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Scaling Two Dimensions

Lesson # 18

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2019 Open Up Resources |

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Let’s change more dimensions of shapes.

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Today’s Goals

I can create a graph representing the relationship between volume and radius for all cylinders (or cones) with a fixed height.
I can explain in my own words why changing the radius by a scale factor changes the volume by the scale factor squared.

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Tripling Statements

Warm Up 18.1

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Work quietly (1-2 min)
Discuss your statements with your partner

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Let’s Talk About It

If a, b, and c are all tripled, the expression becomes 3a+3b+3c, which can be written as 3(a+b+c) by using the distributive property to factor out the 3. So if all the addends are tripled, their sum, m, is also tripled.
Looking at the third statement, if a is tripled, the expression becomes 3a(bc) , which, by using the associative property, can be written as 3(abc) . So if just a is tripled, then n, the product of a, b, and c is also tripled.

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A Square Base

Activity 18.2

MLR8: Discussion Supports

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Work quietly (5-7 min)

Clare sketches a rectangular prism with a height of 11 and a square base and labels the edges of the base s. She asks Han what he thinks will happen to the volume of the rectangular prism if she triples s .

Han says the volume will be 9 times bigger. Is he right? Explain or show your reasoning.

Select previously identified students to share whether they think Han is correct. If possible, begin with students who made sketches of the two rectangular prisms to make sense of the problem.

Ask students: “If this equation was graphed with edge length s on the horizontal axis and the volume of the prism on the vertical axis, what would the graph look like?” Suggest that they complete a table showing the volume of the rectangular prism when s equals 1, 2, 3, 4, and 5 units (the corresponding values of volume are 11, 44, 99, 176, and 275 cubic units) and then sketch a graph using these points. Give students quiet work time and then select a student to display their table and graph for all to see. Ask students what they notice about the graph when compared to the graphs from the previous lesson (the graph is non-linear—the volume increases by the square of whatever the base edge-length increases by).

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Are you ready for more?

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Playing with Cones

Activity 18.3

MLR3: Clarify, Critique, Correct

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Work quietly (4-7 min)

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Let’s Talk About It

If the radius was quadrupled (made 4 times as large), how many times as large would the volume be?
If the radius was halved, how many times as large would the volume be?
If the radius was scaled by an unknown factor a, how many times as large would the volume be?

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Lesson Synthesis

What do these graphs represent? How are these graphs similar? Different?
Think about what happens when a cube’s edge lengths are doubled or tripled. What happens to the volume?
Why do you think changing the radius of a cylinder results in a graph that is not proportional?

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Today’s Goals

I can create a graph representing the relationship between volume and radius for all cylinders (or cones) with a fixed height.
I can explain in my own words why changing the radius by a scale factor changes the volume by the scale factor squared.

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Halving Dimensions

Cool Down 18.4