In plain words
What does the bottom number of a fraction mean?
The bottom number, the denominator, tells you how many equal parts make one whole. In 1/5, the whole is cut into 5 equal parts, and the 5 counts every part, shaded or not. So one shaded part out of five is 1/5.
- The big idea
- The bottom number tells you how many equal parts make one whole, so it counts every part, whether it’s shaded or not.
- The common confusion
- The bottom number counts the shaded pieces, or the pieces left over, so one shaded piece out of five gets written 5/1 or 1/4. It often sounds like this: “One piece is shaded and four aren’t, so that’s (one fourth).”
Why it works
The bottom number of a fraction is called the denominator. It tells you how many equal parts make one whole.
A circle cut into 5 equal parts with 1 part shaded shows 1/5. The 5 counts all the equal parts in the circle, shaded or not, and the 1 counts the shaded part.
The denominator isn’t the number of shaded parts, and it isn’t the number of parts left over either. One shaded part and four unshaded parts make 1/5 of the circle, not 1/4.
The denominator also names the size of the parts. Cut the same circle into 8 equal parts and each part is an eighth, smaller than a fifth. Cut it into 3 and each part is a third, bigger than a fifth.
On a number line, the whole is the distance from 0 to 1. The denominator tells you how many equal parts that distance is cut into, so 1/7 is the end of the first of 7 equal parts.
A worked example
A chocolate bar is snapped into 6 equal pieces and you eat 1, so you ate 1/6 of the bar. The 6 counts every piece the bar was made of, including the 5 you didn’t eat.
Why this confusion is so common
It’s natural to count what you can see shaded, or compare the shaded parts with the unshaded ones: 1 shaded and 4 not, so “1/4”. That comparison is a real idea, a ratio of 1 to 4, but it isn’t the fraction of the whole. A fraction needs two different counts: all the equal parts in the whole (the bottom number) and the parts we’re talking about (the top number).
Try this away from the screen
Take three identical paper strips and fold one into 2 equal parts, one into 4 and one into 8. Shade one part of each strip and label it 1/2, 1/4 and 1/8. Then ask what the bottom number counts, and why the shaded part gets smaller as the bottom number gets bigger.
Helpful things to say
| Instead of | Try |
|---|---|
| “The top is the shaded ones and the bottom is the rest.” | “How many equal parts make the whole? That’s the bottom number.” |
| “Just count the pieces.” | “Which pieces does each number count: all of them, or only the shaded ones?” |
Practice questions
These questions check the same idea with new numbers and new pictures. The best way to use them is one at a time, with pictures you can change and a reply to every answer. If you’d rather read them here, each answer stays hidden until you open it, so whoever is practicing can try first.
Try these questions one at a timeQuestion 1
What fraction of the bar is shaded?
The choices are (one sixth), (one fifth) and (6 over 1).
Show the answer
The answer is (one sixth). The bar is cut into 6 equal parts and 1 is shaded, so the shaded part is (one sixth). The bottom number counts all 6 parts.
Question 2
Drag the point to (one seventh) on this number line.
Show the answer
To answer it, place the point at (one seventh). On a number line, the whole is the distance from 0 to 1. Here it’s cut into 7 equal parts, so each part is (one seventh), and (one seventh) is where the first part after 0 ends.
Question 3
What fraction of the square is shaded?
The choices are (one fourth), (one third) and (4 over 1).
Show the answer
The answer is (one fourth). The two diagonals cut the square into 4 equal triangles. One of them is shaded, so the shaded part is (one fourth) of the square.
Question 4
A cake is cut into 9 equal slices. What fraction of the cake is one slice?
The choices are (one eighth), (one ninth) and (9 over 1).
Show the answer
The answer is (one ninth). The whole cake is 9 equal slices, so the bottom number is 9, and one slice is (one ninth) of the cake.
Questions people ask
What does the denominator of a fraction tell you?
The denominator (the bottom number) tells you how many equal parts make one whole. In 3/8, the whole is cut into 8 equal parts, so each part is one eighth.
Why is 1 shaded part out of 5 written 1/5 and not 1/4?
Because the bottom number counts all the equal parts in the whole, including the shaded one. There are 5 equal parts, so each one is 1/5. Comparing 1 shaded part with 4 unshaded parts gives a ratio, 1 to 4, which is a different idea.
Can the denominator be 1?
Yes. A denominator of 1 means the whole isn’t cut at all, so 5/1 means 5 wholes, which is 5. That’s why 5/1 and 1/5 are very different numbers.
Why does a bigger denominator mean smaller parts?
When the same whole is cut into more equal parts, each part has to be smaller. One eighth of a pizza is less than one fifth of the same pizza.
Words to know
- Denominator
- The denominator is the bottom number of a fraction, and it tells you how many equal parts make one whole.
- Equal parts
- Equal parts are exactly the same size as each other. A fraction only makes sense when the whole has been split into parts like that.
- Unit fraction
- A unit fraction has 1 on top, like 1/2, 1/3 or 1/8, so it’s exactly one of the equal parts of a whole.
- Number line
- A number line is a line where every point stands for a number, and fractions have their own places between the whole numbers.
Curriculum connections
We don’t assume any particular curriculum, but for teachers in the United States, here’s how this lesson lines up with the Common Core State Standards:
- 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.
- 3.NF.A.2.a
Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line.
Related lessons
The Top Number
What does the top number of a fraction mean?
The top number counts the parts you have, so 3/5 means 3 of the 5 equal parts. · About 3 minutes
The Bigger Denominator Trap
Is (one sixth) bigger than (one eighth)?
Cutting the same whole into more pieces makes each piece smaller, which is why 1/8 is less than 1/6. · About 3 minutes
Equal Means Equal
Do the parts of a fraction have to be equal?
Four pieces only make fourths when they’re all the same size, though they don’t have to be the same shape. · About 3 minutes
If you’d rather read plain text, there’s a markdown version of this lesson.