The purpose of this activity is for students to use a diagram to help calculate the product of a three-digit number and a two-digit number. The diagram helps to organize the individual products that can be used to find the full product. During the Activity Synthesis, students connect the diagram to the distributive property when they explain how the sum of the individual products gives the full product (MP7).

• Invite students to share their work for finding the product .
• Display:
• “How does the diagram represent this equation?” (It shows 62 broken into 60 and 2 and 35 broken into 30 and 5.)
• Display:
• “How do you know this equation is true?” (The diagram shows broken into those 4 partial products.)
• “How is finding the product related to finding the product ?” (The products and partial products are the same, except that I also have in .)

The purpose of this activity is for students to write expressions to represent different ways to decompose a product. Then they choose one decomposition with which to find the product. Students consider how certain decompositions are more helpful than others, depending on the specific numbers in the problem. The diagrams used here relate to the partial-products and standard algorithm methods, which students will learn in future lessons.

• “How did you decide to find the value of ?”
• “Where do you see the strategy you used to find the value of represented in the diagrams?”

• Display:
• “How does this expression relate to the product ?” (It represents one of the products in the first diagram.)
• “Why isn’t this expression written in any of the other diagrams?” (The other diagrams are decomposed differently.)
• Invite students to share the diagram they chose to find the product and how it was helpful. As students share, record equations to represent each partial product.
• “What are the advantages or disadvantages of this way to calculate ?” (For completely broken apart partial products, each product is simple to calculate, but I do have 6 different numbers to add up at the end. When I broke the full product into two products, the calculations I used to find each product were harder, but once I had them, there were only two products to add. When I broke the full product into 3 products, this was a good compromise. The products were not too hard to calculate, and there were just 3 of them to add.)