# Fractions of Collections

> **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.

- 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.
- 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: “1/4 of 12 counters is 1 counter. It says 1.”
- Family: [Parts & Wholes](https://slices.org/explore/parts-and-wholes) (slice 5 of 20)
- [Try the lesson yourself](https://slices.org/slices/fractions-of-collections): about 3 minutes, free, with no sign-in.

## In plain words

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 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 ways to talk about it

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

[Try these questions one at a time](https://slices.org/set?t=Practice%3A+Fractions+of+Collections&q=s05-transfer-1%2Cs05-transfer-2%2Cs05-review-1%2Cs05-review-2), with pictures you can change and a reply to every answer.

1. Select 2/3 of these 9 counters. **Answer:** select 6 of the 9 counters. Share the 9 counters into 3 equal groups, which puts 3 in each. 2/3 means 2 of those groups, so that’s 3 + 3 = 6 counters.
2. A class has 20 pencils, and 3/4 of them are sharp. How many pencils are sharp? **Answer:** 15. Share the 20 pencils into 4 equal groups, which puts 5 in each. 3/4 means 3 of those groups, so that’s 5 + 5 + 5 = 15 pencils.
3. This bar stands for 15 stickers, split into 3 equal parts. How many stickers is 1/3 of them? **Answer:** 5. The 15 stickers are shared equally among the 3 parts of the bar, so each part stands for 5 stickers. That makes 1/3 of 15 equal to 5.
4. This pizza is cut into 8 equal slices. How many slices make 1/4 of the pizza? The choices are “1 slice”, “2 slices” and “4 slices”. **Answer:** 2 slices. Put the 8 slices into 4 equal groups and you get 2 slices in each. So 2 slices make 1/4 of the pizza, and one slice on its own is 1/8.

## 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.

## 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.
- [5.NF.B.4.a](https://www.thecorestandards.org/Math/Content/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).)

## Words to know

- [Fraction of a set](https://slices.org/glossary/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](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.
- [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.
- [Numerator](https://slices.org/glossary/numerator): The numerator is the top number of a fraction, and it counts how many of the equal parts you have.

## Related slices

- [What’s the Whole?](https://slices.org/slices/whats-the-whole): Half of a big pizza is more than half of a small one, because a fraction is always a fraction of something.
- [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.
- [Multiplication Can Make Smaller](https://slices.org/slices/multiplication-can-make-smaller): Multiplying by 1/2 means taking half of something, so 1/2 × 1/2 is half of a half, which is 1/4.