# Lesson 6

Using Diagrams to Find the Number of Groups

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

### 6.1: How Many of These in That?

1. We can think of the division expression $$10 \div 2\frac12$$ as the question: “How many groups of $$2\frac 12$$ are in 10?” Complete the tape diagram to represent this question. Then find the answer. ### 6.2: Representing Groups of Fractions with Tape Diagrams

To make sense of the question “How many $$\frac 23$$s are in 1?,” Andre wrote equations and drew a tape diagram.

$$\displaystyle {?} \boldcdot \frac 23 = 1$$

$$\displaystyle 1 \div \frac 23 = {?}$$

1. In an earlier task, we used pattern blocks to help us solve the equation $$1 \div \frac 23 = {?}$$. Explain how Andre’s tape diagram can also help us solve the equation.

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

1. How many $$\frac 34$$s are in 1? 2. How many $$\frac23$$s are in 3? 3. How many $$\frac32$$s are in 5? ### 6.3: Finding Number of Groups

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

1. How many $$\frac38$$-inch thick books make a stack that is 6 inches tall?

2. How many groups of $$\frac12$$ pound are in $$2\frac 34$$ pounds?

2. Write a question that can be represented by the division equation $$5 \div 1\frac12 = {?}$$. Then, find the answer and explain or show your reasoning.

### Summary

A baker used 2 kilograms of flour to make several batches of a pastry recipe. The recipe called for $$\frac25$$ kilogram of flour per batch. How many batches did she make?

We can think of the question as: “How many groups of $$\frac25$$ kilogram make 2 kilograms?” and represent that question with the equations:

$$\displaystyle {?} \boldcdot \frac25=2$$

$$\displaystyle 2 \div \frac25 = {?}$$

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.

We can see there are 5 groups of $$\frac 25$$ in 2. Multiplying 5 and $$\frac25$$ allows us to check this answer: $$5 \boldcdot \frac 25 = \frac{10}{5}$$ and $$\frac {10}{5} = 2$$, so the answer is correct.

Notice the number of groups that result from $$2 \div \frac25$$ 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 $$\frac34$$ cup. How many servings are there in $$3\frac12$$ cups?

$$\displaystyle {?}\boldcdot \frac34 = 3\frac12$$

$$\displaystyle 3\frac12 \div \frac34 = {?}$$

Looking at the diagram, we can see there are 4 full groups of $$\frac 34$$, plus 2 fourths. If 3 fourths make a whole group, then 2 fourths make $$\frac 23$$ of a group. So the number of servings (the “?” in each equation) is $$4\frac23$$. We can check this by multiplying $$4\frac23$$ and $$\frac34$$.
$$4\frac23 \boldcdot \frac34 = \frac{14}{3} \boldcdot \frac34$$, and $$\frac{14}{3} \boldcdot \frac34 = \frac{14}{4}$$, which is indeed equivalent to $$3\frac12$$.