This Warm-up prompts students to carefully analyze and compare multiplication expressions. In making comparisons, students have a reason to use language precisely (MP6). The activity also enables the teacher to hear the terminologies that students know and how they talk about multiplication and equivalent expressions.
Launch
Arrange students in groups of 2–4. Display the expressions for all to see. Give students 1 minute of quiet think time. and ask them to indicate when they have noticed three expressions that go together and can explain why. Next, tell students to share their response with their group and then together to find as many sets of three as they can.
Activity
None
Which three go together? Why do they go together?
A
B
C
D
Student Response
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Building on Student Thinking
Activity Synthesis
Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations, and ensure that the reasons given are correct.
During the discussion, prompt students to explain the meaning of any term that they use, such as “factor,” “product,” “equivalent,” or “commutative,” and to clarify their reasoning as needed. Consider asking:
“How do you know . . . ?”
“What do you mean by . . . ?”
“Can you say that in another way?”
16.2
Activity
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.
This activity builds on the two methods for computing products of decimals introduced in a previous lesson, namely by:
Thinking in terms of division. Multiplying a number by 0.1 or is the same as dividing the number by 10, multiplying by 0.01 or is the same as dividing by 100, and so on.
Thinking in terms of fractions. We can write decimal factors as fractions, compute the product, and then convert the product to a decimal.
Students make sense of both methods and use a preferred method to find the products of other decimals. As they reason about the relationship between whole numbers, decimals, and fractions and apply it to multiply decimals, students practice looking for and making use of the structure of the base-ten system (MP7).
To reason correctly about the products of decimals, students also need to keep careful track of the digits and their place value. In doing so, they practice attending to precision (MP6).
This activity is the first time that students see these two methods for computing products of decimals. The previous lesson referenced in the Activity Narrative is not included in this course.
Launch
Arrange students in groups of 2. Tell students that they will analyze two methods for multiplying decimals. Ask each student to study one method and then explain it to their partner. After making sense of both methods together, each partner applies one method to find a new product.
Give partners 5–6 minutes to complete the first set of questions. Ask students to pause for a brief discussion before moving on to the second half of the activity.
Invite a student who used each method to share their reasoning. If not already mentioned in students' explanations, ask:
“Why might Elena have multiplied by 2.4 and 1.3 by 10?” (To get the decimals into whole numbers, which are easier to multiply.)
“What might be Elena’s reason for dividing 312 by 100?” (She multiplied the original factors by or 100, so the product 312 is 100 times the original product and must be divided by 100).
“How is Noah's method different from Elena's?” (Noah converted each decimal into fractions and multiplied the fractions.)
Give students another 5 minutes of quiet time to work on the second set of questions.
Activity Synthesis
Focus the discussion on how the two methods are mathematically equivalent. Highlight that:
In Noah’s method, the product of the numerators of his fractions (23 and 15) is the same as the product of Elena’s whole numbers.
In both methods, the product of 23 and 15 is divided by 1,000. For Noah, 1,000 is the denominator of his fraction. For Elena, the division by 1,000 reverts the initial multiplication, which was by 1,000.
MLR8 Discussion Supports. Invite students to begin partner interactions by repeating the question, “What did Elena (or Noah) do first to find ? What did Elena (or Noah) do next?” This gives both students an opportunity to produce language. Advances: Conversing
16.3
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 find the product of two decimals using an area diagram, and they analyze two ways to multiply the same pair of decimals numerically, making connections between the strategies along the way. Then they apply the strategies to multiply another pair of decimals.
The key ideas here are:
We can find products of decimals by decomposing the factors by place value, finding partial products, and adding them, just as we have done when multiplying whole numbers.
We can represent the multiplication and the partial products using a diagram or with numerical calculations.
In an upcoming lesson, students will generalize the process and use the algorithm to compute products of decimals.
In the digital version of the activity, students can use an applet to represent multiplication of two decimals with an area diagram and to check their calculations. The applet allows students to specify the factors by place value, represent the ones and tenths as side lengths of a rectangle, and see the area of the rectangle decomposed accordingly. It also displays the corresponding partial products and final product. The digital version may be helpful for students to verify their work and to better visualize the structure of multiplication by place value.
If students don't have individual access, displaying the applet for all to see would be helpful during the synthesis.
Because this activity uses the same numbers as the previous activity, the timing for this activity can be adjusted to 15 minutes.
Activity Synthesis
The goal of the discussion is to highlight the connections between an area diagram, which can help us visualize partial products, and numerical calculations, which can help us record partial products concisely and calculate more efficiently.
Display the diagram and calculations for for all to see.
Ask questions such as:
“What do you notice about the labels of side lengths?” (Each one is decomposed into a whole number and a decimal, or ones and tenths.)
“In the diagram, what do the numbers in each sub-rectangle represent?” (The area of the sub-rectangle) “How do we find them?” (Multiply the length and width of each sub-rectangle.)
“In Calculation A, what do 0.12, 0.6, 0.4, and 2 represent?” (Partial products, or the areas of those sub-rectangles)
“If we didn’t draw a diagram, how could we find those partial products?” (We can multiply each digit in one factor by each digit in the other factor, keeping in mind the place value of each digit.)
Consider annotating each partial product in Calculation A with its factors:
Alternatively, consider separating each step and using color coding to show the digits being multiplied, as shown here:
Lesson Synthesis
The goal of this discussion is to highlight that we can multiply decimals using the same reasoning used with whole numbers: by decomposing the factors by place value, finding partial products, and adding them. An area diagram can help us visualize the parts, and a numerical calculation is a way to record them more efficiently.
Display the diagrams and calculations for and for all to see.
Invite students to make connections by asking questions, such as:
“How are the two diagrams alike?” (The side lengths are decomposed by place value. The rectangles are partitioned the same way.)
“How are the diagrams different?” (The side lengths are whole numbers in one diagram and decimals in the other. The decimal lengths are 1 tenth of the whole-number ones.)
“How are the two calculations alike?” (The structure, setup, and non-zero digits are the same. There are four partial products, found using a similar process.)
“How are the two calculations different?” (The partial products in the whole-number calculation have no decimals. The partial products in the decimal calculation are 1 hundredth of the whole-number calculations.)
“How do we know that ?” (We can think of this as , which equals . We can reason that has to be as much as .)
“How do we know that ?” (We can think of this as , which equals . We can reason that has to be as much as .)
Student Lesson Summary
Here are three other ways to calculate a product of two decimals, such as .
First, we can multiply each decimal by the same power of 10 to obtain whole-number factors.
Because we multiplied both 0.04 and 0.07 by 100 to get 4 and 7, the product 28 is times the original product, so we need to divide 28 by 10,000.
Second, we can write each decimal as a fraction and multiply them.
Third, we can use an area diagram. The product can be thought of as the area of a rectangle with side lengths of 0.04 unit and 0.07 unit.
In this diagram, each small square is 0.01 unit by 0.01 unit. The area of each square, in square units, is therefore , which is .
Because the rectangle is composed of 28 small squares, the area of the rectangle, in square units, must be:
All three calculations show that .
To find the product of two two-digit numbers, such as , we can think of finding the area of a rectangle with those numbers, 3.4 units and 1.2 units, as side lengths.
First, we draw a rectangle and partition each side length by place value, into ones and tenths:
Then, we decompose the rectangle into four smaller sub-rectangles and find their areas.
Area diagram. A rectangle partitioned into 4 rectangles, A, B, C, D. D, vertical side, 1, horizontal side, 3. C, vertical side, 1, horizontal side, 0 point 4. B, vertical side, 0 point 2, horizontal side, 3. A, vertical side, 0 point 2, horizontal side, 0 point 4.
A:
B:
C:
D:
Each multiplication gives a partial product that represents the area of a sub-rectangle. The sum of the four partial products gives the area of the entire rectangle, 4.08 square units.
We can show the same partial-product calculations vertically. Here are two ways:
Two vertical calculations of 3 point 4 times 1 point 2. First calculation, 7 rows. First row: 3 point 4. Second row: multiplication symbol, 1 point 2. Horizontal line. Third row: 0 point 0 8, A. Fourth row: 0 point 6, B. Fifth row: 0 point 4, C. Sixth row: plus 3, D. Horizontal line. Seventh row: 4 point 0 8. Second calculation, 5 rows. First row: 3 point 4. Second row: multiplication symbol, 1 point 2. Horizontal line. Third row: 0 point 6 8, A plus B. Fourth row: plus 3 point 4, C plus D. Horizontal line. Fifth row: 4 point 0 8.
The calculation on the left shows four partial products, one for the area of each sub-rectangle.
The calculation on the right shows two partial products:
0.68 is the value of , or the combined area of A and B.
3.4 is the value of , or the combined area of C and D.
In both calculations, adding the partial products gives a total of 4.08, which is the product of and the area (in square units) of the entire rectangle.
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.
Elena and Noah used different methods to compute . Both calculations were correct.
Elena's method. 2 point 4 times 10 equals 24. 1 point 3 times 10 equals 13. 24 times 13 equals 312. 312 divided by 100 equals 3 point 1 2. Noah's method. 2 point 4 equals the fraction 24 over 10. 1 point 3 equals the fraction 13 over 10. The fraction 24 over 10 times the fraction 13 over 10 equals the fraction 312 over 100. The fraction 312 over 100 equals 3 point 1 2.
Analyze the two methods, then discuss these questions with your partner.
Which method makes more sense to you? Why?
What might Elena do to compute ?
What might Noah do to compute ?
Will the two methods result in the same value?
Compute each product using the equation and what you know about fractions, decimals, and place value. Explain or show your reasoning.
Student Response
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Building on Student Thinking
Launch
Arrange students in groups of 2. Give students 1–2 minutes of quiet work time for the first problem. Pause for a brief whole-class discussion, making sure that all students label each region correctly.
Then give partners 2–3 minutes to analyze the given calculations and discuss the questions. Monitor student discussions to check for understanding. If necessary, pause to have a whole-class discussion on the interpretation of these calculations.
Next, give students 5–6 minutes of quiet time to complete the remaining questions. Tell students that their diagram need not be drawn exactly and that it is fine to estimate appropriate side lengths, but that the labels should reflect the numbers being multiplied.
Some students might find it helpful to use a grid to align the digits in vertical calculations. Provide access to graph paper.
If access to digital devices is available, consider allowing students to use the applet to check their calculations and to explore the products of other decimals.
Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations. For example, use color coding to highlight connections between the partial products in the calculations and in the area diagrams. Supports accessibility for: Visual-Spatial Processing
Activity
None
Here is an area diagram that represents .
A rectangle partitioned vertically and horizontally into 4 rectangles. Top left rectangle, vertical side, 1, horizontal side, 2. Top right rectangle, vertical side, 1, horizontal side, 0 point 4. Bottom left rectangle, vertical side, 0 point 3, horizontal side, 2. Bottom right rectangle, vertical side, 0 point 3, horizontal side, 0 point 4.
Find the region that represents . Label it with its area of 0.12.
Label the other regions with their areas.
Find the value of . Show your reasoning.
Here are two ways of calculating .
Two vertical calculations of 2 point 4 times 1 point 3. Calculation A, 7 rows. First row: 2 point 4. Second row: multiplication symbol, 1 point 3. Horizontal line. Third row: 0 point 1 2. Fourth row: 0 point 6. Fifth row: 0 point 4. Sixth row: plus 2. Horizontal line. Seventh row: 3 point 1 2. Calculation B, 5 rows. First row: 2 point 4. Second row: multiplication symbol, 1 point 3. Horizontal line. Third row: 0 point 7 2. Fourth row: plus 2 point 4. Horizontal line. Fifth row: 3 point 1 2.
Analyze the calculations and discuss these questions with a partner:
In Calculation A, where do the 0.12 and other partial products come from?
In Calculation B, where do the 0.72 and 2.4 come from?
In each calculation, why are the numbers below the horizontal line aligned vertically the way they are?
Find the value of in two ways:
Draw and label a diagram. Show your reasoning.
Calculate numerically, without using a diagram. Be prepared to explain your reasoning.
Student Response
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Building on Student Thinking
When using vertical calculations, students might find the correct partial products but not align them by place value (for example, they might align the rightmost digit of all partial products), resulting in an incorrect sum. Ask them what values the digits in each partial product represent and to consider how they should be added.
