Unit 3, Lesson 1
Minimizing Surface Area
Integrated Math 3
1.1
Warm-up
Instructional Routines
NoneMaterials
NoneActivity Narrative
The purpose of this activity is to introduce students to the central problem of the lesson: how to calculate which cylinder takes the least material to build for a specific volume. Before students begin working with surface areas of cylinders, they are explicitly asked to estimate which cylinder they think needs the least amount of materials to make. They will revisit these cylinders during the Lesson Synthesis. Making a reasonable estimate and comparing a computed value to one’s estimate is often an important aspect of making sense of problems (MP1).
Launch
Tell students to close their books or devices (or to keep them closed), and display the problem stem and image of the circular cylinders for all to see. Ask students to estimate which cylinder they think needs the least amount of material to build, which includes the top, bottom, and sides, and give a brief quiet think time. Invite each student to vote, and display the results of the vote throughout the lesson. If any students think that two more of the cylinders take the same amount of materials to make, record their responses as well.
Activity
NoneHere are four circular cylinders that have the same volume.
- Which cylinder needs the least material to build? Explain your reasoning.
- What information would be useful to know to determine which cylinder takes the least material to build?
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is to make sure students remember how to calculate the volume and surface area of a cylinder.
Select students who chose A, B, C, D, or another option to share their reasoning. If not brought up during the discussion, discuss how to calculate the volume of a cylinder and the surface area of a cylinder , and write these formulas where they can remain displayed throughout the lesson. Consider using a rolled piece of paper to demonstrate the rectangular part of the surface area. If students are unsatisfied with not knowing which of the 4 has the smallest surface area, assure them that we will revisit the problem at the end of the lesson.
Student Lesson Summary
Some relationships cannot be described by polynomial functions. For example, let’s think about the relationship between the radius , in centimeters, and the surface area , in square centimeters, of the set of cylinders with a volume of 330 cm3 (this is a volume of 330 mL). What radius would result in the cylinder with the minimum surface area?
We know these formulas are true for all cylinders with radius , height , surface area , and volume :
Since we are only interested in cylinders with a volume of 330 cm3, we can use the volume formula to rewrite the surface area formula as:
Can you see how? We can use the volume formula rearranged as and then substitute for in the formula for surface area.
We now have an equation giving as a function of for cylinders with a volume of 330 cm3. From the graph of shown here, we can quickly identify that a radius of about 3.75 cm results in a cylinder with minimum surface area and a volume of 330 cm3.
is an example of a rational function. Rational functions are fractions with polynomials in the numerator and denominator. Polynomial functions are a type of rational function with 1 in the denominator.
Graph of a rational function with a point (3 point 46 comma 264 point 312), origin O. Horizontal axis labeled r, scale 0 to 8 by 2’s. Vertical axis labeled S, scale 0 to 600, by 200’s.
In this situation, the height of a cylinder with fixed volume varies inversely with the square of the radius, , which means that, as the value of increases, the value of the height decreases, and vice versa. In later lessons, we’ll learn more about different features of rational functions, like why their graphs can look like they are made of two separate curves.