Scaling One Dimension
This lesson is optional. This is the first of a series of lessons building up to the formula for the volume of a sphere. In order to understand why that formula has the radius raised to the third power, students start studying how the volume of a three-dimensional figure changes when you scale one or more of its dimensions (length, width, height, radius). In this lesson they consider just one of the dimensions.
In the warm-up, students graph a proportional relationship and recall that in a proportional relationship the two quantities change by the same scale factor: when you multiply one of them by a scale factor the other one gets multiplied by the same scale factor. In the first activity, students consider a rectangular prism with two edges of constant length and one edge of variable length. They graph the volume of the prism as a function of the length and see that the volume is proportional to the length. They conclude that when you double the length the volume doubles. Then they investigate the volume of a cone as a function of its height when you keep the radius constant. Again they see that the volume is proportional to the height, and that when you halve the height you halve the volume. In the final activity they use a graph of this proportional relationship to find the radius.
The main purpose of the lesson is to understand that when you scale just one of the dimensions of a three-dimensional figure by a factor, the volume scales by the same factor. A secondary purpose is to see some examples of linear functions arising out of geometry. (A proportional relationship is a particular kind of linear function.)
- Create a graph and an equation to represent the function relationship between the volume of a cylinder and its height, and justify (orally) that the relationship is linear.
- Interpret (in writing) a point on a graph representing the volume of a cone as a function of its height, and explain (orally) how changing one dimension affects the other.
Let’s see how changing one dimension changes the volume of a shape.
- I can create a graph the relationship between volume and height for all cylinders (or cones) with a fixed radius.
- I can explain in my own words why changing the height by a scale factor changes the volume by the same scale factor.
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