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.

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.

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.

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.)

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.

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.