

of a pizza (5 slices), your friend clearly has more pizza. Because the slices are the same size, we only need to look at the top numbers. The fraction with the larger numerator is always the larger fraction. For example, comparing these three fractions is just like counting whole numbers:

\frac{2}{7} < \frac{4}{7} < \frac{6}{7}

Three horizontal bar models stacked vertically, showing shaded segments for the fractions two-sevenths, four-sevenths, and six-sevenths.

As you can see in the diagram, when the divisions are identical, more shaded segments simply mean a larger total amount. Now, let's flip this concept on its head. What happens when the top numbers (the numerators) are the same, but the bottom numbers (the denominators) are different? For instance, how would you compare $1/3$ and $1/6$?