# The Bottom Number

> **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.

- 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.
- 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 1/4.”
- Family: [Parts & Wholes](https://slices.org/explore/parts-and-wholes) (slice 3 of 20)
- [Try the lesson yourself](https://slices.org/slices/the-bottom-number): about 3 minutes, free, with no sign-in.

## In plain words

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 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 ways to talk about it

| Instead of saying | Try saying |
| --- | --- |
| “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

[Try these questions one at a time](https://slices.org/set?t=Practice%3A+The+Bottom+Number&q=s03-transfer-1%2Cs03-transfer-2%2Cs03-review-1%2Cs03-review-2), with pictures you can change and a reply to every answer.

1. What fraction of the bar is shaded? The choices are 1/6, 1/5 and 6/1. **Answer:** 1/6. The bar is cut into 6 equal parts and 1 is shaded, so the shaded part is 1/6. The bottom number counts all 6 parts.
2. Drag the point to 1/7 on this number line. **Answer:** place the point at 1/7. 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 1/7, and 1/7 is where the first part after 0 ends.
3. What fraction of the square is shaded? The choices are 1/4, 1/3 and 4/1. **Answer:** 1/4. The two diagonals cut the square into 4 equal triangles. One of them is shaded, so the shaded part is 1/4 of the square.
4. A cake is cut into 9 equal slices. What fraction of the cake is one slice? The choices are 1/8, 1/9 and 9/1. **Answer:** 1/9. The whole cake is 9 equal slices, so the bottom number is 9, and one slice is 1/9 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.

## Curriculum connections (US Common Core)

- [3.NF.A.1](https://www.thecorestandards.org/Math/Content/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](https://www.thecorestandards.org/Math/Content/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.

## Words to know

- [Denominator](https://slices.org/glossary/denominator): The denominator is the bottom number of a fraction, and it tells you how many equal parts make one whole.
- [Equal parts](https://slices.org/glossary/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](https://slices.org/glossary/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](https://slices.org/glossary/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.

## Related slices

- [The Top Number](https://slices.org/slices/the-top-number): The top number counts the parts you have, so 3/5 means 3 of the 5 equal parts.
- [The Bigger Denominator Trap](https://slices.org/slices/bigger-denominator-trap): Cutting the same whole into more pieces makes each piece smaller, which is why 1/8 is less than 1/6.
- [Equal Means Equal](https://slices.org/slices/equal-means-equal): Four pieces only make fourths when they’re all the same size, though they don’t have to be the same shape.