Lesson 22

The purpose of this How Many Do You See routine is to prompt students to decompose a rectilinear figure to find its area and to recognize that there are many ways to do so. Students are also reminded that area is additive. The reasoning here prepares students to reason flexibly about the area of rectilinear figures later in the lesson.

• Groups of 2
• “How many do you see? How do you see them?”
• Flash the image.

• 30 seconds: quiet think time
• Display the image.
• 1 minute: partner discussion
• Record responses.

• “We’ve seen how helpful it is to decompose the figure into rectangles. In how many ways could we do that here?” (Many ways)
• As students share each way, record the thinking for all to see.
• “Are there ways to partition that are more helpful than others?” (Partitioning into larger rectangles is more efficient than smaller ones. The latter would mean more multiplication and more partial areas to add up.)

In this activity, students solve geometric problems by reasoning about length and area, decomposing and recomposing of rectangles, considering units of measurements, and performing operations.

Each question can be approached in a variety of ways. Consider asking students to create a visual display of their approach and to share it with the class.

The first problem offers students an opportunity to make sense of a problem and persevere in solving it (MP1). They may focus on the area of the banner and poster paper or start thinking about cutting up the poster paper into pieces that can be used for the banner. They will also need to convert between feet and inches at some point in their solution.

• Explain what a banner is or show an example, if needed.
• Give students access to grid paper and inch tiles.

• 5–7 minutes: independent work time
• Monitor for different ways students reason about decomposing the 24 by 36 rectangle:
  • students who discuss the side length that is 8 feet long and consider how many inches long it would be
  • students who use tiles or a drawing to help reorganize the area

Students may see that there are several things to consider about the situation but may be unsure how to begin. Consider asking:

• “What do you know about the problem and the situation? What don't you know?”
• “What are some ways to visualize or show what Jada's teacher plans to do?”
• “How might the relationship between feet and inches help us think about this problem?”

• Select students who use different strategies to share their reasoning. Record and display their strategies.
• If it does not come up as a strategy, consider asking, “How could Jada's teacher cut the paper up and rearrange it to make a banner?”
• Consider discussing any benefits or potential challenges of the different approaches. “Are some strategies more efficient or more prone to error than others?” (When cutting the paper into more pieces, there are more measurements to account for, making it more likely to miss something. Cutting the paper into more pieces is also less efficient for Jada's teacher, as it means more taping as well.)

In this activity, students perform operations on multi-digit numbers to solve situations about perimeter and area. They use operations to convert units of measurements along the way. Converting inches to feet could be done by dividing by 12, but this is not an expectation at this point. Students could perform the conversion with multiplicative reasoning. To convert 180 inches into feet, for example, they could reason \(12 \times {?} = 180\), or \(12 \times 10 = 120\) and \(12 \times 5 = 60\).

In grade 3, students learned that area is additive, and that the area of rectilinear figures can be found by decomposing them into non-overlapping rectangles. Students apply that understanding here, after converting lengths in different units into the same unit.

• Groups of 2
• 1 minute: quiet time to read the opening paragraphs and look at the diagram in the task statement
• “What do you notice? What do you wonder?”
• Explain what a baseboard is or show an example.
• Make sure students recognize that all measurements need to be in the same unit before finding perimeter and area.

• 6–8 minutes: independent work time, problems 1–2
• 2 minutes: partner discussion
• Monitor for students who:
  • choose to consider all units in terms of inches, feet, or yards strategically
  • find a way to convert and keep track of the values systematically
• Pause for a whole-class discussion on the first two questions before students answer the last question.
  • “How did you decide which unit to use? How do we convert from _____ to _____?”
  • “How did you keep track of all the conversions?”
• As students work on the last question, monitor for:
  • different ways students decompose the diagram of the room to find its area
  • equations that show how the area is computed

• Select students to share how they reasoned about the area of the room. Record and display their reasoning.
• If some students found the area using different units, ask how they would find out if the two answers represent the same amount.