The purpose of this lesson is for students to see some examples of linear functions, in this case proportional ones, that arise out of geometry. In an earlier lesson, students considered the relationship between the volume of water inside a graduated cylinder and the height of the water. In that example, the height was a function of the volume, and the relationship was proportional.

In the Warm-up, students study the graph of a proportional relationship and recall that in a proportional relationship, the two quantities change by the same scale factor: when one of them is multiplied by a scale factor, the other one gets multiplied by the same scale factor.

Next, students consider rectangular prisms that share two edge lengths but have a third length that varies. They graph prism volume as a function of the third edge length and see that the volume is proportional to the length. They conclude that when the length doubles, the volume doubles.

Then they investigate volume as a function of the height for cylinders with a fixed radius. Again they see that the volume is proportional to the height, and that when the height is halved, the volume is halved. During this activity, students have opportunities to explain their reasoning and critique the reasoning of others (MP3).

The final activity is optional as it contains content beyond grade level. Consider using this activity to give students additional practice working with the formula for the volume of a cone.