Fractions as Parts of a Set or Collection - Third Grade Mathematics

Fractions do not only describe single objects cut into slices; they also describe parts of a group or collection of separate items. Whether you are counting the fraction of red marbles in a bag, the fraction of cats in an animal shelter, or the fraction of days it rained this week, fractions as parts of a set allow you to analyze collections with clarity. In this chapter, you will master finding fractions of sets and collections.

Understanding a Set as the Whole

In a set model, the "whole" is not one continuous shape, but a collection of distinct, countable individual objects.

The Entire Set (The Whole): All items in the collection combined.
The Denominator:            The TOTAL number of items in the set.
The Numerator:              The number of items that share a specific trait.

Look at this collection of shapes:

Collection:
[ * ]   [ * ]   [ * ]   [ o ]   [ o ]

Total number of shapes = 5  (Denominator = 5)
Number of stars (*) = 3     (Numerator = 3)
Number of circles (o) = 2   (Numerator = 2)

Fraction of shapes that are stars:   3/5 (three-fifths)
Fraction of shapes that are circles: 2/5 (two-fifths)

Notice that 3/5 + 2/5 = 5/5, which represents the entire set!

Finding a Unit Fraction of a Set

Finding a unit fraction of a collection means splitting the collection into equal groups and finding how many items are in just one group. This is the exact same action as division!

Rule: To find 1/b of a total, divide the total by b!
1/b of N = N ÷ b

Let us find 1/3 of 12 apples:
1. Divide the 12 apples into 3 equal groups: 12 ÷ 3 = 4.

2. Look at 1 group: it has 4 apples.
Therefore: 1/3 of 12 = 4 apples!

Group 1: [ @ @ @ @ ]  <- 1/3 of the set (4 apples)
Group 2: [ @ @ @ @ ]  <- 1/3 of the set (4 apples)
Group 3: [ @ @ @ @ ]  <- 1/3 of the set (4 apples)
Total: 12 apples

Finding Non-Unit Fractions of a Set

What if you need to find 2/3 of 12 apples? Now that you know 1/3 of 12 is 4 apples, 2/3 simply means taking 2 of those groups!

Step 1: Find 1/3 of 12 -> 12 ÷ 3 = 4 (size of one group).
Step 2: Multiply by the numerator (2) -> 4 x 2 = 8 apples!
Therefore: 2/3 of 12 = 8 apples.

Two simple steps unlock any fraction of a set:
1. Divide by the denominator to find the value of one share.

2. Multiply by the numerator to count the shares you need!


Sets with Different Categories

Real-world collections often contain multiple subsets.

Consider a basket of 10 fruit:
- 4 are apples

- 3 are bananas

- 2 are oranges

- 1 is a pear


We can state:
- 4/10 of the fruit are apples.

- 3/10 are bananas.

- 2/10 are oranges.

- 1/10 is a pear.
The sum of all parts: 4/10 + 3/10 + 2/10 + 1/10 = 10/10 (the complete fruit basket).

Chapter Practice Exercises

  1. Look at this set of coins: [ Penny ] [ Penny ] [ Nickel ] [ Dime ] [ Penny ] [ Nickel ] a. How many total coins are in the set? b. What fraction of the coins are pennies? c. What fraction of the coins are nickels? d. What fraction of the coins are dimes?
  2. Find the fractional amount of each set: a. 1/2 of 16 counters b. 1/4 of 20 pencils c. 1/3 of 18 stickers d. 1/5 of 25 marbles
  3. Find the non-unit fraction: a. 3/4 of 20 pencils (Hint: find 1/4 first, then multiply by 3) b. 2/3 of 18 stickers
  4. Out of 14 students on a playground, 8 are wearing sneakers and 6 are wearing boots. What fraction of the students are wearing sneakers?
  5. There are 24 birds in a flock. One-fourth of the birds flew away. a. How many birds flew away? b. How many birds stayed on the tree? What fraction of the flock stayed?

Solutions and Step-by-Step Answers

  1. Coin collection: a. Total coins = 6. b. There are 3 pennies, so 3/6 of the coins are pennies. c. There are 2 nickels, so 2/6 of the coins are nickels. d. There is 1 dime, so 1/6 of the coins is a dime.
  2. Unit fractions of sets: a. 1/2 of 16 = 16 ÷ 2 = 8 counters. b. 1/4 of 20 = 20 ÷ 4 = 5 pencils. c. 1/3 of 18 = 18 ÷ 3 = 6 stickers. d. 1/5 of 25 = 25 ÷ 5 = 5 marbles.
  3. Non-unit fractions: a. 1/4 of 20 = 5. So 3/4 of 20 = 5 x 3 = 15 pencils. b. 1/3 of 18 = 6. So 2/3 of 18 = 6 x 2 = 12 stickers.
  4. 8 out of 14 students are wearing sneakers, which is 8/14 of the students.
  5. Flock of 24 birds: a. 1/4 of 24 = 24 ÷ 4 = 6 birds flew away. b. 24 - 6 = 18 birds stayed. The fraction that stayed is 3/4 (or 18/24).