# Lesson 16

Minimizing Surface Area

### Lesson Narrative

In this lesson, students are introduced to rational functions through the context of calculating the minimum surface area for a can with a specific volume. Rational functions are those defined by a fraction with polynomials in the numerator and denominator. This lesson purposely echoes the first lesson of the unit in which students calculated the volume of a box created from a single sheet of paper. This encourages comparing rational and polynomial functions during the lesson before revealing at the end that polynomials are a specific type of rational function. Just as the first lesson gave students a visual and hands-on way of understanding polynomials, this lesson gives students a geometric way of understanding rational functions. Students will see other examples of situations that rational functions model in the following lessons.

Students begin the lesson making sense of the central problem by considering which of 4 cylinders with the same volume and different dimensions would take the least amount of materials to build (MP1). Students use estimation before calculating the answer directly. Making reasonable estimations to put upper and lower bounds on possible outcomes of a problem is an important skill for mathematical modeling (MP4). Next, they consider the inverse relationship between the radius and height for cylinders of a specific volume. They calculate heights for several cylinders given a radius, and this repeated reasoning encourages students to rearrange the formula for the volume of a cylinder to suit their needs (MP8). In the last activity, students build an equation for the function relating the radius and surface area of a cylinder with a specific volume and use the equation and graph to answer the original question: which cylinder would take the smallest amount of materials to build?

### Learning Goals

Teacher Facing

• Create a rational function to model the surface area of a cylinder of known volume.
• Interpret graphs of rational functions in context.

### Student Facing

• Let’s investigate surface areas of different cylinders.

### Required Preparation

Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)

### Student Facing

• I can write a rational function to model different properties of cylinders.

Building On

Building Towards

### Glossary Entries

• rational function

A rational function is a function defined by a fraction with polynomials in the numerator and denominator. Rational functions include polynomials because a polynomial can be written as a fraction with denominator 1.