Lesson 14

The purpose of this warm-up is to elicit the idea that we can find the area of figures even though not all side lengths are given, which will be useful when students find missing side lengths and areas in a later activity. While students may notice and wonder many things about this figure, the missing side lengths are the important discussion points.

• Groups of 2
• Display the image.
• “¿Qué observan? ¿Qué se preguntan?” // “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discutan con su compañero lo que pensaron” // “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Share and record responses.

• “¿Creen que podríamos encontrar el área de esta figura aunque algunas de las longitudes de sus lados sean desconocidas? ¿Por qué sí o por qué no?” // “Do you think we could still find the area of this figure even though some of the side lengths are missing? Why or why not?” (Yes, we could use the side lengths we know to find the missing side lengths.)
• “¿Cómo encontrarían las longitudes de los lados desconocidos?” // “How would you find the missing side lengths?” (I know the top side is 10 cm, and that is the horizontal length of the entire figure. If one part of that length is 4 cm, then the missing length is 6 cm. I know the vertical length of the figure is 9 cm. If one part of it is 2 cm, then the missing length is 7 cm.)

The purpose of this activity is for students to consider a strategy to find a missing side length of a figure composed of rectangles. They find the missing side length by decomposing the figure into non-overlapping rectangles.

• Groups of 2
• Display the image.
• “Consideren esta figura. Piensen por un momento en el razonamiento que usó Tyler para encontrar la longitud del lado desconocido” // “Consider this figure. Take some time to think about Tyler’s reasoning for finding the missing side length.”
• 1–2 minutes: quiet think time

• “Compartan con su compañero sus ideas sobre el razonamiento de Tyler” // “Share your ideas about Tyler’s reasoning with your partner.”
• 2–3 minutes: partner discussion

• Invite students to share their ideas about Tyler’s reasoning.
• Consider asking:
  • “¿Qué estrategias usaron ustedes o su compañero para encontrar la longitud del lado desconocido?” // “What strategies did you or your partner use to find the missing side length?” (Encourage students to share on the displayed figure.)

The purpose of this activity is for students to find the area of a figure composed of rectangles with missing sides by decomposing it into two non-overlapping rectangles. There are several ways to approach these problems and students are given freedom to choose their own strategy to find the area. They share their reasoning with a student who has used a different method and prepare to share this new method during the class discussion (MP3). The synthesis should focus on ways to find missing side lengths and emphasize different ways to find the area of the figure.

At this point, students should be comfortable multiplying side lengths of rectangles to find the area. Discussion should focus on how students decompose the figures and how they find the missing side lengths.

• Groups of 2

• “Ya discutimos cómo encontraríamos la longitud del lado desconocido de esta figura. Ahora, encuentren el área de la figura” // “Now that we’ve discussed how we would find the missing side length of this figure, find the area of this figure.”
• 3–5 minutes: independent work time
• Monitor for a variety of strategies to group students for discussion.
• “Busquen a alguien que haya usado la misma estrategia que ustedes para encontrar el área de la figura. Discutan por qué su estrategia tiene sentido” // “Find someone who used the same strategy that you used to find the area of the figure. Discuss how your strategy makes sense.”
• 2 minutes: partner discussion
• “Ahora, busquen a alguien que haya usado una estrategia diferente. Cada uno explique su estrategia. Asegúrense de que puedan explicarle a la clase la estrategia de su compañero” // “Now, find someone who used a different strategy than you. Explain your strategies to each other. Be sure you could explain your partner’s strategy to the class.”
• 4 minutes: partner discussion
• “Con su compañero actual, encuentren el área de la segunda figura” // “Work with your current partner to find the area of the second figure.”
• 5 minutes: partner work time
• Circulate and consider asking: “¿Qué información conocen que podría ayudarles a encontrar lo que desconocen?” // “What information do you know that might help you find what you are missing?”
• Monitor for a variety of ways that students use to find the area of the second figure, including different decompositions, different strategies for finding missing sides lengths, and different expressions they write.

If students determine incorrect lengths for the missing sides, consider asking:

• “Dime cómo encontraste la longitud del lado desconocido” // “Tell me about how you found the missing side length.”
• “¿Cuáles estrategias de la actividad anterior podrías usar para encontrar la longitud del lado desconocido?” // “What strategies from the previous activity could you use to find the missing side length?”

• Select students to share a variety of strategies for finding the area of the second figure.
• Consider asking:
  • “¿Cuál lado desconocido se tenía que encontrar para usar esta estrategia? ¿Cómo encontraron la longitud del lado desconocido?” // “What missing side had to be found to use this strategy? How did you find the missing side length?”
  • “Después de saber cuáles eran las longitudes de los lados desconocidos, ¿cómo encontraron el área?” // “After you knew the missing side lengths, how did you find the area?”
  • “¿Qué expresión escribiríamos para que corresponda a la descomposición que hicieron de la figura?” // “What is the expression we would write to match how you decomposed the figure?”
  • “¿Hay alguna otra manera en la que podríamos partir esta figura en rectángulos que no se superponen?” // “Is there another way that we could break this figure into non-overlapping rectangles?” (Students may also see the figure as 3 rectangles: \((2 \times 3) + (6 \times 2) + (2 \times 2)\).)