Math Core

When you share a pan of brownies, everyone wants a fair piece. In math, fair pieces are parts with equal area. Each equal part is a unit fraction of the whole shape.

Equal parts have equal area

To divide a shape into equal parts, every part must cover the same amount of space. Remember, the amount of space a shape covers is its area.

This rectangle is cut into 3 equal parts.

A rectangle cut into 3 equal parts. Each part is 1/3 of the whole.

There are 3 equal parts, so each part is $\dfrac{1}{3}$ of the rectangle.

Equal parts and unit fractions

When a shape is divided into $b$ equal parts, each part is $\dfrac{1}{b}$ of the whole.

2 equal parts: each is $\dfrac{1}{2}$ .
4 equal parts: each is $\dfrac{1}{4}$ .
6 equal parts: each is $\dfrac{1}{6}$ .

Many shapes, many ways

You can divide all kinds of shapes. This hexagon is cut into 6 equal triangles. Each triangle is $\dfrac{1}{6}$ of the hexagon.

A hexagon cut into 6 equal parts. Each part is 1/6 of the hexagon.

This square is cut from corner to corner both ways. That makes 4 equal triangles, so each is $\dfrac{1}{4}$ of the square.

A square cut into 4 equal triangles. Each part is 1/4 of the square.

Equal parts don't have to look the same

Parts are equal when they have the same area. They don't have to be the same shape! This rectangle has 8 small squares. It is split into a red part and a yellow part.


🟥	🟥	🟥	🟨
🟥	🟨	🟨	🟨

The red part has 4 squares and the yellow part has 4 squares. The parts have different shapes, but the same area. So each part is $\dfrac{1}{2}$ of the rectangle.

Using area to find the size of each part

If you know the area of the whole shape, you can find the area of each equal part. Divide the whole area by the number of parts.

Worked example: Name the fraction

A rectangle is cut into 4 equal parts. What fraction of the rectangle is each part?

There are 4 equal parts, so each part is $\dfrac{1}{4}$ of the rectangle.

Worked example: Area of each part

A rectangle has an area of 12 square centimeters. It is cut into 4 equal parts. What is the area of each part? What fraction of the whole is each part?

Share the area equally: $12 \div 4 = 3$ .

Each part has an area of 3 square centimeters and is $\dfrac{1}{4}$ of the rectangle.

Worked example: How many parts?

A garden has an area of 20 square meters. It is split into equal beds. Each bed has an area of 5 square meters. How many beds are there, and what fraction of the garden is each bed?

How many 5s make 20? $20 \div 5 = 4$ .

There are 4 beds, so each bed is $\dfrac{1}{4}$ of the garden.

Common mistake

Counting parts is not enough. A shape cut into 3 parts shows thirds only if all 3 parts have the same area. If one part is bigger, the parts are not thirds.

Practice

Practice 1

A shape is cut into 6 equal parts. What fraction of the shape is each part?

Type a number.

Practice 2

A round pizza is cut into 8 equal slices. What fraction of the pizza is one slice?

Type a number.

Practice 3

Which shape is divided into fourths?

Practice 4

A rectangle has an area of 20 square inches. It is cut into 5 equal parts. What is the area of each part, in square inches?

Type a number.

Practice 5

A rectangle of 8 squares is split into a red part and a yellow part.


🟥	🟥	🟨	🟨
🟥	🟥	🟥	🟨

Is each part $\dfrac{1}{2}$ of the rectangle?

Practice 6

A garden is 3 meters wide and 4 meters long. It is divided into 6 equal plots. What is the area of each plot, in square meters?

Type a number.

Practice 7

A rectangle has an area of 24 square units. It is cut into equal parts, and each part has an area of 6 square units. What fraction of the rectangle is each part?

Type a number.