This activity prompts students to make sense of base-ten diagrams for representing multiplication. The representation supports students in grouping tens and ones and encourages them to use place value understanding and to apply the distributive property (MP7).

This activity is an opportunity for students to build conceptual understanding of partial products in a more concrete way. In the next activity, students will notice that working with these drawings can be cumbersome and transition to using rectangular diagrams, which are more abstract.

• Select 2 students who organized the diagram in the first problem in different ways and then compare their diagrams to Tyler’s method.
• Invite selected students to share their method for finding the value of .
• “How are these methods the same? How are they different?”
• “How are these methods like Tyler’s method to find the value of ? How are they different?”

This activity continues to encourage place value reasoning for finding the product of a two-digit factor and a one-digit factor.

Students make sense of two representations that show the two-digit factor decomposed by place value: a base-ten diagram and a rectangle. The latter looks like an area diagram that students have used in IM Grade 3, where the side lengths of a rectangle represent two factors. As the factors become larger, however, it becomes necessary to draw rectangles whose side lengths are not proportional. When rectangles no longer accurately represent area, the term “area diagrams” is not used. Instead, “rectangular diagrams” is used in teacher materials and “diagrams” in student materials.

Students then choose a representation to use to find products and write corresponding expressions. In the Activity Synthesis, they learn that the results of multiplying a part of one factor by the other factor can be called “partial products.” In future lessons, students will use rectangular diagrams to represent multiplication of larger numbers.

• Select 2–3 students to share their responses and reasoning.
• “How are Han’s and Priya’s diagrams similar?” (They both show the 53 as 50 and 3. They multiply the parts and add them to find the total value.)
• “How are they different?” (Han just drew a rectangle, and Priya drew all the tens and ones.)
• “What expressions can you write to show how to multiply ? What is the product?” (  . The product is 318.)
• Explain that the 300 and 18 in both Han’s and Priya’s diagrams are called partial products. Each is found by multiplying a part of one of the factors by the other factor.
• For the last problem, if students do not partition their rectangular diagrams by place value, display the sample diagrams as shown in the student responses without products. Ask:
  • “What expressions can we write to represent each part of the diagram?” ( and or   and .)
  • “How does decomposing the two-digit number into tens and ones help to find the product?” (Finding multiples of ten and facts like is easier than finding a product of say, 36 and 8. Then we can add the smaller products together to find the total product.)