Lesson 8

This warm-up prompts students to carefully analyze and compare features of figures built from rectangular prisms. In making comparisons, students have a reason to use language precisely (MP6), and refer to different measurements of the figures, their volume, or other characteristics. 

• Groups of 2
• Display the image.
• “Escojan una 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 compañero lo que pensaron” // “Discuss your thinking with your partner.”
• 2–3 minutes: partner discussion
• Share and record responses.

• Refer to Figure C. “Hoy vamos a encontrar el volumen de figuras formadas por dos o más prismas rectangulares. ¿Pueden ver dos prismas en la figura C? ¿En dónde?” // “Today we are going to find the volume of figures made up of two or more rectangular prisms. Where do we see two rectangular prisms in Figure C?”
• If possible, draw over the image as students explain where they see the two prisms.
• If no one mentions it, show two different ways that Figure C can be decomposed into two right rectangular prisms, that is, by cutting vertically or horizontally.

The purpose of this activity is for students to combine rectangular prisms to make a new figure and understand that the volume of the new figure is the sum of the volumes of the two rectangular prisms. This will be true no matter how they put the two rectangular prisms together.

• Each group of 2 needs at least 30 connecting cubes.

• Groups of 2
• “Vamos a construir prismas diferentes y los vamos a juntar” // “We are going to build different rectangular prisms and put them together.”
• “Compañeros A, construyan un prisma rectangular con 12 cubos. Compañeros B, construyan un prisma rectangular con 10 cubos” // “Partner A, build a rectangular prism with 12 cubes. Partner B, build a rectangular prism with 10 cubes.”

• 10 minutes: partner work time building prisms and finding volume.
• 2 minutes: partner discussion time
• Monitor for students who build one figure out of two rectangular prisms in different ways.

If students do not see two rectangular prisms that compose the figure, ask them to build the figure with cubes and show how they could decompose the figure into two rectangular prisms.

• Ask previously identified students to share the individual prisms they built and how they put them together.
• “¿Cuál era el volumen de la figura que hicieron cuando juntaron sus prismas? ¿Cómo lo saben?” // “What was the volume of the shape you made when you put your prisms together? How do you know?” (22 cubic units because I used 12 cubes and my partner used 10 cubes, so that’s 22 cubes altogether.)
• Highlight that the different figures students made all have a volume of 22 cubic units.
• Display the figure that Diego and Jada made.
• “¿Cuál es el volumen de la figura que Diego y Jada hicieron? ¿Cómo lo saben?” // “What is the volume of the figure Diego and Jada made? How do you know?” (“30 cubic units. I broke it into two prisms and found their volumes and added them.”)

The purpose of this activity is for students to find the volume of a figure composed of two rectangular prisms and recognize that volume is additive. In the previous activity, students physically put two rectangular prisms together and then decomposed a figure into two rectangular prisms. In this activity, they are given an image of a figure composed of two rectangular prisms and asked to find the volume. They can do this in many ways. During the synthesis, connect the different strategies students used to the decomposition of the figure into two rectangular prisms.

As students experiment with different ways to group the cubes to efficiently count them, applying what they already know about the volume of rectangular prisms, they are looking for and making use of the structure of geometric objects (MP7). As students begin to generalize the idea that volume is additive, they are looking for and expressing regularity in repeated reasoning (MP8).

• Groups of 2

• 5 minutes: individual work time
• 5 minutes: partner discussion
• As students work, monitor for students who:
  • find two (or more) rectangular prisms that can be put together to make the figure and adding their volumes.
  • find the number of cubes in a base layer and multiplying by the number of layers, even though the base layer is not a rectangle (figure d).

If students did not decompose the figures into rectangular prisms, encourage them to build the figures with connecting cubes and ask, “¿Cómo se puede descomponer esta figura en dos prismas rectangulares?” // “how can you decompose the figure into two rectangular prisms?”

• Display Figures c and d.
• Ask selected students to share their way of splitting each figure.
• “¿Quién partió la figura de la misma manera? ¿Quién la partió distinto?” // “Who broke the figure up the same way? Who broke it up differently?”
• “¿Pueden pensar en otras maneras en las que se pueden partir estas figuras?” // “Can you think of other ways you could break up these figures?“ (I can cut them into several layers—3 horizontal layers for c, and 5 vertical layers for d. Each layer is a rectangular prism.)