# Lesson 25

Divide Decimals by Decimals

## Warm-up: Number Talk: Same/Different (10 minutes)

### Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have to divide whole numbers by decimals. These understandings help students develop fluency and will be helpful later in this lesson when students divide decimals greater than 1 by decimals less than 1.

### Launch

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

### Activity

• Keep problems and work displayed.
• Repeat with each problem.

### Student Facing

Find the value of each expression mentally.

• $$20 \div 2$$
• $$2 \div 0.2$$
• $$50 \div 2$$
• $$5 \div 0.2$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “Why do $$50 \div 2$$ and $$5 \div 0.2$$ have the same value?” (Because 5 is 50 tenths and I am dividing that into groups of 2 tenths so that's $$50 \div 2$$.)

## Activity 1: Dividing by a Tenth and a Hundredth (15 minutes)

### Narrative

The purpose of this activity is for students to divide decimal numbers by 0.1 and 0.01. They are given diagrams to help see that there are 10 tenths in each whole and 100 hundredths in each whole. The diagrams are not labeled with the whole so that the same diagram which shows $$1.6 \div 0.1 = 16$$ can be interpreted as whole number division showing $$160 \div 10 = 16$$. This dual way of interpreting one diagram is highlighted in the synthesis. When students interpret the diagram as representing two different equations they attend to precision in the meaning each part of the diagram (MP6).

MLR7 Compare and Connect. Synthesis: After all strategies have been presented, lead a discussion comparing, contrasting, and connecting the different approaches. Ask, “What kinds of additional details or language helped you understand the displays?”, “Were there any additional details or language that you have questions about?”, and “Did anyone solve the problem the same way, but would explain it differently?”

• Groups of 2

### Activity

• 5 minutes: independent work time
• 5 minutes: partner work time
• Monitor for students who:
• Describe how Jada's diagram shows the value of $$1.6 \div 0.1$$ as 16.
• Describe how Jada's diagram also represents the expression $$160 \div 10$$.

### Student Facing

1. To find the value of $$1.6 \div 0.1$$, Jada drew this diagram.
1. Describe how the diagram shows 1.6.

2. Describe how the diagram shows 16 groups of 1 tenth.
3. Describe how the diagram shows the value of $$1.6 \div 0.1$$.
4. Describe how the diagram also represents the expression $$160 \div 10$$.
2. Explain how this diagram represents $$1.3 \div 0.01$$.
1. What is the value of $$1.3 \div 0.01$$? Explain or show your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Ask selected students to share their reasoning for each problem.
• Display: $$1.6 \div 0.1 = 160 \div 10$$
• “How does the first diagram show that this equation is true?” (If each large square is a whole then the number of shaded strips is $$1.6 \div 0.1$$ and if each large square is 100 then the number of those strips is $$160 \div 10$$. The same diagram represents both expressions so they are equal.)
• Display: $$1.3 \div 0.01 =130 \div 1$$
• "How does the second diagram show that this equation is true?" (If each large square is a whole then the number of small pieces represents $$1.3 \div 0.01$$ and if each large square is 100 then the number of small pieces represents $$130 \div 1$$.)

## Activity 2: Divide Decimals by Decimals (20 minutes)

### Narrative

In this activity, students practice finding quotients of decimals divided by 0.1 and 0.01. Students find the value of different expressions without the scaffold of a diagram. Monitor for these approaches:

• diagrams
• whole number quotient facts
• multiples of the divisor
Representation: Internalize Comprehension. Synthesis: Invite students to identify which details were necessary to solve the problem. Display the sentence frame: “The next time I evaluate a division expression containing decimals, I will pay attention to . . . . “
Supports accessibility for: Conceptual Processing, Attention, Organization

### Required Materials

Materials to Copy

• Small Grids

• Groups of 2

### Activity

• 8 minutes: independent work time
• 2 minutes: partner discussion

### Student Facing

Find the value of each expression. Explain or show your reasoning.
1. $$5 \div 0.1$$
2. $$5 \div 0.01$$
3. $$0.5 \div 0.1$$
4. $$0.5 \div 0.01$$
5. $$0.02 \div 0.01$$
6. $$1.53 \div 0.01$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Display:

$$0.5 \div 0.1 = 5$$

$$0.50 \div 0.01 = 50$$

• “How can we use the meaning of decimal place values to explain these equations?” (5 tenths is the same as 50 hundredths so that’s 5 groups of 0.1 or 50 groups of 0.01.)
• “How can we use the meaning of decimal place values to help find the value of $$1.53 \div 0.01$$?” (The three is in the hundredths place so there are 3 one hundredths in three hundredths. The 5 is in the tenths place and there are 10 hundredths in each tenth so that's 50 more hundredths. There are 100 hundredths in one whole. That's 153 hundredths altogether in 1.53.)

## Lesson Synthesis

### Lesson Synthesis

“Today we divided a decimal by a decimal and then found lots of quotients involving decimals.”

Display:

$$1.25 \div 0.01 = 125$$

“How do we know this equation is true?” (If we multiply the dividend and the divisor by 100, we get $$1.25 \div 0.01 = 125 \div 1$$, which is 125. We can also see that there are 100 hundredths in 1, and 25 hundredths in 0.25, so there are 125 hundredths in 1.25.)

“How is dividing with decimals the same as dividing with whole numbers? How is it different?” (I can use multiplication in both cases. I can draw a diagram in both cases. I use place value in both cases. With decimals I need to think carefully about the meaning of each digit. I think the diagrams are more helpful to get started with decimals to visualize the numbers I am working with.)

## Cool-down: Divide by Decimals (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Section Summary

### Student Facing

In this section we learned to divide with decimals. We studied different ways to find a quotient like $$3 \div 0.1$$. We can draw a diagram which shows that there are 10 groups of 0.1 in each whole so there are $$3 \times 10$$ or 30 groups of 0.1 in 3 wholes: $$3 \div 0.1 = 30$$.

We can also think about place value. We know 3 is 30 tenths and 0.1 is 1 tenth, so $$3 \div 0.1$$ is equivalent to $$30 \div 1$$ which has the value 30. We also can use multiplication to find the value of $$3 \div 0.1$$. We know that $$10 \times 0.1 = 1$$ and $$30 \times 0.1 = 3$$ so this also shows that $$3 \div 0.1 = 30$$.