Lesson 6Using Diagrams to Find the Number of Groups

Learning Goal

Let’s draw tape diagrams to think about division with fractions.

Learning Targets

I can use a tape diagram to represent equal-sized groups and find the number of groups.

Warm Up: How Many of These in That?

Problem 1

We can think of the division expression $10 \div 2 \frac{1}{2}$ as the question: “How many groups of $2 \frac{1}{2}$ are in 10?” Complete the tape diagram to represent this question. Then find the answer.

A rectangle made up of 10 squares in a grid and is labeled as 10 like a tape diagram.

Problem 2

Complete the tape diagram to represent the question: “How many groups of 2 are in 7?” Then find the answer.

A rectangle made up of 7 squares in a grid and is labeled as 7 like a tape diagram.

Activity 1: Representing Groups of Fractions with Tape Diagrams

Problem 1

To make sense of the question “How many $\frac{2}{3}$ s are in 1?,” Andre wrote equations and drew a tape diagram.

In an earlier task, we used pattern blocks to help us solve the equation $1 \div \frac{2}{3} = ?$ . Explain how Andre’s tape diagram can also help us solve the equation.

$? \cdot \frac{2}{3} = 1$

$1 \div \frac{2}{3} = ?$

A tape diagram with three equal parts. The first two parts are shaded and are each labeled one third. Above the tape diagram is a bracket labeled 1, and contains all three parts. Below the diagram there is a bracket labeled "1 group of two thirds," and contains the first two parts.

Print Version

To make sense of the question “How many $\frac{2}{3}$ s are in 1?,” Andre wrote equations and drew a tape diagram.

In an earlier task, we used pattern blocks to help us solve the equation $1 \div \frac{2}{3} = ?$ . Explain how Andre’s tape diagram can also help us solve the equation.

$? \cdot \frac{2}{3} = 1$

$1 \div \frac{2}{3} = ?$

Problem 2

Write a multiplication equation and a division equation for each question. Then, draw a tape diagram and find the answer.

How many $\frac{3}{4}$ s are in 1?
How many $\frac{2}{3}$ s are in 3?
How many $\frac{3}{2}$ s are in 5?

Activity 2: Finding Number of Groups

Problem 1

Write a multiplication equation or a division equation for each question. Then, find the answer and explain or show your reasoning

How many $\frac{3}{8}$ -inch thick books make a stack that is 6 inches tall?
How many groups of $\frac{1}{2}$ pound are in $2 \frac{3}{4}$ pounds?

Problem 2

Write a question that can be represented by the division equation $5 \div 1 \frac{1}{2} = ?$ . Then, find the answer and explain or show your reasoning.

Lesson Summary

A baker used 2 kilograms of flour to make several batches of a pastry recipe. The recipe called for $\frac{2}{5}$ kilogram of flour per batch. How many batches did she make?

We can think of the question as: “How many groups of $\frac{2}{5}$ kilogram make 2 kilograms?” and represent that question with the equations:

$\begin{matrix} ? \cdot \frac{2}{5} = 2 \\ 2 \div \frac{2}{5} = ? \end{matrix}$ To help us make sense of the question, we can draw a tape diagram. This diagram shows 2 whole kilograms, with each kilogram partitioned into fifths.

A tape diagram with 10 segments. The first two segments are labeled one-fifth and together as one batch. The diagram is labeled as 2 kg and "? batches" is labeled on the bottom.

We can see there are 5 groups of $\frac{2}{5}$ in 2. Multiplying 5 and $\frac{2}{5}$ allows us to check this answer: $5 \cdot \frac{2}{5} = \frac{10}{5}$ and $\frac{10}{5} = 2$ , so the answer is correct.

Notice the number of groups that result from $2 \div \frac{2}{5}$ is a whole number. Sometimes the number of groups we find from dividing may not be a whole number. Here is an example:

Suppose one serving of rice is $\frac{3}{4}$ cup. How many servings are there in $3 \frac{1}{2}$ cups?

A tape diagram with 2 sets of 3 blue segments, 2 sets of 3 green segments, 1 set of 2 orange segments, and 1 set of 2 white segments. Each segment is one-fourth.

$\begin{matrix} ? \cdot \frac{3}{4} = 3 \frac{1}{2} \\ 3 \frac{1}{2} \div \frac{3}{4} = ? \end{matrix}$ Looking at the diagram, we can see there are 4 full groups of $\frac{3}{4}$ , plus 2 fourths. If 3 fourths make a whole group, then 2 fourths make $\frac{2}{3}$ of a group. So the number of servings (the “?” in each equation) is $4 \frac{2}{3}$ . We can check this by multiplying $4 \frac{2}{3}$ and $\frac{3}{4}$ .

$4 \frac{2}{3} \cdot \frac{3}{4} = \frac{14}{3} \cdot \frac{3}{4}$ , and $\frac{14}{3} \cdot \frac{3}{4} = \frac{14}{4}$ , which is indeed equivalent to $3 \frac{1}{2}$ .