In this activity, students use rectangular diagrams and similar reasoning as in earlier activities to represent the multiplication of 2 two-digit numbers. They analyze a progression of diagrams, starting with those that represent multiplication of two-digit and one-digit numbers (18 and 6), a two-digit number and a ten (18 and 10), and then 2 two-digit numbers (18 and 16).

Students may decompose factors in different ways. For example, those who are familiar with multiples of 25 may find it intuitive to decompose as and , rather than decomposing 25 into .

• Select two students to share their diagrams, solutions, and reasoning for the last problem. 
• “How did you decompose the factors and the diagram?” (I decomposed the 13 into 10 and 3 and the 21 into 20 and 1.) 
• “What expression could we write to show the partial product represented by each part of the diagram?” 
• “How do these partial products help us find the value of ?” (Adding them gives us the value of : )

In this activity, students analyze two ways of decomposing a factor: by place value and not by place value. As they write the corresponding partial products, they see more clearly why it is helpful to decompose each of the factors by place value (MP7). Students may notice that when the factors are decomposed by place value, they end up finding multiples of 10 and multiplying a number by single-digit factors—both of which they can do with some degree of fluency.

This activity uses MLR5 Co-craft Questions. Advances: writing, reading, representing

• “How can both diagrams be used to find the value of  ?” (We can partition each side of a rectangle in many ways without changing the total side lengths.)
• “Why was the partitioning in Diagram A more helpful than in Diagram B?” ( is easier than or to find mentally, because we are multiplying multiples of 10.)