# Lesson 20

Products in the Hundredths Place

## Warm-up: What do you know about $1\times 0.1$ and $0.1 \times 0.1$? (10 minutes)

### Narrative

The purpose of this What Do You Know About ____? is for students to think about $$0.1 \times 0.1$$ before working with this kind of expression more formally in the lesson. Students know that 0.1 is the same as $$\frac{1}{10}$$ and they know how to find products of fractions. The goal of the synthesis is to highlight this before students find products of decimals in the lesson.

### Launch

• Display the expressions.
• “What do you know about $$1 \times 0.1$$ and $$0.1 \times 0.1$$?”
• 1 minute: quiet think time

### Activity

• Record responses.
• “How could we represent these expressions?” (I could use a hundredths grid or area diagram.)

### Student Facing

What do you know about these expressions?

• $$1 \times 0.1$$
• $$0.1 \times 0.1$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “Can you find the value of $$0.1 \times 0.1$$?” (Yes, 0.1 is $$\frac{1}{10}$$ so that’s $$\frac{1}{10} \times \frac{1}{10}$$ and I know that’s $$\frac{1}{100}$$.)

## Activity 1: Products of Tenths (15 minutes)

### Narrative

The purpose of this activity is for students to find products of a number of tenths and a number of tenths written as decimals. Students can think of find these products in many ways including

• using a diagram
• using whole number arithmetic and place value reasoning or properties of operations (MP7)
The goal of the synthesis is to relate these different ways of finding the product.

MLR8 Discussion Supports. Synthesis: During group work, invite students to take turns sharing their responses. Ask students to restate what they heard using precise mathematical language and their own words. Display the sentence frame: “I heard you say . . .” Original speakers can agree or clarify for their partner.
Representation: Internalize Comprehension. Provide students with a graphic organizer, such as a two-column table, to record the multiplication expression using fractions and the corresponding multiplication expression using decimal numbers to show the connection between fractions and decimal numbers and why $$0.1 \times 0.1 = 0.01$$.
Supports accessibility for: Conceptual Processing, Memory

### Required Materials

Materials to Copy

• Small Grids

• Groups of 2

### Activity

• 1–2 minutes: quiet think time
• 6–8 minutes: partner work time
• Monitor for students who:
• use grids
• use whole number facts and place value reasoning

### Student Facing

1. Find the value of each expression. Explain or show your reasoning. Use the grids if they are helpful.

1. $$2 \times 0.3$$

2. $$0.2 \times 0.3$$

2. Kiran says $$0.2 \times 0.4 = 0.8$$. Do you agree with Kiran? Explain or show your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Invite students to share their reasoning for $$0.2 \times 0.4$$.
• Display a student generated diagram of $$0.2 \times 0.4$$ or the diagram from the student solution.
• “How does the diagram show $$0.2 \times 0.4$$?” (There is 2 tenths of 4 tenths of the rectangle shaded.)
• “How did you know that the shaded region has area 0.08 square units?” (There are $$2 \times 4$$ shaded pieces and each one is $$\frac{1}{100}$$ of the full square.)
• Display equation $$0.2 \times 0.4 = 2 \times 4 \times (0.1 \times 0.1)$$.
• “How does the diagram show this equation?” (The shaded part is 2 tenths of 4 tenths of the rectangle so that's $$0.2 \times 0.4$$. It’s $$2 \times 4 \times (0.1 \times 0.1)$$ because there are $$2 \times 4$$ pieces and each one has area $$0.1 \times 0.1$$ or one hundredth of a square unit.)

## Activity 2: Multiply Tenths (20 minutes)

### Narrative

The purpose of this activity is for students to multiply decimals by decimals, building on the strategies they saw in the previous activity. Monitor for these strategies:
• using a diagram
• using whole number products and place value understanding
• using expressions to show their thinking

### Required Materials

Materials to Copy

• Small Grids

### Launch

• Groups of 2
• Make copies of hundredths grid blackline master available.

### Activity

• 5 minutes: independent work time
• 2 minutes: partner discussion
• Monitor for students who:
• use the grids
• multiply two whole numbers and then multiply their product by $$0.01$$

### Student Facing

1. Find the value of each expression. Explain or show your reasoning.

1. $$1.8 \times 0.4$$
2. $$2.5 \times 0.6$$
3. $$3.8 \times 0.7$$
2. How are these products the same? How are they different?
• $$74 \times 6$$
• $$7.4 \times 6$$
• $$7.4 \times 0.6$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Invite students to share their responses and reasoning for the product $$1.8 \times 0.4$$.
• Display student generated diagram or diagram in student solution.
• “How does the diagram show $$1.8 \times 0.4$$ ?” (There is a full group of 0.4 and then there is 8 tenths of another group of 0.4.)
• “How does the diagram show $$18 \times 4 \times 0.01$$?” (There is an 18 by 4 array of pieces and each piece is a hundredth of the whole.)
• Display: $$1.8 \times 0.4 = (18 \times 4) \times 0.01$$
• Invite students to share their responses about the products $$74 \times 6$$, $$7.4 \times 6$$ and $$7.4 \times 0.6$$.
• “How can you use the whole number product to find decimal products?” (I just think about how many tenths or hundredths I have.)

## Lesson Synthesis

### Lesson Synthesis

“Today we found products of decimals using diagrams and thinking about place value.”

Display:
$$4.5 \times 8.1 = 45 \times 0.1 \times 81 \times 0.1$$

“How do we know this is true?” ($$4.5 = 45 \times 0.1$$ and $$8.1 = 81 \times 0.1$$ so $$4.5 \times 8.1 = 45 \times 0.1 \times 81 \times 0.1$$)

Display:
$$4.5 \times 8.1 = 45 \times 81 \times 0.01$$

“How do we know this is true?” (If we change the order of factors in the expression $$45 \times 0.1 \times 81 \times 0.1$$, we get $$45 \times 81 \times 0.1 \times 0.1$$ and that is equal to $$45 \times 81 \times 0.01$$.)

“How is this helpful for finding the value of $$4.5 \times 8.1$$?” (I can just find the whole number product and then say I have that many hundredths.)

## Cool-down: Tenths (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.