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Unit: Being Green – Minimizing the Surface Area of a Soda Can
I. UNIT OVERVIEW & PURPOSE:
This unit is designed for high school students to understand the relationship between surface area and volume through a social justice application. Students will work in teams as they are introduced to the calculus topic of optimization to minimize the surface area of a cylinder using the volume as a constraint. First, students will measure a soda can and calculate the volume and surface area. Then they will use an Excel spreadsheet to test new dimensions and choose the one which provides the minimum surface area. Students will design a model using their chosen dimensions. Finally, students will prepare an argument for why or why not soda companies should consider their new design.
II. UNIT AUTHOR:
Lauren LaVenture Lord Botetourt High School Botetourt County Public Schools
III. COURSE:
Mathematical Modeling: Capstone Course
IV. CONTENT STRAND:
Geometry
V. OBJECTIVES:
The student will: calculate surface area and volume of a cylinder use technology (Excel spreadsheet) to test and judge optimized dimensions create a net and three dimensional model construct an argument for or against changing the dimensions of a soda can
VI. MATHEMATICS PERFORMANCE EXPECTATION(s):
MPE 2: Collect and analyze data, determine the equation of the curve of best fit, make predictions, and solve real-world problems using mathematical models. Mathematical models will include polynomial, exponential, and logarithmic functions.
MPE 6: The student will use formulas for surface area and volume of three-dimensional objects to solve real-world problems. MPE 7:The student will use similar geometric objects in two- or three-dimensions to a) compare ratios between side lengths, perimeters, areas, and volumes; b) determine how changes in one or more dimensions of an object affect area and/or volume of the object; c) determine how changes in area and/or volume of an object affect one or more dimensions of the object; and d) solve real-world problems about similar geometric objects.
VII. CONTENT:
Through this unit students will see how surface area and volume are related but can be manipulated independently. Students are introduced to a calculus concept of optimization while minimizing the surface area in an effort to reduce waste and live greener. Students will make an argument, targeted to the soda company, to persuade them to change the dimensions of the soda can or maintain the current dimensions.
VIII. REFERENCE/RESOURCE MATERIALS:
Unit handout and rubric for students (attached in the lesson) Soda cans Rulers Calculators Microsoft Excel (example spreadsheet attached in the lesson) Construction paper Scissors Tape Compasses Internet
Participation Rubric (Source: HA Program, Auburn U. Mark Burns, Instructor)
Worksheets created by Kuta Infinite Geometry Software Website: Keep America Beautiful www.kab.org
IX. PRIMARY ASSESSMENT STRATEGIES:
Assessments will be in the form of: Mathematical accuracy Oral presentation of argument Participation Rubric for overall unit performance (attached) All specific questions for the assessments are attached in the respective lessons
X. EVALUATION CRITERIA:
Mathematical accuracy is graded on correctness for the following: Surface area of soda can Volume of soda can Minimized surface area dimensions (Excel) Construction of model *keys provided for each in the lessons The group argument will be assessed on preparation and persuasive strength. There is no right or wrong side. Participation of each individual group member will be evaluated by each member and the instructor using a rubric. Each student will receive a final unit grade using a rubric which contains all of the points mentioned above. The rubric can be seen in the following lessons.
XI. INSTRUCTIONAL TIME:
Regular Schedule: about 5-6 days Block Schedule: about 3- 4 days
Lesson 2: Minimizing the Surface Area
Objectives: Students will use technology (Excel spreadsheet) to test and judge optimized dimensions
Standards: MPE 2 and AII.9: Collect and analyze data, determine the equation of the curve of best fit, make predictions, and solve real-world problems using mathematical models. Mathematical models will include polynomial, exponential, and logarithmic functions. MPE 7 and G.14: The student will use similar geometric objects in two- or three-dimensions to b) determine how changes in one or more dimensions of an object affect area and/or volume of the object; c) determine how changes in area and/or volume of an object affect one or more dimensions of the object; and d) solve real-world problems about similar geometric objects. NCTM Geometry: analyze properties and determine attributes of two- and three-dimensional objects;
Materials: Microsoft Excel spreadsheet – at least 1 per group
Process:
- Before class starts create the Excel spreadsheet as shown below and send the file to each group or each student
- Before students begin review what they learned from the previous lesson: the volume of the soda can needs to remain 17.3 in^3 but we want to minimize the surface area by adjusting the dimensions of radius and height. Explain the height column of the spreadsheet is a formula for height given a radius value and the constant volume. Optional: students can derive the height formula and compare to the spreadsheet or students can create the spreadsheet as a team.
Formulas: Height column = (17.29/(3.14159(B3)^2)) Surface Area column = =(23.14159B3^2)+(23.14159B3C3)
- Students will type a “test” radius value and the spreadsheet will fill in the correct values for height and surface area. Students will continue to test radius values until they are confident they have found the minimal surface area to the nearest tenth. Below is an example of a team’s spreadsheet.
- Students should create a scatterplot to compare radius values to surface area to verify their minimal dimensions.
- Students will have Excel calculate the curve of best-fit and students should answer the following questions (answers provided) Is the equation of best-fit a line? No What type of equation is the best-fit? Quadratic What is the equation? y = 22.94x^2 - 64.218x + 81. Locate your minimal surface area, what is this point in respect to the quadratic? Minimum Explain, in your own words, the relationship between the graph of the quadratic and finding the optimized surface area of a soda can.
y = 22.94x^2 - 64.218x + 81. R² = 0.
37
38
39
40
0 0.5 1 1.5 2
Surface Area
Radius (in) Poly. (Radius (in))
Figure 1: This is the net of the model with the specific lengths labeled
Figure 2: This is the net folded and taped up to represent the optimized dimensions of the soda can
Lesson 4: Choose a Side and Present an Argument
Objectives: Students will construct an argument for or against changing the dimensions of a soda can
Standards: MPE 7 and G.14: The student will use similar geometric objects in two- or three-dimensions to b) determine how changes in one or more dimensions of an object affect area and/or volume of the object; c) determine how changes in area and/or volume of an object affect one or more dimensions of the object; and d) solve real-world problems about similar geometric objects.
NCTM Geometry: Analyze properties and determine attributes of two- and three-dimensional objects; Visualize three-dimensional objects and spaces from different perspectives and analyze their cross sections; Use geometric models to gain insights into, and answer questions in, other areas of mathematics; Use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture.
Materials: Soda cans Models Internet for research Presentation software such as Microsoft Power Point Participation rubric – each student will need one for each group member including themselves (appendix 4) Unit rubric 1 for each student (appendix 5)
Process:
- Students should first compare and contrast their model with the soda can. This list can include appearance, marketing, shelf space, packaging, ease of use, material cost, material waste and any other characteristics students notice. See figure 3 on the next page for a visual comparison.
- As a team, students must decide if soda companies should change the dimensions of the soda can or keep the current dimensions.
- As a team, students will prepare an argument for their decision. The argument should be targeted to the soda company and so should appeal to the concerns of a business (cost, productivity, marketing). Students should have access to the internet to conduct any necessary research. Students should make use of mathematics and the difference in material cost between the two options or in the shelf space difference between the two options. For example: the optimized dimensions reduce the surface area and likewise the material to make the soda can by 2.6 in^2 per can. Also, the optimized soda can will increase the shelf space by 2.
Appendix 1: Student Unit Handout
Being Green: Minimizing the Surface Area of a Soda Can
You and your teammates are concerned citizens. You see so many soda cans being thrown away or tossed on the side of the road and you want to do something about it. Sure you could promote a recycling program (which is a great idea) but you want to make a difference that will help the cans carelessly strewn along the roadway. You and your teammates will test if there is another design for the soda can that is still cylindrical and still holds 12 fluid ounces, but reduces waste. Once the new design is made, you and your team will need to construct a convincing argument for the soda company to either alter the current soda can or keep it the same.
- Find the dimensions of the current soda can and calculate the volume and surface area
- Use an Excel spreadsheet to find the optimal dimensions that will hold the current volume constant but minimize the surface area
- Construct a model using the optimized dimensions – you will first construct a net.
- Compare and contrast the soda can and the model. Consider the following: appearance, marketing, shelf space, packaging, ease of use, material cost, material waste and any other characteristics
- Decide, as a team, if you should convince the soda company to use the new dimensions or keep the current dimensions
- Construct an argument targeted to the soda company. Consider cost, productivity and marketing. You will have class time to conduct research. Your argument should use the shelf space difference between the two options.
- Present your argument as a team to the class
Acceptable Mathematical Accuracy
Accurately measured the soda can Accurately calculated the volume Accurately calculated the surface area Accurately found the optimized dimensions Accurately constructed the model
Acceptable: 10 points Borderline: 8 points Inadequate: 5 points Unacceptable: 2 points
Argument Created a list of at least 6 comparisons Argument has at least 3 statements At least one statement is mathematically based beyond the project Argument was designed to persuade a soda company
Acceptable: 15 points Borderline: 12 points Inadequate: 8 points Unacceptable: 4 points
Presentation Each member had a role in the presentation Presentation was clear and persuasive
Acceptable: 10 points Borderline: 8 points Inadequate: 5 points Unacceptable: 2 points Participation Determined by the average ratings given by yourself, teammates, and your teacher in the following categories: contribution, attitude, encouragement of others, listening to others, making decisions, knowledge, organization
Acceptable: 5 points Borderline: 4 points Inadequate: 3 points Unacceptable: 2 points
Appendix 2: Kuta Infinite Geometry Software Surface Area and Volume Worksheet
Answer Key:
Answer Key:
Appendix 4: Participation Rubric Source: HA Program, Auburn U. Mark Burns, Instructor
Partner Being Evaluated: ___________________ Name of Evaluator: _________________
Quantitative Evaluation (Circle your response in each category.)
Contributed
at the meetings
Criticized with
no solutions offered
Initiated ideas
Sat back & let others
do the hard work
Accepted
responsibility
Kept quiet,
hoping to avoid tasks
Delivered
on promises
Late, or didn't
get work done
Positive attitude
Complained a lot
Organized
Scattered and
unproductive
Prepared & helped
make decisions
Unprepared and
disruptive
Knowledgeable
Weak conceptual/practical
Background
Encouraged
everyone’s participation
Tried to dominate
group discussions
