In plain words
Do the parts of a fraction have to be equal?
Yes. A fraction like 1/4 means one of 4 equal parts of a whole, so 4 pieces of different sizes are not fourths. The parts must be the same size, but they don’t have to be the same shape.
- The big idea
- Fractions need equal shares of one whole. Four parts are fourths only when all four are the same size, even if their shapes differ.
- The common confusion
- Any four pieces are fourths, even when the pieces are different sizes. It often sounds like this: “I cut it into four pieces, so each piece is a fourth.”
Why it works
A fraction describes equal shares of a whole. Fourths are what you get when one whole is split into 4 parts that are all the same size.
Counting pieces is not enough. A rectangle cut into 4 strips of different widths has 4 parts, but they are not fourths, because some strips are bigger than others.
To check, compare the sizes. When all 4 parts are equal, each one is 1/4 of the whole: exactly 4 of them fill it, with nothing left over.
Equal parts do not have to be the same shape. Cut the left half of a square into two small squares and the right half into two triangles: all four parts are fourths, because each half was cut into 2 equal parts.
The same rule holds for every fraction. Thirds are 3 equal parts of a whole, fifths are 5 equal parts, and so on.
A worked example
Three friends share a pizza cut into 3 slices of different sizes. Everyone gets one slice, but not everyone gets the same amount, so the slices are not thirds. Cut the pizza into 3 equal slices and each friend gets exactly 1/3.
Why this confusion is so common
Children usually meet fractions by counting: shade 1 of 4 pieces and call it one fourth. Because most pictures in books are already cut into equal pieces, the word “equal” quietly drops out of view. Everyday language doesn’t help either: people talk about “the bigger half”. Asking “are the parts the same size?” every time brings the missing condition back.
Try this away from the screen
Take two identical paper squares. Fold one in half and in half again the other way to make 4 small squares. Fold the other along both diagonals to make 4 triangles. Cut out one small square and one triangle. They look different, but cut the small square along its diagonal and the two halves fit exactly onto the triangle: the same amount of paper, so both are fourths.
Helpful things to say
| Instead of | Try |
|---|---|
| “There are 4 pieces, so each one is a fourth.” | “Are all 4 parts the same size? How could we check?” |
| “Fourths always look like this.” | “These parts are different shapes. Do they cover the same amount of the whole?” |
Practice questions
For grown-ups helping out: the answers are tucked away so learners can try first.
Which shape is cut into fourths?
Show the answer
Answer: Shape B. The two diagonals cut the square into 4 matching triangles, so each one is 1/4. The other shapes have 4 parts, but not 4 equal parts.
Three friends share this pizza, one slice each. Does each friend get 1/3 of the pizza?
Show the answer
Answer: No. Thirds are 3 equal parts of one whole. These slices are different sizes, so the friends don’t all get the same amount, and they don’t all get 1/3.
These 4 parts are not all the same shape. Is each part 1/4 of the square?
Show the answer
Answer: Yes. Each half of the square is cut into 2 equal parts: two small squares on one side, two triangles on the other. All 4 parts are the same size, so each is 1/4.
Which strip of paper is cut into thirds?
Show the answer
Answer: Strip B. Thirds are 3 equal parts. Only one strip has 3 pieces of the same width.
Questions people ask
Do equal parts have to be the same shape?
No. Equal parts must be the same size (the same area), not the same shape. A square cut into two small squares and two triangles is still cut into fourths, as long as each part covers exactly one fourth of the square.
How can I tell whether parts are equal?
Compare them directly: fold one part onto another, cut them out and stack them, or count the squares on grid paper. Slices of a circle that meet in the middle are equal when the angles where they meet are equal. Looking alone can mislead, so it is worth checking.
Can one piece be 1/4 if the other pieces are different sizes?
Yes. A piece is one fourth of the whole when exactly 4 pieces of its size would fill the whole, however the rest is cut. But the pieces only count as fourths together when every one of them is that size.
When do children learn about equal parts?
In the US Common Core standards, children split circles and rectangles into equal shares in grade 2, and in grade 3 they describe each equal part as a unit fraction such as 1/4. Every later fraction idea builds on it.
Words to know
- Equal parts
- Parts that are exactly the same size. A fraction only makes sense when the whole is split into equal parts.
- Whole
- The one complete thing, length or group that a fraction is a part of.
- Fraction
- A number that describes equal parts of a whole. Every fraction also has its own point on the number line.
Curriculum connections
We don’t assume any particular curriculum. For teachers in the United States, here’s how this slice lines up with the Common Core State Standards:
- 3.G.A.2
Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole. For example, partition a shape into 4 parts with equal area, and describe the area of each part as 1/4 of the area of the shape.
- 2.G.A.3
Partition 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.
- 3.NF.A.1
Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.
Related slices
The Bottom Number
What does the bottom number of a fraction mean?
What the bottom number of a fraction counts: all the equal parts that make one whole. · About 3 minutes
What’s the Whole?
Is half of a large pizza the same amount as half of a small pizza?
Why half of a big pizza is more than half of a small one: a fraction depends on its whole. · About 3 minutes
The Top Number
What does the top number of a fraction mean?
What the top number of a fraction counts: how many of the equal parts you have. · About 3 minutes
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