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).

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.

This activity prepares students to use area diagrams to find products of decimals in upcoming lessons. The reasoning here revisits students’ work in grade 5, during which they used area diagrams to find products of two numbers, including whole numbers, fractions, and decimals through tenths.

In addition to using the structure of base-ten numbers in their reasoning, students also use the structure of the area diagram to help them find products of decimals (MP7). 
As students work, monitor for the different ways students reason about the area of each unit square and the area of the large rectangle given particular side lengths. Identify a few students with correct reasoning so they can share later.

Some students might think that (just like ). If this happens, have them write 0.1 in fraction form so they see that , or 0.01.

The goal of this discussion is to highlight that the sides of a rectangle can represent lengths in different values, so its area can represent many different products. (This idea is not unlike how a collection of base-ten blocks can represent 103, 10.3, 1.03, and so on.)

If time permits, consider discussing the idea of the same diagram representing different values in the context of measurement units. Display a gridded 2-by-4 rectangle with the side length of squares marked “1 cm.” Ask students:

• “What is the area of each square in cm²?” (1) “What is the area of the rectangle in cm²?” (8) “What equation can represent the relationship between the side lengths and area?” ( )
• “What is the area of each square in mm²?” (100) “What is the area of the rectangle in mm²?” (800) “What equation can represent the relationship between the side lengths and area?” ( )
• “What is the area of each square in m²?” (, or 0.0001) “What is the area of the rectangle in m²?” (, or 0.0008) “What equation can represent the relationship between the side lengths and area?” ( )

Reiterate that the diagram hasn’t changed, but the side lengths and area of the same diagram can represent different values depending on the unit of measurement we use.

Tell students that they will continue to use area diagrams to multiply decimals in an upcoming lesson.