Lesson 21

Multiply More Decimals

Warm-up: Estimation Exploration: Central Park (10 minutes)

Narrative

The purpose of this estimation exploration is for students to apply what they know about area and multiplication of decimals to a situation where the side length of the rectangle are decimals. Students will approximate the length and width to obtain a product of decimals which they can find mentally. This prepares them for work in the lesson where they will find more complex products of decimals since making an estimate is a good way to check work. 

Launch

  • Groups of 2
  • Display the image.
  • “What is an estimate that’s too high?” “Too low?” “About right?”
  • 1 minute: quiet think time

Activity

  • “Discuss your thinking with your partner.”
  • 1 minute: partner discussion
  • Record responses.

Student Facing

Central Park is a large park in Manhattan. It is about 3.85 kilometers long and 0.79 km wide. What is the area of Central Park?

Record an estimate that is:

Image of map of Central Park.
too low about right too high
\(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\)

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

  • Invite students to share their estimates.
  • “How do you know the area is greater than 2 square kilometers?” (I know that \(3 \times 0.7\) is 21 tenths or 2.1 and it’s more than that.)
  • “How do you know the area is less than 3.2 square kilometers?” (I know 3.85 is less than 4 and 0.79 is less than 0.8. Then \(4 \times 0.8\) is 32 tenths or 3.2.)

Activity 1: Multiply More Decimals (20 minutes)

Narrative

In previous lessons, students have found products of a whole number and a decimal and products of two decimals. They used diagrams, place value reasoning, and expressions to explain their reasoning. The purpose of this activity is for students to find both kinds of products with larger numbers. For each product, students show that an expression using whole number products is equal to the given decimal product. Then they calculate the decimal product. Students may use a strategy, other than the given equivalent expressions, to make the calculations. For example, they might decompose the numbers by place value and use the distributive property (partial products). All of these methods focus on the place value of each digit in the products (MP7).

MLR2 Collect and Display. Collect the language students use to explain why the expressions are equivalent. Display words and phrases such as: equal, multiply, decimal, equivalent, same value, divide, tenths, hundredths. During the synthesis, invite students to suggest ways to update the display. Invite students to borrow language from the display as needed.
Advances: Conversing, Reading

Launch

  • Groups of 2

Activity

  • 1–2 minutes: quiet think time
  • 8–10 minutes: partner work time

Student Facing

  1. Explain or show why each pair of expressions is equivalent.

    1. \(7.2 \times 5.3\) and \((72 \times 53) \times 0.01\)
    2. \(6.5 \times 2.8\) and \((65 \times 28) \div 100\)
    3. \(31 \times 0.44\) and \((31 \times 44) \times \frac{1}{100}\)
  2. Find the value of the products in the previous problem.

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

  • Invite students to explain why \(7.2 \times 5.3\) is equivalent to \(72 \times 53 \times 0.01\).
  • Display equations: \(7.2 = 72 \times 0.1\) and \(5.3 = 53 \times 0.1\)
  • “How do you know the equations are true?” (Multiplying 72 or 53 by 0.1 changes the tens to ones and the ones to tenths.)
  • Display equation: \(7.2 \times 5.3 = (72 \times 53) \times 0.01\)
  • “How do you know the equation is true?” (I use the equations for 7.2 and 5.3 and multiply 0.1 and 0.1 to get a hundredth or 0.01.)
  • “How can you use the equation to find the value of \(7.2 \times 5.3\)?” (I can multiply 72 and 53 and then multiply that by 0.01.)

Activity 2: Choose Your Strategy (15 minutes)

Narrative

The purpose of this activity is for students to find products of decimals and whole numbers with no scaffold. As in the previous activity, the products are either a whole number and a decimal to the hundredths or two decimals to the tenths. Students may use any strategy including partial products or using products of whole numbers and place value understanding. The final problem, the product of a three-digit decimal number and a two-digit decimal number, is new but all of the strategies students have used to multiply two-digit decimals apply here as well.

Engagement: Develop Effort and Persistence. Some students may benefit from feedback that emphasizes effort, and time on task. For example, Invite students that used a different method for each problem to explain their choice(s).
Supports accessibility for: Conceptual Processing, Language

Launch

  • Groups of 2

Activity

  • 5-8 minutes: independent work time
  • 2-5 minutes: partner discussion

Student Facing

Find the value of each product.

  1. \(7.3 \times 4.2\)
  2. \(38 \times 0.55\)
  3. \(285 \times 0.17\)

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

  • Invite students to share their responses for the first two products.
  • “How did you use your understanding of place value to find the products?” (I used whole number products and then remembered that I have that many hundredths so I had to multiply that product by 0.01.)
  • “How is the last product the same as the other products you have found? How is it different?” (It is also a whole number and a decimal and the decimal has tenths and hundredths. The whole number is bigger. I can use the same methods but the product is more complicated since one factor has three digits.)

Lesson Synthesis

Lesson Synthesis

“Today we found products of whole numbers and decimals.”

“How is finding products of whole numbers and decimals the same as finding products of whole numbers? How is it different?” (I have to find the products of the digits in both cases. I can use the same strategies for finding those products. When there are decimals, I need to remember that those whole number products of digits might be tenths or hundredths.)

Cool-down: Explain Why Expressions are Equal (5 minutes)

Cool-Down

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Student Section Summary

Student Facing

In this section, we learned to use place value relationships to multiply a whole number and and a decimal. For example,

\(6 \times 0.14 = 0.84\)

because 6 groups of 14 hundredths is \(6 \times 14\) or 84 hundredths.

We also found products like \(1.7 \times 0.3\). We can use a diagram to see that this is \(17 \times 3\) hundredths or \(0.51\).

Two diagrams. Each squares. Length and width, 1.