In plain words
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.
- The big idea
- When you cut the same whole into more equal pieces, each piece gets smaller.
- The 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 (one eighth) is bigger.”
Why it works
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 so 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 things to say
| Instead of | Try |
|---|---|
| “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
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
Which is bigger: (one fifth) or (one ninth)?
Show the answer
The answer is (one fifth). 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.
Question 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”.
Show the answer
The answer is “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 (one third) bigger than (one tenth).
Question 3
Which is bigger: (one fourth) or (one twelfth)?
Show the answer
The answer is (one fourth). Twelve equal pieces of the same bar are much smaller than four equal pieces, so (one fourth) is bigger than (one twelfth).
Question 4
Which of these is the smallest?
The choices are (one half), (one fifth) and (one ninth).
Show the answer
The answer is (one ninth). The circle cut into the most pieces has the smallest pieces, so (one ninth) 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.
Words to know
- Denominator
- The denominator is the bottom number of a fraction, and it tells you how many equal parts make one whole.
- 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
- 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.
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.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.
Related lessons
Same Top, Compare the Pieces
How do you compare fractions with the same numerator?
Both 3/5 and 3/8 have 3 pieces, but fifths are bigger than eighths, so 3/5 is the bigger fraction. · About 3 minutes
The Bottom Number
What does the bottom number of a fraction mean?
The bottom number counts all the equal parts that make up the whole, whether they’re shaded or not. · About 3 minutes
The Bigger Numerator Trap
Which is bigger, (three eighths) or (two thirds)?
Even though 3/8 has the bigger top number, 2/3 is the bigger fraction, because thirds are much bigger pieces than eighths. · About 3 minutes
If you’d rather read plain text, there’s a markdown version of this lesson.