In previous lessons, students have found products of a whole number and a decimal and products of two decimals. They used diagrams, place value reasoning, and expressions to explain their reasoning. The purpose of this activity is for students to find both kinds of products with larger numbers. For each product, students show that an expression using whole number products is equal to the given decimal product. Then they calculate the decimal product. Students may use a strategy, other than the given equivalent expressions, to make the calculations. For example, they might decompose the numbers by place value and use the distributive property (partial products). All of these methods focus on the place value of each digit in the products (MP7).

• Invite students to explain why is equivalent to .
• Display equations: and
• “How do you know the equations are true?” (Multiplying 72 or 53 by 0.1 changes the tens to ones and the ones to tenths.)
• Display equation:
• “How do you know the equation is true?” (I use the equations for 7.2 and 5.3 and multiply 0.1 and 0.1 to get a hundredth or 0.01.)
• “How can you use the equation to find the value of ?” (I can multiply 72 and 53 and then multiply that by 0.01.)

The purpose of this activity is for students to find products of decimals and whole numbers with no scaffold. As in the previous activity, the products are either a whole number and a decimal to the hundredths or two decimals to the tenths. Students may use any strategy including partial products or using products of whole numbers and place value understanding. The final problem, the product of a three-digit decimal number and a two-digit decimal number, is new but all of the strategies students have used to multiply two-digit decimals apply here as well.

• Invite students to share their responses for the first two products.
• “How did you use your understanding of place value to find the products?” (I used whole number products. Then I remembered that I have that many hundredths, so I had to multiply that product by 0.01.)
• “How is the last product the same as the other products you have found? How is it different?” (It is also a whole number and a decimal and the decimal has tenths and hundredths. The whole number is bigger. I can use the same methods but the product is more complicated since one factor has three digits.)