In this activity, students recognize and use the fact that multiplying by 0.1, 0.01, and 0.001 is equivalent to multiplying by , , and , respectively. In all cases, the essential point to understand is that in the base-ten system, the value of each place is the value of the place immediately to its left. Writing the decimals 0.1, 0.01, and 0.001 in fraction form will help students recognize how the number of decimal places in the factors affect the number of decimal places in the product. Though students are not expected to make a generalization at this point, the reasoning here prompts students to look for and make use of structure (MP7).

Through repeated reasoning, students see how the number of decimal places in the factors can help them place the decimal point in the product (MP8).

The purpose of this discussion is to highlight the number of decimal places in a product of two numbers in base-ten. Consider asking some of the following questions:

• “How does the size of a product compare to the size of the factor when the factor is multiplied by 0.1? (The product is one tenth the size of the factor.) “What does that mean in terms of the number of decimal places?” (All the digits move one place to the right, so the product has one more decimal place compared to the factor.)
• “How does the size of a product compare to the size of the factor when the factor is multiplied by 0.01?” (The product is one hundredth the size of the factor.)  “What does that mean in terms of the number of decimal places?” (All the digits move two places to the right, so the product has two more decimal places compared to the factor).
• “Can you predict the outcome of multiplying 750 by 0.1 or 0.01 without calculating?” (Multiplying 750 by 0.1 would produce 75. Multiplying 750 by 0.01 would produce 7.5.)

In this activity, students continue to think about products of decimals. They use what they know about the fractions and , as well as the commutative and associative properties, to identify and write multiplication expressions that could help them find the product of two decimals.

While students may be able to start by calculating the value of each decimal product, the goal is for them to look for and use the structure of equivalent expressions (MP7), and later generalize the process to multiply any two decimals.

As students work, listen for the different ways that students decide on which expressions are equivalent to . Identify a few students or groups with differing approaches, and ask them to share later.

Invite students to share their responses and reasoning for the last question. Ask questions such as:

• “Why might it be helpful to write or as ? (Using whole numbers or fractions can help make the multiplication simpler. Even if there is division, it is division by a power of 10.)
• “We can write as , and as . Why is it okay to rearrange the factors?” (It doesn’t change the product.)
• “What do you notice about the number of decimal places in the product of ?” (The number is in the thousandths. It has 3 decimal places.)