Lesson 7What Fraction of a Group?

Learning Goal

Let’s think about dividing things into groups when we can’t even make one whole group.

Learning Targets

I can tell when a question is asking for the number of groups and that number is less than 1.
I can use diagrams and multiplication and division equations to represent and answer “what fraction of a group?” questions.

Warm Up: Estimating a Fraction of a Number

Problem 1

Estimate the quantities:

What is $\frac{1}{3}$ of 7?
What is $\frac{4}{5}$ of $9 \frac{2}{3}$ ?
What is $2 \frac{4}{7}$ of $10 \frac{1}{9}$ ?

Problem 2

Write a multiplication expression for each of the previous questions.

Activity 1: Fractions of Ropes

Problem 1

The segments in the applet represent 4 different lengths of rope. Compare one rope to another, moving the rope by dragging the open circle at one endpoint. You can use the yellow pins to mark off lengths.

open applet in presentation mode

Complete each sentence comparing the lengths of the ropes. Then, use the measurements shown on the grid to write a multiplication equation and a division equation for each comparison.

Rope B is times as long as rope A.
Rope C is times as long as rope A.
Rope D is times as long as rope A.

Print Version

Here is a diagram that shows four ropes of different lengths.

The lengths of 4 lines representing ropes on a grid are labeled A, B, C, and D. Rope A is 4 units. Rope B is 20 units. Rope C is 9 units. Rope D is 3 units.

Complete each sentence comparing the lengths of the ropes. Then, use the measurements shown on the grid to write a multiplication equation and a division equation for each comparison.

Rope B is times as long as rope A.
Rope C is times as long as rope A.
Rope D is times as long as rope A.

Problem 2

Each equation can be used to answer a question about Ropes C and D. What could each question be?

$? \cdot 3 = 9$ and $9 \div 3 = ?$
$? \cdot 9 = 3$ and $3 \div 9 = ?$

Activity 2: Fractional Batches of Ice Cream

One batch of an ice cream recipe uses 9 cups of milk. A chef makes different amounts of ice cream on different days. Here are the amounts of milk she used:

Monday: 12 cups
Tuesday: $22 \frac{1}{2}$ cups

Thursday: 6 cups
Friday: $7 \frac{1}{2}$ cups

Problem 1

How many batches of ice cream did she make on these days? For each day, write a division equation, draw a tape diagram, and find the answer.

Monday
Tuesday

Problem 2

What fraction of a batch of ice cream did she make on these days? For each day, write a division equation, draw a tape diagram, and find the answer.

Thursday
Friday

Problem 3

For each question, write a division equation, draw a tape diagram, and find the answer.

What fraction of 9 is 3?
What fraction of 5 is $\frac{1}{2}$ ?

Lesson Summary

It is natural to think about groups when we have more than one group, but we can also have a fraction of a group.

To find the amount in a fraction of a group, we can multiply the fraction by the amount in the whole group. If a bag of rice weighs 5 kg, $\frac{3}{4}$ of a bag would weigh ( $\frac{3}{4} \cdot 5)$ kg.

A tape diagram labeled 5 kg and 1 bag. It's made up of 4 segments. The first three segments are yellow and are labeled three-fourths of a bag and (three-fourths times 5) kg.

Sometimes we need to find what fraction of a group an amount is. Suppose a full bag of flour weighs 6 kg. A chef used 3 kg of flour. What fraction of a full bag was used? In other words, what fraction of 6 kg is 3 kg?

This question can be represented by a multiplication equation and a division equation, as well as by a diagram.

$? \cdot 6 = 3$ $3 \div 6 = ?$

A tape diagram of 6 equal parts. Above the diagram, a brace from the beginning of the diagram to the end of the diagram is labeled "6 kilograms."Below the diagram, a brace from the beginning of the diagram to the end of the diagram is labeled 1 bag. A third brace that contains the first three parts is labeled "three ." Below the diagram, a fourth brace which also contains the first three parts is labeled "question mark bag."

We can see from the diagram that 3 is $\frac{1}{2}$ of 6, and we can check this answer by multiplying: $\frac{1}{2} \cdot 6 = 3$ .

In any situation where we want to know what fraction one number is of another number, we can write a division equation to help us find the answer.

For example, “What fraction of 3 is $2 \frac{1}{4}$ ?” can be expressed as $? \cdot 3 = 2 \frac{1}{4}$ , which can also be written as $2 \frac{1}{4} \div 3 = ?$ .

The answer to “What is $2 \frac{1}{4} \div 3$ ?” is also the answer to the original question.

A tape diagram of 12 parts labeled 3 cups and 1 group. 9 parts are red and labeled "? group" and "two and one-fourth or nine-fourth cups".

The diagram shows that 3 wholes contain 12 fourths, and $2 \frac{1}{4}$ contains 9 fourths, so the answer to this question is $\frac{9}{12}$ , which is equivalent to $\frac{3}{4}$ .

We can use diagrams to help us solve other division problems that require finding a fraction of a group. For example, here is a diagram to help us answer the question: “What fraction of $\frac{9}{4}$ is $\frac{3}{2}$ ?,” which can be written as $\frac{3}{2} \div \frac{9}{4} = ?$ .

A tape diagram of 9 parts labeled "two and one-fourth or nine-fourths" and "1 group". 6 segments are yellow and labeled "? group" and "three-halves or six-fourths"

We can see that the quotient is $\frac{6}{9}$ , which is equivalent to $\frac{2}{3}$ . To check this, let’s multiply. $\frac{2}{3} \cdot \frac{9}{4} = \frac{18}{12}$ , and $\frac{18}{12}$ is, indeed, equal to $\frac{3}{2}$ .