Lesson 18Scaling Two Dimensions
Learning Goal
Let’s change more dimensions of shapes.
Learning Targets
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.
Warm Up: Tripling Statements
Problem 1
Activity 1: A Square Base
Problem 1
Clare sketches a rectangular prism with a height of 11 and a square base and labels the edges of the base
Han says the volume will be 9 times bigger. Is he right? Explain or show your reasoning.
Are you ready for more?
Problem 1
A cylinder can be constructed from a piece of paper by curling it so that you can glue together two opposite edges (the dashed edges in the figure).
If you wanted to increase the volume inside the resulting cylinder, would it make more sense to double
, , or does it not matter? If you wanted to increase the surface area of the resulting cylinder, would it make more sense to double
, , or does it not matter? How would your answers to these questions change if we made a cylinder by gluing together the solid lines instead of the dashed lines?
Activity 2: Playing with Cones
Problem 1
There are many cones with a height of 7 units. Let
Write an equation that expresses the relationship between
and . Use 3.14 as an approximation for . Predict what happens to the volume if you triple the value of
. Graph this equation.
What happens to the volume if you triple
? Where do you see this in the graph? How can you see it algebraically?
Print Version
There are many cones with a height of 7 units. Let
Write an equation that expresses the relationship between
and . Use 3.14 as an approximation for . Predict what happens to the volume if you triple the value of
. Graph this equation.
What happens to the volume if you triple
? Where do you see this in the graph? How can you see it algebraically?
Lesson Summary
There are many rectangular prisms that have a length of 4 units and width of 5 units but differing heights. If
The equation shows us that the volume of a prism with a base area of 20 square units is a linear function of the height. Because this is a proportional relationship, if the height gets multiplied by a factor of
What happens if we scale two dimensions of a prism by a factor of
For example, think about a prism with a length of 4 units, width of 5 units, and height of 6 units. Its volume is 120 cubic units since
A similar relationship holds for cylinders. Think of a cylinder with a height of 6 and a radius of 5. The volume would be
Why does the volume multiply by