Investigation 2: Investigating Fractions with Manipulatives

New Concept

Improper fractions are fractions that are equal to or greater than $1$. In a fraction equal to $1$, the numerator equals the denominator (as in $\frac{5}{5}$). In a fraction greater than $1$, the numerator is greater than the denominator (as in $\frac{7}{5}$).

What’s next

This is your introduction to fractions representing more than one whole. Next, you'll use visual models to convert improper fractions into mixed numbers.

Property

Use comparison symbols to show the relationship between fractions:

• < means is less than
• = means is equal to
• > means is greater than

Examples

Using fraction manipulatives, we can see a $\frac{1}{2}$ piece is larger than a $\frac{1}{3}$ piece, so $\frac{1}{2} > \frac{1}{3}$.
Comparing three $\frac{1}{6}$ pieces to three $\frac{1}{8}$ pieces shows that $\frac{3}{6}$ is larger, so $\frac{3}{6} > \frac{3}{8}$.
Drawing two identical rectangles shows that $\frac{2}{3}$ of the area is larger than $\frac{3}{5}$ of the area, so $\frac{2}{3} > \frac{3}{5}$.

Explanation

Think of it like a pizza party! Would you rather have $\frac{1}{2}$ of a pizza or $\frac{1}{3}$? The fraction pieces show you which slice is bigger, so you know which fraction is greater. The bigger the denominator, the more slices the pizza is cut into, making each individual slice smaller. It’s a tasty way to visualize math!

Property

Improper fractions are fractions that are equal to or greater than 1. In a fraction equal to 1, the numerator equals the denominator. In a fraction greater than 1, the numerator is greater than the denominator.

Examples

The fraction $\frac{3}{3}$ is an improper fraction because the numerator equals the denominator, meaning it is equal to 1.
The fraction $\frac{4}{3}$ is an improper fraction because the numerator is greater than the denominator, meaning it is greater than 1.
If you have 7 slices from a pizza cut into 6, you have $\frac{7}{6}$ of a pizza, which is more than one whole pizza.

Explanation

Imagine you're super hungry and eat more than one whole pizza! An improper fraction is just a way to say you have at least one whole thing, plus some extra slices. The top number (numerator) gets bigger than the bottom number (denominator) because you have more pieces than what makes up a single whole item.

Property

An improper fraction can be converted to a mixed number, which consists of a whole number and a proper fraction combined.

Examples

The improper fraction $\frac{5}{4}$ is shown as five $\frac{1}{4}$ pieces, which form one whole circle and one $\frac{1}{4}$ piece, so $\frac{5}{4} = 1 \frac{1}{4}$.
To represent $\frac{3}{2}$, you can use three $\frac{1}{2}$ pieces. Two pieces form a whole circle, leaving one piece, so $\frac{3}{2} = 1 \frac{1}{2}$.
The improper fraction $\frac{7}{6}$ means you have seven $\frac{1}{6}$ pieces. Six make a whole circle, leaving one extra piece, so $\frac{7}{6} = 1 \frac{1}{6}$.

Explanation

An improper fraction like $\frac{5}{4}$ is like having five slices from a pizza cut into four. You can group four of those slices to make one whole pizza, with one slice left over. So, $\frac{5}{4}$ is the same as having 1 whole pizza and $\frac{1}{4}$ of another. It is just two different ways to describe the same delicious amount!