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 nonroutine problems. (A) compare and analyze various methods for solving a reallife 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 dataanalysis 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 CocaCola 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. CocaCola 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 CocaCola 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 breakeven 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 yintercept 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, breakeven 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. IH35, Austin, TX or FAX Attn: Tom Hughes, Computer files: Send as attachments to 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 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,
8 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 CocaCola 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, CocaCola 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 57 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
9 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
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