# Lesson 6

Relate Division and Multiplication

## Warm-up: Number Talk: Multiply and Divide (10 minutes)

### Narrative

The purpose of this Number Talk is for students to demonstrate strategies and understandings they have for multiplying a fraction by a whole number. These understandings help students develop fluency and will be helpful later in the lesson when students explore the relationship between multiplication and division.

### Launch

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$3 \times \frac{1}{2}$$
• $$3 \times \frac{2}{2}$$
• $$3 \times \frac{3}{2}$$
• $$5 \times \frac{3}{2}$$

### Activity Synthesis

• Display equation: $$\frac{15}{2} = 7 \frac{1}{2}$$
• “How do we know this is true?” (They are both equal to $$15 \div 2$$ or 7 is $$\frac{14}{2}$$ so $$7 \frac{1}{2}$$ is $$\frac{15}{2}$$).

## Activity 1: The Race (20 minutes)

### Narrative

The purpose of this activity is for students to apply what they learned in earlier lessons to represent a division situation. During the synthesis, students connect what they know about division to multiplication when they see that the situation of 2 people equally sharing the distance in a 3 mile race can also be described as each person running one-half of the three mile race. Students connect the language they used to describe the situation to multiplication and division expressions. This relationship between multiplication and division is the focus of the next several lessons.

### Launch

• Display the image from the student workbook.
• “This image shows two children who are running in a relay race. They are on the same team. They each have to run the same distance. The boy holding the baton is finishing his turn. When he hands the baton to his teammate, his teammate will take her turn.”

### Activity

• Groups of 2
• 2 minutes: independent think time
• 5–8 minutes: partner work time

MLR2 Collect and Display

• Circulate, listen for and collect the language students use to describe how to represent the situation and how far each person ran.
• Listen for:
• 3 miles divided into 2 parts
• half of 3 miles
• one and one half miles
• three halves
• Record students’ words and phrases on a visual display and update it throughout the lesson.

### Student Facing

1. Lin and Han ran a 3 mile relay race as a team. They each ran the same distance. Draw a diagram to represent the situation.
2. Take turns describing to your partner how your diagrams represent the situation.
3. How far did each person run?

### Activity Synthesis

• “These are the words and phrases that you used to describe the situation. Are there any other words or phrases that are important to include on our display?”
• As students share responses, update the display, by adding (or replacing) language, diagrams, or annotations.
• Display: $$3 \div 2$$, $$1\frac{1}{2}$$, $$\frac{3}{2}$$, and $$3\times \frac{1}{2}$$
• “How does each of these expressions represent the situation?” (3 miles was divided into 2 equal parts, each person ran $$1\frac{1}{2}$$ miles or $$\frac{3}{2}$$ miles which can be thought of as $$3\times \frac{1}{2}$$.
• Consider giving students time to record their answers in their journal.
• Display: “one half of 3 miles”
• “This is another way to describe how far each person ran.”
• Display: $$\frac{1}{2}\times3$$
• “We can also use this multiplication expression to represent one-half of 3.”
• “In the next activity, you will describe how different diagrams represent each of these expressions.”

## Activity 2: Where Do You See It? (15 minutes)

### Narrative

The purpose of this activity is for students to build on the previous activity to interpret diagrams as representing different expressions. Both diagrams in the activity represent solutions to the running situation in the previous activity. In this activity, the diagrams are interpreted without the running context. As students use the diagrams to interpret expressions, they begin to see relationships between the expressions $$\frac{3}{2}$$, $$1\frac{1}{2}$$$$3 \times \frac{1}{2}$$, $$\frac{1}{2}\times3$$, and $$3 \div 2$$. In this activity, students make sense of different ways to interpret a given diagram and relate the operations of division and multiplication (MP7).

MLR1 Stronger and Clearer Each Time. Synthesis: Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their response to “How is each expression represented in the diagrams?” Invite listeners to ask questions, to press for details and to suggest mathematical language. Give students 2–3 minutes to revise their written explanation based on the feedback they receive.

• Groups of 2

### Activity

• 5 minutes: independent work time
• 5–8 minutes: partner discussion
• Monitor for students who choose different diagrams to represent the same expression.

### Student Facing

For each expression, choose one of the diagrams and describe how the diagram represents the expression. Be prepared to explain why you chose that diagram.

1. $$3 \times \frac{1}{2}$$
2. $$3 \div 2$$
3. $$\frac{1}{2} \times 3$$

### Student Response

If students do not explain where they see each expression in one of the diagrams, refer to the expressions and ask: “How are these expressions the same? How are they different?” Then, refer to one of the numbers in one of the expressions and ask students to describe how the diagrams represent the number.

### Activity Synthesis

• “What is the same about the diagrams?” (They both show 3 rectangles that are divided into 2 equal pieces.)
• “What is different about the diagrams?” (The shaded pieces are next to each other in one of the diagrams.)
• Ask previously selected students to share their explanations.
• Display expression: $$3 \div 2$$
• “Which diagram did you choose for the expression $$3 \div 2$$? Why?” (I chose diagram A because I see the 3 miles and the 2 equal parts. I chose diagram B because I can see the 2 equal parts that make 3 miles and can see that they make $$1 \frac{1}{2}$$ miles.)
• Display expressions: $$3 \times \frac{1}{2}$$, $$3 \div 2$$, $$\frac{1}{2} \times 3$$
• “How do we know all the expressions are equal?” (Because the same diagram represents all of them or because they all equal $$\frac{3}{2}$$ or $$1\frac{1}{2}$$.)

## Lesson Synthesis

### Lesson Synthesis

Record answers to the questions below for all to see and save responses to revisit during the synthesis of an upcoming lesson.

“When did we use multiplication today?” (In the first activity, we found one half of three miles and we can show that with $$\frac{1}{2}\times3$$. The diagrams from the second activity represent $$3 \times \frac{1}{2}$$ and $$\frac{1}{2}\times3$$.)

“When did we use division today?” (During the first activity, we divided 3 miles into 2 equal parts to figure out how far each person ran. The diagram from the second activity represents $$3 \div 2$$, and $$\frac {3}{2}$$.)

“What did we learn about the relationship between multiplication and division?” (We can use both of them to solve and represent the same problem.)

“What do you still wonder about the relationship between multiplication and division?” (Can we always use multiplication or division to solve a division problem?)