The goal of this activity is for students to examine different ways to write the product of a three-digit number and a two-digit number as a sum of partial products. Students match sets of partial products, which can be put together to make the full product. Students are provided blank diagrams, familiar from the previous lesson, that they may choose to use to support their reasoning. In the Activity Synthesis, students relate the expressions and diagrams to equations to prepare them to analyze symbolic notation for partial products in the next activity.

When students relate partial products and diagrams to the product , they look for and identify structure (MP7).

• Invite previously selected students to share their strategies. As students share, record their reasoning with equations.
• Display:
• “How do you know this equation is true?” (I can put the 30 and 5 together since they are both multiplied by 245. I see that is the top row of the diagram and is the bottom row. Together, that’s the whole diagram.)

The purpose of this activity is for students to consider two different ways of recording partial products in an algorithm with which they worked in a previous course. The numbers are the same as in the previous activity to allow students to make connections between the diagram and the written strategies. Students examine two different ways to list the partial products in vertical calculations, corresponding to working from left to right and from right to left. Regardless of the order, the key idea behind the algorithm is to multiply the values of each digit in one factor by the values of each digit in the other factor. 

• “Where do you see the values of these equations represented in Clare's and Andre's work?”
• Write the equation . “How is this equation represented in Andre's and Clare's work?”

• “Both of these strategies use an algorithm that lists the partial products. An algorithm is a set of steps that works every time as long as the steps are carried out correctly.”
• “How are both approaches the same?” (Both multiply ones and tens by hundreds, tens, and ones.)
• “How are the approaches different?” (One starts with the hundreds, and the other starts with the ones. One goes from left to right, and the other goes from right to left.)
• “Why is it important to list the products in an organized way?” (That way I know I found all the partial products. I did not leave some out or find some twice.)
• Display:
• Display students’ work to show the list of equations from the second problem, or use the list in the Student Responses.
• “How does each expression relate to the product ?” ( is the product of the 3 in the tens place of 35 and the 2 in the hundreds place of 245.)