Lesson 20
Products in the Hundredths Place
Warmup: What do you know about $1\times 0.1$ and $0.1 \times 0.1$? (10 minutes)
Narrative
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
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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)
Advances: Listening, Speaking
Supports accessibility for: Conceptual Processing, Memory
Required Materials
Materials to Copy
 Small Grids
Launch
 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

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

\(2 \times 0.3\)

\(0.2 \times 0.3\)

 Kiran says \(0.2 \times 0.4 = 0.8\). Do you agree with Kiran? Explain or show your reasoning.
Student Response
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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
 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

Find the value of each expression. Explain or show your reasoning.
 \(1.8 \times 0.4\)
 \(2.5 \times 0.6\)
 \(3.8 \times 0.7\)
 How are these products the same? How are they different?
 \(74 \times 6\)
 \(7.4 \times 6\)
 \(7.4 \times 0.6\)
Student Response
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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.)
Cooldown: Tenths (5 minutes)
CoolDown
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