Math Models: Sample lesson
Tom Hughes, 1999
Models of a Vending Machine Business
Lesson Overview
Students take on different roles in simulating
starting a vending machine business in their
school that will generate a profit.
TEKS
(1) The student uses a variety of strategies and
approaches to solve both routine and non-routine
problems.
(A) compare and analyze various methods for solving
a real-life problem.
(C) select a method to solve a problem, defend the
method, and justify the reasonableness of the results.
Mathematics Overview
simultaneous linear equations, depreciation,
amortization, statistical probability
(sampling)
Materials
Handouts with data, business roles and other
information necessary for students to get
started with the simulation.
(3) The student develops and implements a plan for
collecting and analyzing data in order to make
decisions.
(A) formulate a meaningful question, determine the
data needed to answer the question, gather the
appropriate data, analyze the data, and draw
reasonable conclusions.
(B) communicate methods used, analysis conducted,
and conclusions drawn for a data-analysis project by
written report, visual display, oral report, or multimedia presentation.
(C) determine the appropriateness of a model for
making predictions from a given set of data.
(6) The student uses algebraic formulas, graphs, and
amortization models to solve problems involving
credit.
(A) analyze methods of payment available in retail
purchasing and compare relative advantages and
disadvantages of each option
(C) use amortization models to investigate
automobile financing and compare buying and leasing
a vehicle.
Lesson Resources
spreadsheet program and/or
graphing calculator
Related Resources
Guides for small businesses
Amortization tables
contacts with local vending businesses.
Tom Hughes, 1999
1
1. Setup (to set the stage and motivate the students to participate)
1.1. Most schools have snack and soda machines, so this should be a familiar business context.
Have the class estimate the gross income for the vending machines in the school. This is how
much money is spent on snacks and soda each day by the students. Students may want to
interview friends outside of class as to how many snacks and soda they buy per week and
what brands. It may be very powerful to compare these statistical predictions with the actual
buying patterns at the school assuming the school does having vending machines.
1.2. While there are many different approaches to this kind of simulation, one possibility is to
have each group of students take on the role of the partners in the business. Depending on
their inclination and the instructor’s, they could represent the student council, the business
manager of the school, or someone starting up their own business by providing service to the
school.
1.3. To set up the mathematics for this simulation, you may wish to have students brainstorm
about the different kinds of expenses and service issues they might have to consider in the
course of the simulation.
1.4. To focus attention on TEK #6, let the students know that one decision they must make is
whether to purchase the machines via a loan (which leads to the concept of amortization)
versus leasing the machines. Choosing to buy the machines will probably make the
simulation more complicated and interesting if you have sufficient class time to deal with it.
Teacher's notes (to personalize the lesson for your classroom):
2. Lesson Outline
2.1. Set up the lesson and outline the general parameters of the business.
2.1.1. One vending machine operator uses the rule of thumb that soda machines have about 5
times the business as snack machines (and also have no issues of spoilage).
2.1.2. Another rule of thumb is that 30% of the people who go by the vending machines in a
business situation will buy something in the course of a day.
2.1.3. New machines cost from $2500 to $6000 and have an effective lifetime of at least 7
years.
2.1.3.1.
There are many vending machine companies with Web pages (search for
vending machines). This may be a rich source of information for student projects.
2.1.3.2.
Students should try to find the cost of specific machines that exist in the school or
in other places where they might buy soda or snacks.
Tom Hughes, 1999
2
2.1.4. Coca-Cola rents soda machines at a fairly low cost ($20 per month) but you must then
buy your soda from them. They absorb all repair costs but will take out any machines
that have been vandalized more than twice.
2.1.5. Repair costs can vary greatly. Coca-Cola charges $56 per hour plus travel. One of the
most common parts to repair or replace is the dollar bill validator which costs $125 - $175
each plus labor.
2.1.6. Coca-Cola sells soda for $7.70 a case (24 cans) if you buy by the truckload. One truck
has 24 pallets with 96 cases per pallet. If you buy 5 truckloads per month, their price
drops to $6.00 per case.
2.1.7. If you own your own machines you can buy soda by the case from places like Sam’s
Club for slightly less than $7.00 per case (but they don’t deliver).
2.1.7.1.
This is a good area for student investigation into the costs of soda in your area.
Mathematically, exploring a cost difference of $1.00 per case in terms of the business
model may make students more sensitive to multiplicative factors.
2.2. Do a basic cost/profit calculation as part of a break-even analysis.
2.2.1. This can start off as a very simple analysis, comparing the gross profits on the machines
(gross income minus cost of the snacks and soda) versus the cost of buying or leasing the
machines. This introduces the use of simultaneous equations and the business concept of
a breakeven analysis. The students may find this most meaningful as a graph of income
vs. number of cases sold with the fixed expense plotted as a horizontal line.
2.2.2. An alternative set of simultaneous equations (both dependent on volume) is to plot
both the gross income and the expenses versus the sales volume.
2.2.3. Depending on how much time you wish to spend on the lesson, you can have students
add in estimates for repair and other service factors while revising their business model.
2.3. Using depreciation and amortization, decide whether to lease or buy machines.
2.3.1. While some sample costs are given above, different groups could work with different
sets of parameters that would lead to different conclusions.
Teacher's notes (to personalize the lesson for your classroom):
3. Guiding Questions (to engage students in mathematical thinking during the lesson)
3.1. What factors determine the amount of gross income?
3.1.1. Volume of sales. Is this price dependent? (an opportunity for a survey question)
3.1.2. Price per unit. Do you sell sodas at $.50, $.75, $1.00? What is the impact on income?
3.1.3. What products sell best? In what proportions?
3.1.4. How many individual machines are needed? Justify your answer with math.
Tom Hughes, 1999
3
3.2. What factors determine the expenses of operating this kinds of business.
3.2.1. Which factors depend on volume of sales and which are volume independent? This
can be a good example of slope and y-intercept in a linear equation.
3.3. How do you find a breakeven point?
3.3.1. Graphs are probably the best tool for the initial analysis.
3.3.2. This is a great opportunity to have students use a graphical representation of an
inequality. Students could color in the regions of profitability and net loss in green and
red, for example.
3.4. Can you believe survey results?
3.4.1. If you can spend the time you may want to work with probability.
3.4.2. If students find that half of their survey population buys one soda per day, how does
that scale up for the whole school? A better analysis may categorize students as drinking
no sodas, 1 per day, 2 per day, etc. This creates a nice problem in estimating the daily
sales volume in the school.
3.5. What are the basic issues of leasing versus owning?
3.5.1. Given a cost per machine and an interest rate, students can work out an amortization
schedule on a spreadsheet or iteratively on a calculator.
3.5.2. A more complex problem can be created by adding in repair costs and depreciation
factors.
3.6. What if …?
3.6.1. Students need to explore the boundaries of this situation. What if sales volume doubles
over the next two years? What if more machines are needed? What if sales don’t match
expectations? This may be a good brainstorming activity for students which leads to
them creating their own problems.
Teacher's notes (to personalize the lesson for your classroom):
4. Summary Questions (to direct students' attention to the key math in the lesson)
4.1. What were the best mathematical tools for analyzing this business situation?
4.2. What is the best way to present your analysis so others will be convinced ?
4.3. Can you trust surveys to give you realistic probabilities that something will happen?
Teacher's notes (to personalize the lesson for your classroom):
5. Assessment Task(s) - (to evaluate what students have learned)
5.1. Final report detailing the students’ financial analysis of the business.
Tom Hughes, 1999
4
5.2. Class presentations
5.3. A quiz on specific pieces of the analysis: graphs of simultaneous equations, spreadsheets, etc.
5.4. Concept map of business financial terms and of this business in particular focusing on the
relationship to expense and income.
Teacher's notes (to personalize the lesson for your classroom):
6. Additional activities and projects
6.1. Get someone from a small business to talk about how they analyze a situation. Having
someone from a vending machine business would be ideal.
6.2. Do a presentation for another class in the school. An economics class would be ideal.
6.3. Find college students involved in economics or business to come to your class as consultants.
Teacher's notes (to personalize the lesson for your classroom):
7. Extension(s) (to lead students to connect the mathematics learned to other situations, both within
and outside the classroom.)
7.1. Have students interview someone in the business community about how they deal with
forecasts of increased and decreased demand.
7.2. Have students talk to someone who creates or makes use of market surveys.
Teacher's notes (to personalize the lesson for your classroom):
8. Author’s notes
8.1. Overview
8.1.1. Types of decisions students are expected to make in this simulation:
8.1.1.1.
Lease the machines or buy them (do a cost comparison of leasing vs.
amortization).
8.1.1.2.
Survey of students to help decide what products will sell best and at what
volume..
8.1.1.3.
Overall breakeven analysis at different sales volumes and pricing schemes
8.2. Objectives
8.2.1. Students will learn the basic elements of small business finance: depreciation,
amortization, break-even analysis.
8.2.2. Students will use graphs and spreadsheets to evaluate situations and make
mathematically informed decisions.
Tom Hughes, 1999
5
8.2.3. Students will conduct surveys to determine product preferences, predicted volume,
pricing, and other issues.
8.2.4. Students will interpret this statistical data by scaling their results up to the whole school.
8.2.5. Students will integrate these concepts by making a final report and/or class
presentation.
8.3. Materials required; activity preparation
8.3.1. Students will benefit from basic knowledge of a spreadsheet program with access to
computers as needed.
8.3.2. The teacher may wish to locate sources of data or create data for this simulation beyond
what is provided in this document.
8.3.2.1.
There are many vending machine companies that have Web sites.
8.3.2.2.
Most schools have vending machines and the companies involved may be
willing to share basic facts about the business with the students.
8.3.2.3.
Many of the issues in this simulation are common to many small businesses.
Having a business owner talk about how they set up a business model may provide
some motivation for students.
8.3.2.4.
There are many publications for small business owners. Finding examples of the
same mathematics in these publications may provide some motivation.
9. Prepared class materials (at the end of this document)
9.1 Project overview for students
9.2 Introduction to surveying
10. If you use this lesson you can help other teachers:
10.1 Send samples of student work:
10.1.1 Paper copies: Mail to: Tom Hughes, UT Dana Center, 2901 N. IH-35, Austin, TX 78722
or FAX Attn: Tom Hughes, 512-232-1855
10.1.2 Computer files: Send as E-mail attachments to TBH1@mail.utexas.edu
10.2 Send copies of class handouts you have created (as above).
10.3 Send copies of assessment materials: Rubrics, quizzes, presentation requirements, etc.
10.4 Brainstorm ideas for this project and others on the Math Models listserv (shared mailing list).
Not on the MM mailing list? Send E-mail to Cynthia Schneider cschneider@mail.utexas.edu
at the Dana Center and she will put you on the list.
Tom Hughes, 1999
6
10.5 Suggestions for improving the project or the materials can be sent in by any of the means
above.
Tom Hughes, 1999
7
Models of a Vending Machine Business
Have you ever wanted to run your own business? Most schools and businesses have snack and
soda machines in the building and they make money. How much do you think students in your
school spend every day on soda and snacks? Could you underbid the current service company and
still make some money? It takes some money to start a business and banks are willing to lend
money when you have a good business plan. Such a plan has to forecast many variables including
the sales volume, the costs for equipment and supplies, and the net profit. In determining the
numbers needed to make these forecasts, you are creating a financial model of the business.
Whether you decide to get into vending or not, the methods and tools you will use in this project
are useful in every business.
Here’s some basic information about the vending machine business. You will probably want to get
more information since prices can change quickly if labor or transportation costs go up or down.
1) Soda machines do about 5 times the business of snack machines with no spoilage.
2) About 30% of the people who walk by a vending machine regularly will buy something in the
course of a day.
3) New machines cost from $2500 to $6000 and last 7 or more years.
4) Big soda companies such as Coca-Cola and Pepsi rent machines at fairly low cost but you must
buy your soda from them. They will pay for repairs but will take out any machine vandalized
more than twice.
5) Repair costs are $50-$60 per hour plus travel. One of the most common repairs is the dollar bill
validator which is easy to replace but costs $125 -$175.
6) While you should check current prices, Coca-Cola was recently charging $7.70 per case (24 cans)
if you buy a truckload of 24 pallets with 96 cases per pallet. If you buy 5 truckloads per month,
the price drops to $6 per case. Supply stores like Sam’s Club may sell cases for as low as $7 but
the prices change more often and they don’t deliver.
Getting started: Here’s a brief list of the kinds of questions you need to answer for yourself (and
the bank). Brainstorm with other members of your group to think of specific questions you need
to ask and how you might go about getting anwers for these questions.
1) What is the sales volume in your school? What products? What if you raised or lowered
prices?
2) What are the costs of equipment? Is it better to rent or to buy and pay off the loan in 5-7 years?
3) What are the costs of the consumables (soda, candy, etc)?
4) What are the repair costs? Is vandalism a problem? Can you buy insurance? Should you?
Tom Hughes, 1999
5) Are they any labor or transportation costs? Should you advertise or have promotions?
Collecting the information you need for your business
It’s rarely practical to ask every customer of a business what products they like best or whether
they would buy more if the price went down. The alternative is to survey a sample of the
population and try to scale up those results to the whole. For example, assume there are 2000
students in your school. If you asked 20 students at random whether they preferred Dr. Pepper or
Coke and found out that 10 of them preferred Dr. Pepper but that only 15 of the 20 drank soda at
school, what could you infer about the whole school? Is asking 20 people enough? What if you
did two or three samples of 20 people each? Would you feel more confident about the results?
How would you get answers to other questions?
Do any work involved in answering these questions on a separate sheet of paper.
1) In your group, design a survey that you can give in one or two minutes to other students in the
school. Try to first brainstorm all the questions you might want to ask and then narrow the list
down to the most important.
2) Each person in the group should survey 10 to 20 randomly chosen students in the school before
the next class period.
3) Organize your data into table and graphs and try to determine what the data tell you about the
whole school.
4) As a group, share your results with the other groups in the class. Did they ask the same
questions? What results did they get?
5) Do you have enough information now to estimate the daily sales of snacks and soda in your
school? If not, how would you get this information?
6) Based on your answers in #5, how much money is spent each day in your school on snacks and
soda? Using just the cost of the soda and snacks, what is the gross profit per day?
7) What other information do you need to estimate the expenses associated with this business?
Tom Hughes, 1999