This Math Talk focuses on multiplication of a whole number and a decimal. It encourages students to think about properties of operations and to rely on what they know about place value to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students decompose and multiply decimals beyond tenths later in the lesson.
To multiply decimals to the hundredths, students need to look for and make use of the structure of base-ten numbers (MP7).
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
Invite students to share their strategies, and record and display their responses for all to see.
Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards. Supports accessibility for: Memory, Organization
Activity
None
Find the value of each expression mentally.
Student Response
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Building on Student Thinking
Activity Synthesis
To involve more students in the conversation, consider asking:
“Who can restate ’s reasoning in a different way?”
“Did anyone use the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to ’s strategy?”
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I because . . . .” or “I noticed so I . . . .” Some students may benefit from the opportunity to rehearse with a partner what they will say before they share with the whole class. Advances: Speaking, Representing
17.2
Activity
Optional
Standards Alignment
Building On
Addressing
6.NS.B.3
Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
This activity gives students another opportunity to use partial products to multiply decimals and to connect area diagrams and vertical calculations. More attention is given to how to record the partial products more concisely (as shown in Calculation B in this activity and earlier ones), paving the way to multiplying decimals using the standard algorithm.
In these problems, the side lengths and areas of the rectangle are left blank. Students fill in these fields by decomposing the factors by place value and multiplying them. Like many of the activities in this unit, the work involves understanding the structure of the base-ten system used in multiplication calculations (MP7) and making sense of it from arithmetic and geometric perspectives.
Students also make another key connection about several previously developed ideas here. They see that we can write the non-zero digits of decimal factors as whole numbers, use vertical calculations to multiply them, and then attend to the decimal point in the product afterward. In other words, they notice the algorithm for multiplication can streamline the reasoning processes that they have used up to this point.
Launch
Keep students in groups of 2. Give groups 1–2 minutes to make sense of the first set of questions and the area diagram.
Before students begin labeling and computing, discuss questions such as:
“How might you decompose each factor?”
“How would you decide what number to use for each part of a side length?”
“How would you know what value to record for the regions A, B, C, and D?”
If students propose decomposing each factor or side length in ways other than by place value, consider using an example to discuss the practicality of doing so. (For instance, suppose 2.5 is decomposed into 1.5 and 1, and 1.2 is decomposed into 0.7 and 0.5, what computations would give us the values of A, B, C, and D?)
Give students 3–4 minutes of quiet work time and then time to share their responses with their partner. Follow with a whole-class discussion.
Activity
None
Label the area diagram to represent and to find that product.
Decompose each factor by place value and write the ones and tenths in the blank boxes.
Label Regions A, B, C, and D with their areas. Show your reasoning.
Find the product that the area diagram represents. Show your reasoning.
Here are two ways to calculate . Each number with a box gives the area of one or more regions in the area diagram.
Two vertical calculations of 2 point 5 times 1 point 2. Calculation A, 7 rows. First row: 2 point 5. Second row: multiplication symbol, 1 point 2. Horizontal line. Third row: 0 point 1 0. Fourth row: 0 point 4. Fifth row: 0 point 5. Sixth row: plus 2. Horizontal line. Seventh row: 3 point 0 0. Calculation B, 5 rows. First row: 2 point 5. Second row: multiplication symbol, 1 point 2. Horizontal line. Third row: 0 point 5. Fourth row: plus 2 point 5. Horizontal line. Fifth row: 3 point 0.
In the boxes next to each number, write the letter(s) of the corresponding region(s).
In Calculation B, which two numbers are being multiplied to give 0.5?
Which numbers are being multiplied to give 2.5?
Student Response
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Building on Student Thinking
Some students might think that the partial products in Calculation A must be listed in a particular order for it to be correct. Invite students to list the partial products in any way that makes sense to them and compare their results with those of others who list them a different way.
Activity Synthesis
Display the solution shown here for all to see. Give students a minute to make comparisons and to share their observations or questions.
If no students mentioned listing the partial products in Calculation A in a different order from what’s shown, ask students to discuss this idea. Clarify that the order doesn’t affect the result, but it is conventional to start with the smallest place value in each factor and work toward the larger ones. It means starting with (the area of A) and ending with (area of D).
Consider annotating each partial product with the two values being multiplied, as shown:
Next, draw students’ attention to the partial products in Calculation B. Discuss the calculations that produce the partial products 0.5 and 2.5. If not mentioned by students, point out that each partial product comes from multiplying each digit in the second factor by the entirety of the first factor.
The 0.5, represented by the area of A + B in the diagram, is the product of 0.2 and 2.5.
The 2.5, represented by the area of C + D in the diagram, is the product of 1 and 2.5
To highlight this connection, consider shading or coloring the corresponding areas in the diagram and circling these values in the calculations.
17.3
Activity
Standards Alignment
Building On
Addressing
6.NS.B.3
Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
In this activity, students see that they can find the product of any pair of decimals by using the standard algorithm to multiply a related pair of whole numbers and then placing the decimal point. For example, to calculate , they can first find and then place the decimal point such that the result has two decimal places.
Students begin by analyzing an example and formulating a draft explanation for the placement of the decimal point in a product of decimals. Then, they take turns sharing their initial ideas with different partners. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and to critique the reasoning of others (MP3). As students revise their writing, they have an opportunity to attend to precision in the language that they use to describe their thinking (MP6).
Later, students apply the same method to multiply other decimals in and out of context. They also have an opportunity to check their reasoning using an area diagram.
This activity uses the Stronger and Clearer Each Time math language routine to advance writing, speaking, and listening as students refine mathematical language and ideas.
Launch
Arrange students in groups of 3–4. Display the calculations in the first question. Read aloud the problem stem. Give students 2–3 minutes of quiet time to make sense of the calculations and to write an explanation for the placement of the decimal point.
Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response for why the decimal point of the product of is where it is. In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help clarify and strengthen their partner’s ideas and writing.
If time allows, display these prompts for feedback:
“ makes sense, but what do you mean when you say. . . ?”
“Can you describe that another way?”
Close the partner conversations, and give students 2–3 minutes to revise their first draft. Encourage students to incorporate any good ideas and words they got from their partners to make their next draft stronger and clearer.
After Stronger and Clearer Each Time, give students 8–10 minutes to complete the activity individually or with their group.
Some students might find it helpful to use a grid to align the digits in vertical calculations. Provide access to graph paper.
Representation: Develop Language and Symbols. Activate or supply background knowledge. Review the standard multiplication algorithm. To help students recall the terms “factor” and “product,” ask, “What is the name for each part of a multiplication calculation?” Supports accessibility for: Memory, Language
Activity Synthesis
Focus the discussion on how students calculated the products of decimals in the last two questions. Ask questions such as:
“How did you know where to place the decimal point in the product of ?” (Because 165 is 10 times 16.5 and 7 is 10 times 0.7, the result of , or 1,155, needs to be divided by , or 100. This means moving the digits of 1,155 two places to the right, giving 11.55.)
“How can finding help you find ?” ( is 1,000 times the product of , so we can divide the former by 1,000—or move the digits 3 places to the right—to get the latter.)
If time permits, invite students to reflect on their reasoning, asking questions such as:
“Which method—drawing an area diagram or using vertical calculations—do you prefer when finding products such as ? Why?” (Drawing an area diagram, because the visual representation helps to break up the calculation into smaller, more manageable pieces: , , and . Vertical calculation, because it is quicker to just multiply whole numbers and place the decimal point.)
17.4
Activity
Optional
Standards Alignment
Building On
Addressing
6.NS.B.3
Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
This activity allows students to practice using the methods in this lesson to calculate products of decimals, including a problem in a real-world context. While students can choose to use area diagrams to help organize their work and support their reasoning, they should also practice using the multiplication algorithm on decimals.
Launch
Keep students in groups of 3–4. Give them 5–7 minutes of quiet work time and then time to discuss their responses with their group. If time is limited, consider asking students to convert only one length in the last question from meters to feet.
Provide continued access to graph paper for students who wish to use a grid to align the digits in vertical calculations.
Activity
None
Calculate each product. Show your reasoning. If you get stuck, consider drawing a diagram.
A rectangular playground is 18.2 meters by 12.75 meters.
Find its area in square meters. Show your reasoning.
If 1 meter is approximately 3.28 feet, what are the approximate side lengths of the playground in feet? Show your reasoning.
Student Response
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Building on Student Thinking
Activity Synthesis
Invite students to share their responses and reasoning. Display their calculations and diagrams (if used) for all to see. For each product, discuss how students determined the number of decimal places that the product should have.
When discussing the area of the rectangular playground, students may record the product of the side lengths as 232.050 or 232.05. If so, discuss why both values are correct.
Lesson Synthesis
Focus the discussion on the steps that can be taken to multiply any two decimals and why they work. Discuss questions, such as:
“We can calculate by first calculating . How are the factors in the two expressions related?” (19 is 10 times 1.9, and 8 is 100 times 0.08.)
“Once we have the product of 19 and 8, which is 152, how do we know where to put the decimal point?” (0.9 has one decimal place and 0.08 has two, so the product will have three decimal places.)
“Why does it make sense for this product to have three decimal places?” (We multiplied the factors by 10 and 100 to get 19 and 8, so the product, 152, needs to be divided by 1,000. Doing that moves the digits three places to the right, giving it three decimal places.)
“How can we check if the product we have is correct?” (We can use another strategy. We can estimate to see if it makes sense. For example, 1.9 is a little less than 2 and 0.08 is a little less than 0.1. The answer must be less than 0.2.)
Student Lesson Summary
To multiply two decimals, such as , we can multiply the whole numbers that have the same digits, , and then use what we know about place value to place the decimal point.
Multiplying 125 and 7 gives 875.
We know that 125 is 100 times 1.25, and 7 is 10 times 0.7, so the product of 125 and 7 is 1,000 times the product of 1.25 and 0.7.
This means we need to divide 875 by 1,000, which moves the digits 3 places to the right and gives 0.875.
Let’s find the product of 8.4 and 4.3!
First, we multiply 84 and 43.
84 is 10 times 8.4, and 43 is 10 times 4.3, so the product of 84 and 43 is 100 times the product of 8.4 and 4.3.
Dividing 3,612 by 100 moves the digits 2 places to the right, giving 36.12.
Notice that:
The factor 1.25 has 2 decimal places, the factor 0.7 has 1 decimal place, and the product 0.875 has 3 decimal places.
The factors 8.4 and 4.3 each have 1 decimal place, and the product 36.12 has 2 decimal places.
In general, to find the product of decimals, we can first multiply the corresponding whole numbers. Then we can place the decimal point so the product has as many decimal places as the sum of decimal places in the factors.
Standards Alignment
Building On
5.NBT.B.7
Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
A common way to multiply decimals is to multiply whole numbers, then place the decimal point in the product.
Vertical calculation of 25 times 12, 5 rows. First row: 25. Second row: multiplication symbol, 12. Horizontal line. Third row: 50. Fourth row: plus 250. Horizontal line. Fifth row: 300. Horizontal calculation, 25 times 12 is equal to 300. Horizontal calculation, 2 point 5 times 1 point 2 is equal to 3 point 0 0.
Here is an example for .
Use what you know about place value to explain why the decimal point of the product is placed where it is.
Use the method shown in the first question to calculate .
Complete this area diagram and use it to check your calculation for .
Decompose each factor by place value and write the numbers in the boxes on each side of the rectangle.
Write the area of each lettered region in the diagram. Then find the area of the entire rectangle. Show your reasoning.
About how many centimeters are in 6.25 inches if 1 inch is about 2.5 centimeters? Show your reasoning.
Student Response
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Building on Student Thinking
When completing the diagram to represent , students might recognize that 16.5 needs to be decomposed into three parts but be unsure how to do so. Encourage them to think about the place value of each digit in 16.5. Ask: “What values do the 1, 6, and 5 represent? How do those values correspond to the three parts on the long side of the diagram?”
