# Lesson 18

Scaling Two Dimensions

### Lesson Narrative

This is lesson optional. The previous lesson explored some proportional relationships that arise when we consider the volume of a rectangular prism or cone as a function of one of its dimensions, such as side length or height. Students studied what happens to the volume of the figure when you scale that dimension. In this lesson they see what happens to the volume when you scale two of the dimensions. They consider a rectangular prism on a square base where you keep the height constant and vary the side length of base, and a cone where you keep the height constant and vary the radius of the base. In both cases you are really varying two dimensions, because both the length and the width of the base change at the same time. As in the previous lesson, they consider what happens when you scale the side length or the radius by a particular factor, and this time they discover that the volume scales by the square of the factor. For example, if you triple the side length of the square base of the prism, you multiply the volume by 9, which is \(3^2\). In general, if you scale the side length by \(a\), you multiply the volume by \(a^2\).

The main purpose of this lesson is to understand that if you scale two of the dimensions of a three-dimensional figure by the same factor, the volume scales by the square of that factor. A secondary purpose is to see some interesting examples of non-linear functions arising from geometry.

### Learning Goals

Teacher Facing

- Compare and contrast (orally) graphs of linear and nonlinear functions.
- Create an equation and a graph representing the volume of a cone as a function of its radius, and describe (orally and in writing) how a change in radius affects the volume.
- Describe (orally and in writing) how changing the input of a certain nonlinear function affects the output.

### Student Facing

Let’s change more dimensions of shapes.

### Learning Targets

### Student Facing

- 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.