# The Bigger Denominator Trap

> **Is 1/6 bigger than 1/8?** Yes. 1/6 is bigger than 1/8. When the same whole is cut into 8 equal pieces instead of 6, each piece has to be smaller, so one eighth is less than one sixth.

- Big idea: When you cut the same whole into more equal pieces, each piece gets smaller.
- Common confusion: A larger denominator means a larger piece, so 1/8 must be bigger than 1/6.
- It often sounds like this: “Eight is more than six, so 1/8 is bigger.”
- Family: [Fractions Have Size](https://slices.org/explore/fraction-size) (slice 6 of 20)
- [Try the lesson yourself](https://slices.org/slices/bigger-denominator-trap): about 3 minutes, free, with no sign-in.

## In plain words

Start with two wholes that are exactly the same size, such as two identical bars or two identical pizzas.

Cut one into 6 equal pieces and the other into 8. With 8 pieces, the same whole is shared among more pieces, so each eighth is smaller than each sixth.

So 1/6 is bigger than 1/8. The bigger bottom number gives the smaller piece, as long as the wholes are the same size.

The same reasoning works for every fraction with 1 on top: 1/2 is bigger than 1/3, which is bigger than 1/4, and so on. The more pieces you cut a whole into, the smaller each one gets.

## A worked example

Share one pizza among 6 friends and an identical pizza among 8 friends. Each of the 8 friends gets a smaller slice, so 1/8 of the pizza is less than 1/6 of it.

## Why this confusion is common

We all spend years learning that 8 is bigger than 6, and for whole numbers that is always true. Fractions are the first numbers where the bigger digit can mean the smaller amount, because the denominator describes the size of the parts. Researchers call this whole-number bias, and it is one of the most common sources of fraction confusion.

## Try this away from the screen

Fold one strip of paper into halves, a second into fourths and a third into eighths. Line the strips up and compare one part from each, then ask what happens to the size of a part each time you fold again.

## Helpful ways to talk about it

| Instead of saying | Try saying |
| --- | --- |
| “Remember: the bigger the bottom number, the smaller the fraction.” | “If we cut the same whole into more pieces, what happens to each piece?” |
| “Eight is bigger than six, so it is the other way around.” | “Would you rather have one slice when a pizza is shared by 6 people, or one slice when it is shared by 8?” |

## Practice questions

[Try these questions one at a time](https://slices.org/set?t=Practice%3A+The+Bigger+Denominator+Trap&q=s06-transfer-1%2Cs06-transfer-2%2Cs06-review-1%2Cs06-review-2), with pictures you can change and a reply to every answer.

1. Which is bigger: 1/5 or 1/9? **Answer:** 1/5. Splitting the space from 0 to 1 into 9 equal parts makes smaller spaces than splitting it into 5, so one fifth reaches further from 0.
2. Two pizzas are the same size, but one is cut into 3 equal slices and the other into 10. Which slice is bigger? The choices are “A slice of the pizza cut into 3”, “A slice of the pizza cut into 10” and “The slices are the same size”. **Answer:** A slice of the pizza cut into 3. The pizzas are the same size, so the one cut into fewer slices has bigger slices. That makes 1/3 bigger than 1/10.
3. Which is bigger: 1/4 or 1/12? **Answer:** 1/4. Twelve equal pieces of the same bar are much smaller than four equal pieces, so 1/4 is bigger than 1/12.
4. Which of these is the smallest? The choices are 1/2, 1/5 and 1/9. **Answer:** 1/9. The circle cut into the most pieces has the smallest pieces, so 1/9 is the smallest of the three.

## Questions people ask

### Why is 1/8 smaller than 1/6 if 8 is bigger than 6?

The 8 tells you how many equal pieces the whole is cut into. More pieces from the same whole means each piece is smaller, so one eighth is less than one sixth.

### Is a bigger denominator always a smaller fraction?

No, only when the numerators are the same and the wholes are the same size. 1/8 is less than 1/6, but 7/8 is more than 1/6, because 7/8 is made of many more pieces.

### How can I explain why 1/8 is smaller than 1/6?

Use something you can share, like a pizza, a chocolate bar or a strip of paper. Share one among more people and ask what happens to each share. Then let the picture do the convincing, so the person you’re helping doesn’t have to take a rule on trust.

## Curriculum connections (US Common Core)

- [3.NF.A.3.d](https://www.thecorestandards.org/Math/Content/3/NF/A/3/d/): Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

## 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.
- [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.
- [Whole](https://slices.org/glossary/whole): The whole is the complete thing that a fraction is part of. It might be a pizza, a length on a number line or a group of objects.

## Related slices

- [Same Top, Compare the Pieces](https://slices.org/slices/same-top-compare-the-pieces): Both 3/5 and 3/8 have 3 pieces, but fifths are bigger than eighths, so 3/5 is the bigger fraction.
- [The Bottom Number](https://slices.org/slices/the-bottom-number): The bottom number counts all the equal parts that make up the whole, whether they’re shaded or not.
- [The Bigger Numerator Trap](https://slices.org/slices/bigger-numerator-trap): Even though 3/8 has the bigger top number, 2/3 is the bigger fraction, because thirds are much bigger pieces than eighths.