Title: Can You Can a Can?
A discovery/exploration lesson investigating the production of a cylindrical can of a given volume with the least amount of material, thus minimizing the cost of production. Links to Outcomes:
Problem Solving
Students will use problem solving to investigate the possibilities for dimensions of a cylindrical can, given its volume. • Communication
Students will discuss the geometric concepts of area and volume and present conclusions in a written report. • Reasoning
Students will gather data, make conjectures, and build arguments based on the interpretation of a graph. • Connections
Students will make connections between algebra and geometry, in a real-world situation. • Measurement
Students will use measurement to obtain data which will be used to draw conclusions. • Geometry
Students will use the geometric properties of a figure to write an equation for a mathematical model. • Algebra
Students will represent a real-world situation as an equation and reach conclusions from interpreting its graph. • Technology
Students will use a computer software package and/or a graphics calculator in their investigation. • Cooperation
Students will demonstrate the ability to investigate mathematics in groups of two or three. Brief Overview:
Why is a Campbell’s soup can shaped as it is? Why not pack soup in a tuna can? In this lesson, the student will investigate volume and surface area of a can to determine the dimensions of the “perfect” can - i.e., a can which requires the least amount of material for a given volume. Data will be collected to compare several sizes of cans - is the can a “perfect can?” If not, why? Grade/Level:
Duration/Length:
This lesson is expected to take 1 to 1½ days, depending on the ability and/or level of the class. Prerequisite Knowledge:
Students should be familiar with formulas to find area of a circle, surface area of a cylinder, and volume of a cylinder. Students should be skilled in using Derive and/or the TI-82 calculator. Objectives:
• measure height and circumference of a cylindrical can. • relate circumference of a circle to radius in order to find the radius of the can. • use the formula for volume of a cylinder to compute the volume of a can. • use the formulas for surface area and volume of a cylinder to write an equation which models surface area as a function of dimensions of the cylinder. • use Derive and/or the TI-82 calculator to make computations and graph model. • use the graph to determine if the can is a “perfect” can. • investigate the rationale for a can’s not being “perfect.” • pool data and make conjectures about the dimensions of the cans found on the grocers’ Materials/ Resources/ Printed Materials:
Development/Procedures:
Students should be instructed to bring to class a canned food item the day the activity is
presented. Tell students that, whether large or small, cans they bring must be “regular”
cylindrical cans (i.e. cans with perfectly straight sides and a flat top and bottom). On the day
the activity is to be presented, the teacher should bring two cans of roughly equal volume (not
weight): a short, squat can (like tuna fish comes in) and a taller can (a soup can, for example).
The teacher should determine the radius and height of the two cans. Activity 1 is a guided
demonstration led by the teacher. Activity 2 is a related assignment intended to be completed
by the student, working individually or with a partner. Activity 3 is an extension that
investigates the rationale for the size and shape of a particular can.
Activity 1: Investigating the relationship between dimensions and surface area for a
cylinder of given volume.
1. Background: The teacher should ask the class to imagine that they are going into
Purpose: To determine if the can that the student brought is a “perfect” can, i.e., if the can
was constructed using the minimum material for the volume it contains.
Name _________________________________
Date ________________________
The Perfect Can
For Activity 2, the teacher will circulate around the room to ensure that students are on task
and to answer any questions about the activity. Student worksheets for this activity will be
collected and assessed.
• Students will investigate why cans are sized and shaped as they are by contacting the manufacturer. (Activity 3)
• Students will go to a grocery store and make a list of dimensions and volumes of the cans in one area, say canned fruits. Other students will go to other food sections. Pool data and make conjectures about why cans are produced as they are. Perform statistical analysis appropriate to the course level (i.e., frequency distribution, deviation from the perfect can, etc.). Authors: