Lesson 2

Estimate Volume with Cube Models

Est. Class Sessions: 1–2

Developing the Lesson

Build Cube Models. Set up a display of the classroom objects collected for this lesson. Provide opportunity for students to look at the items that are part of the display. After allowing for several minutes of observation, tell them they will work with their partner to build a model of one of the objects with connecting cubes, so they can estimate the volume of their object in cubic units. Remind students that the volume is the amount of space that an object takes up or how much it would take to fill it all the way up. Hold up one object and ask students to predict its volume in cubic units.

Ask:

X
  • How many connecting cubes do you think it will take to build a model of this?
  • How did you decide on that number?
  • About how many cubes long do you predict it
    will be?
  • About how many cubes wide do you predict it
    will be?
  • About how many cubes high do you predict it
    will be?
  • If you predict that it takes ______ connecting cubes, then what will you estimate that the volume of the object will be? Why? (The volume is the number of connecting cubes in the model.)
  • What units should you use for volume? (cubic units)

Direct students to the Cube Models pages in the Student Activity Book. Have each student pair select one object from the collection of classroom objects. Once each pair of students has an object, ask students to look at the display of the Math Practices page. Tell students that as they work together to estimate the volume of their object they will focus on Math Practices 3 and 6.

Ask:

X
  • As you are building your cube models, how can you check to make sure your model is reasonable? (Possible responses: You need to make sure you model is close to the same size as the real object.)
  • Do your model and your partner’s model have to be exactly the same in order to be reasonable? Explain your thinking. (Possible response: Your model does not have to be the same as your partner’s model but they will probably be close since you are both building the same thing.)
  • What do you need to remember about using labels when you record the volume of your model and the estimated volume of your object? (You need to make sure to write cubic units for the volume.)

Read Questions 1–2 aloud to students, providing time for them to draw their picture and build their cube model. Remind them that their model should be as close to the actual size of the object as possible. After students finish their models, they count the numbers of cubes used to build it and record the volume of their model in Question 3.

Compare Cube Models. Ask each student to place their cube models side by side with the real object and compare them.

Ask them to consider whether the volume of their model is slightly greater than, slightly less than, or the same as the volume of the object using prompts similar to the following:

X
  • Compare your model to the object by checking it from all sides. Look at both from the top. Do they look similar? How are they the same and how are they different?
  • Get at eye-level and look at them from the front. Do they look similar? How are they the same and how are they different?
  • Still looking at both at eye-level, compare them from each side. Do they look similar? How are they the same and how are they different?
  • Compare the volume of your model with the volume of your partner’s model. Why is it possible for your model to have a different volume than your partner’s model even though you used the same object? (Possible response: Since you each built your own model one of you may have made your model a little differents; for example, of you might have made the model a little longer than the other person did.)
  • How can you decide if your cube model is a reasonable representation of your object? (Your cube model should be about the same length, width, and height of the object you are building.)
  • What label do you use when you are telling someone the volume of your model? (cubic units)
X

Use the Cube Models pages of the Student Activity Book with the Feedback Box to assess students’ abilities to measure and estimate volume by building models and counting cubic units [E3]; justify a solution using visual and spatial reasoning [E6]; check for reasonableness [MPE3]; and use labels [MPE6].

Targeted Practice. Place a collection of objects with irregular shapes in a center and have students estimate volume by building cube models and counting cubic units.

After students have considered the similarities and differences of their model and the real object, ask students to answer
Question 4 on the Cube Models pages of the Student Activity Book. This question asks students to use their cube model to estimate the volume of the actual object and explain their reasoning in doing so. Before students begin, review Math Practice Expectations 3 and 6 on the display of the Math Practices page. Remind students that as they are working they should check to make sure their estimate of their object’s volume is reasonable and that they are labeling the volume to show what the number means.

X

Two students explain their visual and spatial

reasoning in determining their differing but

reasonable estimates of the volume of the

same object.

Teacher: John, will you tell me about your model?

John: Okay. I built a stapler. I used 27 cubes. But it doesn't look like a stapler.

Teacher: That's okay. It won't look exactly like the object. We’re estimating. Let’s talk about the size. How would you compare your model with the stapler? Is it the same length?

John: It's a little bit too long. But if I take some cubes off at the end, then it is too short. So I decided that it was better to be a little too long.

Teacher: That’s a good point, John. Sometimes the cubes don’t match the object exactly. We can’t cut a cube in half so we just have to decide which we think is better. I bet other students had the same problem. What about the height of your model, John? How tall is your model?

John: Three cubes tall. But, it’s weird. At one end, it’s the same size but at the other end, it’s taller.

Teacher: Why is that, John?

John: Well, the stapler kind of slants down.

Teacher: Right, and that was kind of hard to represent using the cubes, wasn’t it?

John: I couldn't. I had to just keep a straight line.

Teacher: What about the width? How wide did you make your model?

John: I just made it one cube. I could have made it two cubes but then it would be too much, I think.

Teacher: So again, you had to make a decision about it. Good work, John. Now putting all that together, would you say that your model has exactly the same volume as the stapler or would you say that it is a little bigger or a little smaller? Is it greater than or less than?

John: Well, I don't know. It's bigger this way [points to length] but it's smaller this way [points to width]. I guess it's a little bit bigger because it's also a little taller.

Teacher: It's hard to say exactly, isn't it? That's why we're calling our work estimates. But I think your reasoning is good, so let's say that your model is a little bit bigger than the stapler. You told me that you used 27 cubes to build your model. So what would be a good estimate for the volume of the stapler?

John: Well, I don't think it's a lot bigger so I still say 27.

Teacher: You say the volume of the stapler is 27. Twenty-seven of what, John?

John: 27 cubes.

Teacher: Yes, it took 27 cubes to make it. But when we talk about the volume being 27, what do we say?

John: I know, we have to say cubic units.

Teacher: That's right. John, we'd say that your estimate of the volume of the stapler is 27 cubic units. Who else made a model of a stapler?

Tanya: I did.

Teacher: How does your estimate of volume compare to John's?

Tanya: I got 25 cubes.

Teacher: Do you mean 25 cubic units?

Tanya: Yes, 25 cubic units.

Teacher: Now, Tanya your estimate is a little smaller than John's. Why do you think that happened? Did one of you make a mistake?

Tanya: No. I did it differently. See, where the top slants down right here? I thought it would be better to take away some cubes there to make it more like the stapler. So that's why I only have 25.

Teacher: That's a reasonable judgment to make, Tanya. So your estimate, based on your model, is 25 cubic units. Remember, estimates are our best guesses. They are not exact. So I'd say both Tanya's and John's estimates are reasonable.

Ask:

X
  • Do you think the volume of your cube model is the same as, slightly less than, or slightly greater than that of the volume of the real object? Tell why. (Responses may vary.)

Responses to the prompt above should include reasoning that explains how the similarities and differences between cube models and real objects were considered in evaluating the volume of the two. Comparing the volume of the cube model to that of the real object helps students understand the concept of estimating. Discussion of these comparisons will help students understand that not only the shape of the object but all dimensions should be considered to make a reasonable estimate and why estimates are not exact. See the Sample Dialog.

Adventure Book:
Student Activity Book: R-10 433 434
Answers
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SG_Mini
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X
SG_Mini
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X
SG_Mini
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Master: