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.
• “Escojan uno que sea diferente. Prepárense para compartir por qué es diferente” // “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

• “Discutan con su pareja cómo pensaron” // “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.

• “¿Por qué no tenía sentido que el rectángulo de la imagen C estuviera dividido por la mitad?” // “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: “Encontremos al menos una razón por la que cada uno es diferente” // “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
• “¿En qué lugares de su comunidad ven arte?” // “Where are some places that you see art in your community?”
• 30 seconds: quiet think time
• Share responses.
• “Estos problemas se tratan de un estudiante que visita un festival de arte local. Ahí es donde los artistas locales exhiben y venden su trabajo” // “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.
• “Hablen con su compañero sobre cómo podrían resolver estos problemas” // “Turn and talk to your partner about how you might solve these problems.”
• 1–2 minutes: partner discussion

• “Resuelvan estos problemas. Expliquen o muestren cómo razonaron” // “Solve these problems. Explain or show your reasoning.”
• 6–8 minutes: independent work time
• As students work, consider asking:
  • “¿De qué forma pueden representar la manera en la que piensan en el problema?” // “How can you represent your thinking about the problem?”
  • “¿En qué parte de su trabajo pueden ver _____?” // “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
• “Ahora van a hacer un póster que muestre la forma como pensaron en el tercer problema. Van a trabajar con un compañero que haya resuelto el problema de la misma forma que ustedes” // “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

If students don’t find a solution to the problems, consider asking: “¿De qué se trata este problema?” // “What is this problem about?” and “¿Cómo podrías representar el problema?” // “How could you represent the problem?”

• 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
• “Antes de que empiecen el recorrido por el salón, ¿qué representaciones esperan ver cuando observen el trabajo de los demás estudiantes?” // “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.

• “Mientras visitan los pósteres con su compañero, discutan en qué se parecen y en qué son diferentes las ideas que se muestran en los pósteres. Además, dejen una nota adhesiva en la que describan por qué creen que la solución tiene o no tiene sentido” // “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:
  • “¿Cómo se ven los factores en el diagrama?” // “How are the factors shown in the diagram?”
  • “¿Cómo se ve el producto en el diagrama?” // “How is the product shown in the diagram?”
  • “¿Cómo podríamos representar esta estrategia con expresiones?” // “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.