Lesson 3

The purpose of this Number Talk is for students to multiply three factors. Strategies for multiplying three factors will be helpful as students find the volume of rectangular prisms in this lesson and upcoming lessons. Since the first problem is only 2 factors, it is not important to gather multiple strategies in order to leave more time for the other problems. Students may connect the third factor in the final three problems to ‘adding another layer of cubes’ in a prism when finding volume. Invite these students to share their observations during the synthesis.

• Display one problem.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: individual think time

• Record answers and strategy.
• Keep expressions and work displayed.
• Repeat with each expression

• “How did changing one of the factors impact the product?” (Increasing one of the factors made the product get bigger by 6.)
• Consider asking:
  • “How are problems 2–4 like the work we did with prisms yesterday?”
  • “Who can restate _______’s reasoning in a different way?”
  • “Did anyone have the same strategy but would explain it differently?”

The purpose of this activity is for students to build and determine the volume of rectangular prisms from images. In the activity synthesis, students look at two related prisms to encourage them to think about 8 cubes as a layer.

• Create a set of cards from the blackline master for each group of 4.
• Each group of 2 needs 48 connecting cubes.

• Groups of 2
• Give students access to connecting cubes.
• “You are going to build rectangular prisms shown on cards. Before we look at the cards, what information in the image will help you build the prism?” (The number of cubes in each layer and the number layers.)
• Share responses.
• “Now, you will each pick a card, build the prism, and find its volume. Explain how you found the volume to your partner and then pick another card.”

• Monitor for students who discuss:
  • how many cubes are in each layer.
  • how the cubes are arranged in each layer.
  • how many layers are in the prism.
• Mathematical Community: As students work, monitor for examples of the “Doing Math” actions.

Some students may still count each individual cube to determine the volume of the prism. Ask, “How many cubes are in each layer? How many layers are there?”

• Display Card E
  • “How did you build this rectangular prism?” (I counted 8 cubes and then made a stack of 8 cubes.)
  • “What is the volume of this rectangular prism? How do you know?” (8 cubes because I counted 8 cubes.)
• Display Card A
  • “How did you build this rectangular prism?” (I saw that there are 8 cubes on the top and bottom in 2 rows of 4. Then I built it up until it was 3 cubes high.)
  • “What is the volume of this rectangular prism? How do you know?” (24 cubes because there are 3 sets of 8 cubes and \(3\times8=24\))

The purpose of this activity is for students to find the volume of larger rectangular prisms shown in images. In previous lessons, students built rectangular prisms out of cubes and counted the cubes to determine the volume of the rectangular prisms. In this activity, the prisms were intentionally chosen to encourage students to use the layered structure of the prism to determine the volume of the prism (MP7). When students connect this structure to the operation of multiplication and use expressions and equations to find the volume, they decontextualize the geometric structure to solve the problems (MP2).

• Give students access to connecting cubes.

• Groups of 2
• Display the first prism.
• “How would you describe this prism?” (There are 5 layers and each layer has 8 cubes or there are 2 layers and each layer has 20 cubes in it.)
• Give students access to connecting cubes.

• 10 minutes: independent work time
• 2 minutes: partner discussion
• Monitor for students who use the layered structure to determine the volume.

If students do not notice that each prism has one more layer of 20 cubes, consider asking, “What is the same about each of these prisms? What is different?”

• Invite students to share how they found the volume of the rectangular prisms.
• Display image of the first prism.
• “The expression \(5\times8\) represents the volume of the prism. Where do you see 5 groups of 8 cubes in this prism?” (The top and bottom are 2 by 4 so there are 8 cubes on top and bottom. There are 5 of those layers so that is \(5\times8\).)
• “What do these rectangular prisms have in common?” (They are all 5 cubes high. They can all be broken into 5 equal layers or groups. They also all have a side that is 4 cubes long. They can all be broken into layers with 20 cubes.)
• If time permits, ask “Who saw the layers differently?” (For the first prism, I used the side layer of 10 cubes. I multiplied 10 by 4 layers to get 40.)