# Models of a Vending Machine Business

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1 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 contacts with local vending businesses. Related Resources Guides for small businesses Amortization tables Tom Hughes,

2 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 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 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 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. 2. Lesson Outline 2.1. Set up the lesson and outline the general parameters of the business 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) 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 New machines cost from \$2500 to \$6000 and have an effective lifetime of at least 7 years There are many vending machine companies with Web pages (search for vending machines). This may be a rich source of information for student projects 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,

3 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 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 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 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) 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 Do a basic cost/profit calculation as part of a break-even analysis 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 An alternative set of simultaneous equations (both dependent on volume) is to plot both the gross income and the expenses versus the sales volume 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 Using depreciation and amortization, decide whether to lease or buy machines While some sample costs are given above, different groups could work with different sets of parameters that would lead to different conclusions. 3. Guiding Questions (to engage students in mathematical thinking during the lesson) 3.1. What factors determine the amount of gross income? Volume of sales. Is this price dependent? (an opportunity for a survey question) Price per unit. Do you sell sodas at \$.50, \$.75, \$1.00? What is the impact on income? What products sell best? In what proportions? How many individual machines are needed? Justify your answer with math. Tom Hughes,

4 3.2. What factors determine the expenses of operating this kinds of business 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 How do you find a breakeven point? Graphs are probably the best tool for the initial analysis 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 Can you believe survey results? If you can spend the time you may want to work with probability 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 What are the basic issues of leasing versus owning? Given a cost per machine and an interest rate, students can work out an amortization schedule on a spreadsheet or iteratively on a calculator A more complex problem can be created by adding in repair costs and depreciation factors What if? 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. 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? 5. Assessment Task(s) - (to evaluate what students have learned) 5.1. Final report detailing the students financial analysis of the business. Tom Hughes,

5 5.2. Class presentations 5.3. A quiz on specific pieces of the analysis: graphs of simultaneous equations, spreadsheets, etc Concept map of business financial terms and of this business in particular focusing on the relationship to expense and income. 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 Do a presentation for another class in the school. An economics class would be ideal Find college students involved in economics or business to come to your class as consultants. 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 Have students talk to someone who creates or makes use of market surveys. 8. Author s notes 8.1. Overview Types of decisions students are expected to make in this simulation: Lease the machines or buy them (do a cost comparison of leasing vs. amortization) Survey of students to help decide what products will sell best and at what volume Overall breakeven analysis at different sales volumes and pricing schemes 8.2. Objectives Students will learn the basic elements of small business finance: depreciation, amortization, break-even analysis Students will use graphs and spreadsheets to evaluate situations and make mathematically informed decisions. Tom Hughes,

6 Students will conduct surveys to determine product preferences, predicted volume, pricing, and other issues Students will interpret this statistical data by scaling their results up to the whole school Students will integrate these concepts by making a final report and/or class presentation Materials required; activity preparation Students will benefit from basic knowledge of a spreadsheet program with access to computers as needed The teacher may wish to locate sources of data or create data for this simulation beyond what is provided in this document There are many vending machine companies that have Web sites Most schools have vending machines and the companies involved may be willing to share basic facts about the business with the students 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 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: Paper copies: Mail to: Tom Hughes, UT Dana Center, 2901 N. IH-35, Austin, TX or FAX Attn: Tom Hughes, Computer files: Send as attachments to TBH1@mail.utexas.edu 10.2 Send copies of class handouts you have created (as above) Send copies of assessment materials: Rubrics, quizzes, presentation requirements, etc Brainstorm ideas for this project and others on the Math Models listserv (shared mailing list). Not on the MM mailing list? Send to Cynthia Schneider cschneider@mail.utexas.edu at the Dana Center and she will put you on the list. Tom Hughes,

7 10.5 Suggestions for improving the project or the materials can be sent in by any of the means above. Tom Hughes,

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