Lesson 15

This warm-up prompts students to compare four area diagrams that have been decomposed into two areas, each representing a product. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another. During the synthesis, ask students to explain the meaning of any terminology they use, such as side lengths, area, parts, and decompose.

• Groups of 2
• Display the image.
• “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.

• “Why didn’t it make sense for the rectangle in C to be split in half?” (Because one part of the rectangle should have been larger than the other because 70 is greater than 28. The part with 28 should have been smaller than the part with 70.)
• Consider asking: “Let’s find at least one reason why each one doesn’t belong.”

The purpose of this activity is for students to solve problems that involve multiplication where one factor is a teen number. Students may solve and represent the problem any way they choose. In problem 3, look for different ways in which students are using area diagrams to highlight in the posters for the gallery walk in the next activity. Students reason abstractly and quantitatively when they interpret the stories and represent them with diagrams, expressions, or equations (MP2).

• Groups of 2
• “Where are some places that you see art in your community?”
• 30 seconds: quiet think time
• Share responses.
• “These problems are about a student visiting a local art festival. This is where local artists come to display and sell their work.”
• Give students access to grid paper and base-ten blocks.
• “Turn and talk to your partner about how you might solve these problems.”
• 1–2 minutes: partner discussion

• “Solve these problems. Explain or show your reasoning.”
• 6–8 minutes: independent work time
• As students work, consider asking:
  • “How can you represent your thinking about the problem?”
  • “Where can you see _____ in your work?”
• Monitor for:
  • different ways that students use the area diagram in the third problem
  • students who represent the third problem in the same way to pair to create a poster together
• “Now you are going to create a poster to show your thinking on the third problem. You are going to work with a partner who solved the problem in the same way you did.”
• Give each group tools for creating a visual display.
• 6–8 minutes: partner work time

• Display posters around the room.

The purpose of this activity is for students to see how other students solved one of the problems that involves a factor of a teen number. While students look at each other’s work, they will leave sticky notes describing why they think the answer does or does not make sense (MP3). The synthesis will look specifically at examples of how students used the area diagram to represent the problem.

• Groups of 2
• “Before you begin the gallery walk, what are some representations you expect to see as you look at other students’ work?” (I expect to see rectangles because some of the problems were about the area of rectangles. Base-ten blocks because we were multiplying teen numbers.)
• Share responses.
• Give students sticky notes.

• “As you visit the posters with your partner, discuss what is the same and what is different about the thinking shown on each poster. Also, leave a sticky note describing why you think the solution does or does not make sense.”
• 8–10 minutes: gallery walk
• Monitor for different uses of the area diagram to highlight, specifically:
  • a fully gridded area diagram with no labels and no decomposition
  • a gridded area diagram that was gridded, but also decomposed into parts or labeled along the sides or in the parts of the rectangle
  • a partitioned rectangle that was drawn with no grid, but labeled with side lengths or the area of the parts of the rectangle

• Select 2–3 posters that used a diagram to represent the problem. Discuss them in the order shown above.
• For each poster, ask:
  • “How are the factors shown in the diagram?”
  • “How is the product shown in the diagram?”
  • “How could we represent this strategy with expressions?”
• If students don’t use an ungridded diagram, display one that represents the problems and ask the same questions.