Early Math · Grades 1, 2
Equal Shares: Halves, Thirds and Fourths
Quick answer
Equal shares are parts of the same size. Two equal shares are halves, three are thirds, and four are fourths, also called quarters. The whole is two halves, three thirds or four fourths. Cutting the same whole into more shares makes each share smaller, so a fourth of a pizza is less than a half of it. Equal shares of the same whole do not need to be the same shape: a square cut corner to corner or straight across still makes two halves.
What you'll learn
- Cut circles and rectangles into two, three and four equal shares
- Name shares as halves, thirds, fourths and quarters, and the whole as two halves, three thirds or four fourths
- Explain why more equal shares are smaller, and why equal shares can have different shapes
Fair shares
Two friends share a sandwich. To be fair, each one should get the same amount. So cut it into equal shares: parts that are the same size.
The rectangle is cut into two equal shares. The circle is cut into two parts, but one is much bigger. Those are not equal shares.
Halves, thirds and fourths
We name equal shares by how many there are.
- equal shares are halves. Each one is a half of the whole.
- equal shares are thirds. Each one is a third of the whole.
- equal shares are fourths, or quarters. Each one is a fourth of the whole.
Put all the shares back, and you get the whole. The whole is two halves, three thirds or four fourths.
Why more shares are smaller
Think of one pizza. Share it with friend, and you each get a half. Share it with friends, and you each get a fourth.
It is the same pizza. When more people share it, each share has to be smaller. The more equal shares you cut, the smaller each share is.
So a fourth of a pizza is smaller than a half of the same pizza.
Equal shares can look different
Here are three squares, all the same size. Each one is cut into fourths.
The shares have different shapes: strips, small squares and triangles. But in each square, the shares are the same size. So each one is a fourth.
Equal shares of the same whole must be the same size, not the same shape.
Worked examples
Common mistakes
Practice problems
-
Is this circle cut into halves?
Is it cut into halves? Answer
No
Full solution
The two parts are not the same size. Halves must be equal shares.
-
What part of this circle is shaded?
What part is shaded? Answer
A third
Full solution
There are equal shares and is shaded. That is a third of the circle.
-
How many fourths make a whole?
Answer
Full solution
A whole cut into fourths has equal shares. All make the whole.
-
Which is smaller, a half of a cake or a third of the same cake?
Answer
A third
Full solution
A third comes from cutting shares, and a half from cutting . More shares are smaller shares.
-
How many equal shares is this rectangle cut into? What is each share called?
How many shares? Answer
shares; each one is a half.
Full solution
The line goes across the middle, making equal shares. Two equal shares are halves.
-
What part of the circle is shaded?
What part is shaded? Answer
Three fourths
Full solution
There are equal shares and are shaded. That is three fourths, or three quarters.
-
How many thirds make a whole?
Answer
Full solution
The whole is three thirds.
-
Two squares are the same size. One is cut into strips and one into small squares. Is each share a fourth in both?
Answer
Yes
Full solution
Each square has equal shares. The shapes differ, but the shares are the same size, so each is a fourth.
-
Name a way to cut a rectangle into two halves.
Answer
Down the middle, across the middle, or corner to corner.
Full solution
Any cut through the middle that makes shares of the same size gives halves.
-
Leo cuts an apple into pieces, one big and one small. He says he has two halves. What went wrong?
Hint
What must be true of halves?
Answer
Halves must be the same size. His pieces are not.
Full solution
A half is one of equal shares. A big piece and a small piece are shares, but not equal ones.
So they are not halves.
Frequently asked questions
What are equal shares?
Parts of a whole that are all the same size. Cutting a sandwich down the middle makes two equal shares.
What is a half?
One of two equal shares of a whole. Two halves make the whole.
Is a fourth the same as a quarter?
Yes. Both mean one of four equal shares of a whole.
Which is bigger, a half or a fourth?
A half of the same whole. Cutting into more shares makes each share smaller, so a fourth is smaller than a half.
Do equal shares have to be the same shape?
No. A square cut into four thin strips and one cut into four small squares both make fourths, because each share is the same size.
Standards alignment
This lesson covers the following Common Core State Standards for Mathematics.
- CCSS.MATH.CONTENT.1.G.A.3GeometryPartition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.
- CCSS.MATH.CONTENT.2.G.A.3GeometryPartition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.