In plain words
What is 1/4 of 12?
1/4 of 12 is 3. If you share the 12 into 4 equal groups, each group has 3, and one of those groups is one fourth of the whole collection.
- The big idea
- A whole can be a group of things, so 1/4 of 12 counters is one of 4 equal groups, which is 3 counters.
- The common confusion
- Fractions only describe pieces of one object, so 1/4 of 12 counters must be 1 counter, or 4 counters. It often sounds like this: “(one fourth) of 12 counters is 1 counter. It says 1.”
Why it works
A fraction can describe part of a group as well as part of one object. When the whole is a group, the equal parts are equal groups.
To find 1/4 of 12 counters, share the 12 counters into 4 equal groups. Each group has 3 counters, so 1/4 of 12 is 3.
The bottom number tells you how many equal groups to make, and the top number tells you how many of those groups to take. For 2/3 of 9, make 3 equal groups of 3 and take 2 of them, which is 6.
Two common slips are thinking that 1/4 of 12 is 1 counter, or 4 counters. One counter is 1/12 of the group, and 4 counters are 4/12, which is 1/3.
Finding a fraction of a number is also multiplying: 1/4 of 12 is 1/4 × 12 = 3. Sharing into equal groups shows why the answer is smaller than 12.
A worked example
Four friends share 12 cookies equally. Each friend gets 3 cookies, so 3 cookies is 1/4 of the cookies. Three of the friends together have 3/4 of the cookies, which is 9.
Why this confusion is so common
Most early fraction pictures show one object cut into pieces, so many people come away thinking a fraction is always a piece of one thing. With a group, “one fourth” sounds as if it should be one object, or four. The shift is to see the whole group as the whole, and equal groups as the equal parts.
Try this away from the screen
Count out 12 buttons, coins or pieces of pasta. Ask the learner to find 1/4 of them by dealing them into 4 equal piles, one at a time, like dealing cards. Then find 1/3 and 1/2 of the same 12, and ask why 1/3 of 12 is more than 1/4 of 12.
Helpful things to say
| Instead of | Try |
|---|---|
| “Divide by the bottom, times by the top.” | “How many equal groups do we need? How many of those groups do we take?” |
| “A fourth is one piece.” | “A fourth of what? If the whole is 12 counters, what does one fourth look like?” |
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
Select (two thirds) of these 9 counters.
Show the answer
To answer it, select 6 of the 9 counters. Share the 9 counters into 3 equal groups, which puts 3 in each. (two thirds) means 2 of those groups, so that’s 3 + 3 = 6 counters.
Question 2
A class has 20 pencils, and (three fourths) of them are sharp. How many pencils are sharp?
Show the answer
The answer is 15. Share the 20 pencils into 4 equal groups, which puts 5 in each. (three fourths) means 3 of those groups, so that’s 5 + 5 + 5 = 15 pencils.
Question 3
This bar stands for 15 stickers, split into 3 equal parts. How many stickers is (one third) of them?
Show the answer
The answer is 5. The 15 stickers are shared equally among the 3 parts of the bar, so each part stands for 5 stickers. That makes (one third) of 15 equal to 5.
Question 4
This pizza is cut into 8 equal slices. How many slices make (one fourth) of the pizza?
The choices are “1 slice”, “2 slices” and “4 slices”.
Show the answer
The answer is “2 slices”. Put the 8 slices into 4 equal groups and you get 2 slices in each. So 2 slices make (one fourth) of the pizza, and one slice on its own is (one eighth).
Questions people ask
How do you find a fraction of a number?
Share the number into as many equal groups as the bottom number says, then take as many groups as the top number says. For 2/3 of 9, sharing 9 into 3 equal groups puts 3 in each, and 2 of those groups make 6.
Why is 1/4 of 12 not 1 or 4?
1/4 means one of 4 equal parts of the whole. When the whole is 12 counters, the equal parts are 4 groups of 3, so 1/4 of 12 is 3. One counter is 1/12 of the group, and 4 counters are 4/12, which is 1/3.
Is “of” the same as multiplying?
Yes. 1/4 of 12 means 1/4 × 12, which is 3. Thinking in equal groups shows why multiplying by a fraction less than 1 gives an answer smaller than the number you started with.
What if the group doesn’t share evenly?
Then the answer is not a whole number of objects. 1/4 of 10 is 10/4, or 2 and a half, so you would need to cut one object in half. That is why first examples use numbers that share evenly.
Words to know
- Fraction of a set
- A fraction of a set is a fraction of a group of objects, and you find it by sharing the group out into equal groups.
- 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.
- Denominator
- The denominator is the bottom number of a fraction, and it tells you how many equal parts make one whole.
- Numerator
- The numerator is the top number of a fraction, and it counts how many of the equal parts you have.
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.
- 5.NF.B.4.a
Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = (ac)/(bd).)
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
Multiplication Can Make Smaller
Why does multiplying by a fraction make a number smaller?
Multiplying by 1/2 means taking half of something, so 1/2 × 1/2 is half of a half, which is 1/4. · About 4 minutes
If you’d rather read plain text, there’s a markdown version of this lesson.