Lesson 6

The purpose of this warm-up is to elicit students’ prior knowledge of area and the idea that a rectangle can be decomposed into smaller rectangular regions. Students look at 4 different area diagrams they used in grade 3. The reasoning here will be useful when students use diagrams to multiply two- and one-digit numbers in a later activity. While students may notice and wonder many things about the number of units within the area of the gridded region, focus on the connections between the diagrams with a grid and those without.

• Groups of 2
• Display the image.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “What do you know about the 13 and 4 in the first ungridded rectangle?” (They are side lengths. They correspond to the number of squares across and the number of squares down in the first rectangle.)
• “What about the 10, 3, and 4 in the second ungridded rectangle?” (The 10 and 3 are numbers that add up to 13. The 4 is the length of the shorter side.)
• “How are the four diagrams related? Name as many connections as you see.” (Sample responses:
  • They all represent the same rectangle. They have the same side lengths and the same area.
  • The first two show the number of square units that fit in the rectangle. The last two don’t show it but we can tell by multiplying the side lengths.
  • The shaded portion in the second gridded rectangles shows the \(4 \times 10\) portion in the fourth rectangle.)
• If no students mentioned the area of the rectangles in their analysis, ask: “How might we find the area of the rectangles?” (For the gridded rectangles, we can count the unit squares or multiply the number of units across and down. For the other two, we can multiply the side lengths.)

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.

• Groups of 3–4

• 5 minutes: independent work time on the first problem
• 2–3 minutes: partner discussion
• 3–4 minutes: independent work time on the second problem
• Monitor for students who use different methods to find the value of \(4 \times 36\) in the first problem.

• Select 2 students who organized the diagram in the first problem in different ways and then compare to what Tyler did.
• Invite selected students to share their method for finding the value of \(4 \times 36\).
• “How are these methods the same? How are they different?”
• “How are these methods like what Tyler did to find the value of \(9 \times 18\)? 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 grade 3, where the side lengths of a rectangle represents 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 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.

• Groups of 2

• “Take a few quiet minutes to work on the activity. Afterward, share your thinking with your partner.”
• 6–7 minutes: independent work time
• 3–4 minutes: partner discussion
• As students work on the last problem, monitor for those who:
  • recognize that base-ten drawings are cumbersome and a less efficient way to find the product
  • draw diagrams that show the two-digit side length partitioned by tens and ones
  • write expressions that show place value reasoning
  • use the distributive property (find the partial products, then add to find the value of the product)

• 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 \(6 \times 53\)? What is the product?” ( \(6 \times 50) + (6 \times 3)\). 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?” (\(6 \times 40\) and \(6 \times 8\) or  \(9 \times 60\) and \(9 \times 7\).)
  • “How does partitioning the two-digit number into tens and ones help to find the product?” (Finding multiples of ten and facts like \(6 \times 8\) is easier than finding a product of say, 36 and 8. Then we can add the smaller products together to find the total product.)