Introduction to Fractions

Every fraction has two roles built into it, and understanding these roles is what makes fractions make sense.

The numerator (top number) tells you how many parts you have. The denominator (bottom number) tells you how many equal parts make up the whole.

For example, in $\frac{3}{5}$, the denominator 5 tells us the whole is split into 5 equal parts. The numerator 3 tells us we are taking 3 of those parts.

Here’s a useful comparison:

• $\frac{3}{5}$ → 3 parts out of 5 total
• $\frac{5}{3}$ → 5 parts out of 3 total (this is more than a whole!)

Students often mix these up, thinking the bigger number always means a bigger fraction. But $\frac{3}{5}$ is less than 1, while $\frac{5}{3}$ is greater than 1—even though 5 is larger in both cases.

Mental model: The denominator sets the “size of each slice,” and the numerator counts how many slices you have. Changing the denominator changes the size of each piece.

Fractions become much clearer when you can see them. Drawing them forces you to think about equal parts, which is the foundation of fractions.

To visualize $\frac{2}{3}$, divide a shape into 3 equal parts (this matches the denominator), then shade 2 of them (this matches the numerator).

Now compare:

• $\frac{2}{3}$ → divide into 3 parts, shade 2
• $\frac{2}{5}$ → divide into 5 parts, shade 2

Even though the numerator is the same, the fractions are different because the size of each piece changed.

A common mistake is drawing unequal sections or forgetting that the denominator controls how many total parts there are.

Mental model: Drawing a fraction is like following instructions: “Cut into this many equal pieces, then shade this many.” If either step is wrong, the fraction is wrong.

Equivalent fractions look different but represent the same amount. This works because you are changing the number of pieces while keeping the total amount the same.

For example, $\frac{1}{2}$ means half of something. If you cut that same whole into 4 pieces instead of 2, each piece is smaller—but you now need 2 pieces to represent the same amount: $\frac{2}{4}$.

Compare:

• $\frac{1}{2}$ → 1 out of 2 large pieces
• $\frac{2}{4}$ → 2 out of 4 smaller pieces

Same total, different-sized pieces.

A common mistake is multiplying only the numerator or only the denominator. That changes the value. You must multiply (or divide) both by the same number to keep the fraction equivalent.

Mental model: Equivalent fractions are like resizing a picture without changing its shape—you’re changing the scale, not the amount.

Whole numbers and fractions are not separate ideas—they are connected. Any whole number can be written as a fraction.

For example, 3 means “3 wholes.” But each whole can be thought of as 1 out of 1 equal part. So 3 wholes is $\frac{3}{1}$.

This is useful because it lets you combine whole numbers and fractions using the same system.

Compare:

• $3 = \frac{3}{1}$ (3 wholes)
• $\frac{3}{2}$ (1 and a half)

Students often think fractions are always smaller than 1, but that’s not true. If the numerator is larger than the denominator, the fraction is greater than 1.

Mental model: A whole number is just a fraction where each whole is counted as one complete piece.

Fractions are everywhere because real life involves sharing, measuring, and comparing.

If a cake is cut into 8 equal slices and you eat 3, you’ve eaten $\frac{3}{8}$. In cooking, recipes might call for $\frac{1}{2}$ cup or $\frac{3}{4}$ teaspoon. In construction, measurements often include fractions of units.

Fractions also help you compare: is $\frac{3}{4}$ of a pizza more than $\frac{2}{3}$? These questions show up constantly in real decisions.

A common mistake is ignoring what the “whole” is. $\frac{1}{2}$ of a small pizza is not the same as $\frac{1}{2}$ of a large pizza.

Mental model: A fraction is always tied to a specific whole. Change the whole, and the meaning of the fraction changes.