In plain words
Can a fraction be bigger than 1?
Yes. A fraction is bigger than 1 whenever its top number is bigger than its bottom number. 6/4 means six fourths, and since four fourths make one whole, the other two make half of another, so 6/4 is 1 1/2.
- The big idea
- A fraction can be bigger than one whole. Four fourths fill one whole, so six fourths is a whole and two fourths more.
- The common confusion
- Fractions are always less than one, so a fraction such as 6/4 cannot be a real number. It often sounds like this: “(six fourths) can’t be real. There are only 4 fourths in a pizza.”
Why it works
A fraction counts equal parts of one whole. In 6/4, the 4 says the whole is cut into 4 equal parts (fourths), and the 6 says we have six of those parts.
Four fourths fill exactly one whole, so 4/4 = 1. Nothing stops us from counting on, though, and the next two fourths start to fill a second whole.
So 6/4 is one whole and two fourths more. Two fourths is the same as one half, so 6/4 = 1 2/4, which is 1 1/2.
On a number line, make fourth-size hops from 0. The fourth hop lands on 1, and the sixth hop lands halfway between 1 and 2, because fractions keep going past 1 just as whole numbers do.
A fraction is less than 1 when its top number is smaller than its bottom number, equal to 1 when the two match, and more than 1 when the top number is bigger.
A worked example
A recipe needs 6/4 cups of flour, and your scoop holds 1/4 cup, so you fill it six times. The first four scoops make one full cup and the last two make half a cup more, so 6/4 cups is 1 1/2 cups.
Why this confusion is so common
Most of us first meet fractions as parts of one shape, such as one pizza or one bar, and you cannot shade more than all of a single shape. From that, it is reasonable to conclude that fractions are always less than one. The name “improper fraction” does not help, because it makes these numbers sound like mistakes. Building fractions from unit pieces, and placing them on a number line, shows that fractions keep going past 1.
Try this away from the screen
Cut a few paper circles into fourths and ask the learner to make 6/4 using only fourth pieces. When the first circle is full, where do the extra pieces go? Then try 5/3 with thirds and 8/4 with fourths, and ask what they notice about 8/4.
Helpful things to say
| Instead of | Try |
|---|---|
| “The top number can’t be bigger than the bottom number.” | “What happens if we keep counting fourths after the whole is full?” |
| “6/4 is improper, so we need to fix it.” | “6/4 is a perfectly good number. Can you also say it in wholes and fourths?” |
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
Place (seven fourths) on the number line.
Show the answer
To answer it, place the point at (seven fourths). (seven fourths) is seven fourths. Four of them take you to 1, and the other three take you three fourths of the way from 1 to 2.
Question 2
If each bar is one whole, how much do these (one third) pieces make altogether?
The choices are (five thirds), (five sixths) and (two thirds).
Show the answer
The answer is (five thirds). Three thirds fill the first whole, and two more thirds sit in the second. That makes five thirds, or (five thirds), which is one whole and (two thirds) more.
Question 3
Which of these is the largest?
The choices are (seven eighths), (three thirds) and (five fourths).
Show the answer
The answer is (five fourths). (seven eighths) is less than one whole, and (three thirds) is exactly one whole. (five fourths) is one whole and one more fourth, so it’s the largest.
Question 4
Two pizzas are each cut into 6 equal slices, and Mia’s family eats 8 slices. How much pizza is that?
The choices are “(eight sixths) of a pizza”, “(eight twelfths) of a pizza” and “(six eighths) of a pizza”.
Show the answer
The answer is “(eight sixths) of a pizza”. Each slice is (one sixth) of a pizza, so 8 slices are (eight sixths) of a pizza. That’s one whole pizza and (two sixths) more.
Questions people ask
Is 6/4 a real fraction?
Yes. 6/4 means six fourths, and four of them make one whole while the other two make half of another, so 6/4 is 1 1/2. It sits halfway between 1 and 2 on the number line.
What is an improper fraction?
It’s a fraction whose top number is equal to or bigger than its bottom number, such as 5/5 or 7/4, so it’s worth one whole or more. Despite the name, there’s nothing improper about it.
How do you turn 7/4 into a mixed number?
See how many groups of four fourths fit into seven fourths. There’s one group, which makes one whole, with three fourths left over, so 7/4 = 1 3/4.
Is 4/4 a fraction or a whole number?
It’s both, because 4/4 names the same number as 1, just as 8/4 names the same number as 2. Any fraction whose top and bottom numbers match, such as 3/3 or 10/10, is equal to one whole.
Words to know
- Improper fraction
- An improper fraction has a top number as big as its bottom number or bigger, like 6/4 or 5/5, so it’s worth one whole or more.
- Mixed number
- A mixed number is a whole number and a fraction written together, like 1 2/4, which means one whole and two fourths more.
- 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.
- 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.
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.c
Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram.
- 4.NF.B.3
Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
- 3.NF.A.2.b
Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
Related lessons
What’s the Whole?
Is half of a large pizza the same amount as half of a small pizza?
Half of a big pizza is more than half of a small one, because a fraction is always a fraction of something. · About 3 minutes
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
Counting Spaces, Not Ticks
How do you find (three fourths) on a number line?
On a number line you count the spaces between the marks, starting from 0, rather than the marks themselves. · About 3 minutes
One Number, Not Two
Where is (two fifths) on a number line?
The fraction 2/5 is a single number, and it sits two fifths of the way from 0 to 1. · About 3 minutes
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