Unit 1, Lesson 7
Scaling Solids
Integrated Math 3
7.1
Warm-up
Materials
NoneActivity Narrative
This Math Talk focuses on finding volume. It encourages students to think about efficient strategies to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students analyze the effect of scaling on cube volumes.
In this activity, students have an opportunity to notice and make use of structure (MP7) as they relate the word “cube” to both exponents and geometric solids.
Launch
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
- Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
- Invite students to share their strategies, and record and display their responses for all to see.
- Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
This is the first time students do the Math Talk instructional routine in this course, so it is important to explain how it works before starting.
Explain that a Math Talk has four problems, revealed one at a time. For each problem, students have a minute to quietly think and should give a signal when they have an answer and a strategy. The teacher then selects students to share different strategies (likely 2–3, given limited time) and might ask questions, such as “Who thought about it in a different way?” The teacher then records the responses for all to see, and might ask clarifying questions about the strategies before revealing the next problem.
Consider establishing a small, discreet hand signal that students can display when they have an answer they can support with reasoning. Signals might include a thumbs-up or a certain number of fingers that tells the number of responses they have. Using signals is a quick way to see if students have had enough time to think about the problem. It also keeps students from being distracted or rushed by hands being raised around the class.
Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization
Supports accessibility for: Memory, Organization
Activity
NoneFind the volume of each cube mentally.
Activity Synthesis
To involve more students in the conversation, consider asking:
- “Who can restate ’s reasoning in a different way?”
- “Did anyone use the same strategy but would explain it differently?”
- “Did anyone solve the problem in a different way?”
- “Does anyone want to add on to ’s strategy?”
- “Do you agree or disagree? Why?”
- “What connections to previous problems do you see?”
Remind students that when we multiply a number by itself 3 times, that’s called "cubing" the number. We can use an exponent to indicate cubing. For example, . Ask students why we call it “cubing.”
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. Examples:
Advances: Speaking, Representing
- First, I _____ because _____.
- I noticed _____, so I _____.
Advances: Speaking, Representing
Lesson Synthesis
In this lesson, students studied the effect of dilation on the surface area and volume of solids. Here are some questions for discussion:
- “If we dilate a rectangular pyramid by a factor of 5, how do the side lengths change? How does the surface area change? How about the volume?” (The side lengths are all multiplied by 5. The surface area is multiplied by 25, which is 52, and the volume is multiplied by 125, which is 53.)
- “What are some situations in which a solid is filled with something different from its surface material? What makes up the interior and the surface in these situations?” (Sample responses: A can of soup has metal on its surface and soup filling its interior. A tire has rubber on its surface and air inside. Phone screens are related to surface area, while batteries, processors, and other components are related to volume.)
Add the following theorem to the class reference chart, and ask students to add it to their reference charts:
When any solid is dilated using a scale factor of , all lengths are multiplied by , all areas are multiplied by , and all volumes are multiplied by . (Theorem)
Student Lesson Summary
In earlier activities, we saw that if we dilate a two-dimensional shape, the area of the dilated shape is the area of the original shape multiplied by the square of the scale factor. What happens when we dilate three-dimensional solids?
Here is a rectangular prism with side lengths 3, 4, and 5 units. When we dilate the prism using a scale factor of 3, the lengths become 9, 12, and 15 units.
Because these are three-dimensional shapes, we can look at both volume and surface area. The volume of the original prism is 60 cubic units because . The volume of the dilated prism is 1,620 cubic units because . The volume became 27 times larger! Why? Because the side lengths tripled, when we calculated the volume we were really finding . The volume was multiplied by the cube of the scale factor, or by .
Now let’s look at surface area. How will the surface area change when the prism is dilated?
In the original prism, 2 faces have area 12 square units, 2 have area 20 square units, and 2 have area 15 square units for a total surface area of 94 square units. The corresponding faces of the dilated prism have areas 108, 180, and 135 square units. The surface area of the new prism totals 846 square units, 9 times that of the original. Just like the area of two-dimensional shapes, the surface area of the prism changed by the square of the scale factor.
In general, when we dilate any three-dimensional solid by a scale factor of , the surface area is multiplied by , and the volume is multiplied by .