This lesson covers at least two 45-minute class periods. It may need to be extended over a third period, depending upon the speed with which the students are able to find and process the data.
1. Students will research real-life data on cost information and use this data to generate 2-variable equations that reflect the total cost of producing a product.
2. Students will accurately graph these cost curves using appropriate scale on the coordinate planes.
3. Students will accurately explain the methods they have used through writing. (Writing across the curriculum)
Digital projector connected to computer with Microsoft Powerpoint (if possible)
Attached Powerpoint presentation -- to be projected if possible, or printed if not
Computers with Internet access if possible
Begin the class by reviewing the various definitions of slope and the slope-intercept form of an equation. Next, have students remind you of the definitions of fixed cost and variable cost, as well as the fact that variable cost can be represented by the value of an equation's slope, while the fixed cost can be represented by the equation's y-intercept.
Inform the students that it's now time to start using the material we've been learning by applying it to their project. We're going to run through an example of how to research the potential costs of producing a product together, and then they will research actual products with their partners and determine cost curves for them. Every partner group will research products that we are thinking about buying, and then we can make an informed decision as a class as to which products we want to sell (based on how much they cost to produce).
Use the attached Microsoft Powerpoint presentation to present how to ascertain the variable costs from a potential product and to turn that information into a cost curve. Ensure that all student questions are answered.
Next, demonstrate how to choose an appropriate scale for the graph of this equation (considering the number of units sold and the amount of money involved). Demonstrate as well that using the slope to graph this equation is not feasible (because of the nature of the decimal), so that the students will simply need to find another point (using the equation and plugging in an arbitrary number of units to get a cost) and then connecting the y-intercept to this point.
After students have practiced this, have them brainstorm as a class different products that they would like to sell for their business. Assign each student pair a different product (or assign multiple pairs to a single product if they don't brainstorm enough to go around). If there is time remaining in class and you have the computers available, student pairs can begin researching costs on the Internet during class. If not, this should be homework for the next class.
Students should list all the necessary items that they would need to buy in order to sell the product. For instance, if they want to sell hot chocolate, not only will they need milk and hot chocolate powder, but they will also need styrofoam cups and marshmallows, if they choose to sell those as well. They will also need to make note of how many units each resource will make in order to ascertain the unit cost. (For instance, if cups come in packages of 51, but a tin of cocoa powder only makes 25 cups, they need to note this information!) They need to bring both the cost and number produced for each resource to the following class. Impress upon them that if they do not do this, they will be unable to proceed in the project and that this assignment is part of the portfolio that they will be graded on.
When the next class convenes, have sutdents report their findings on the cost information. Critique and correct anything that sounds unreasonable. Next, have student pairs work together to divide out the unit costs for their product and add them together to find a single unit variable cost for their product. Remind them that the fixed costs for the business will be $15. Next, have the student pairs use their equations to accurately graph the cost curve on graph paper. Once they are finished, they should check it with you and make any corrections necessary (pay particular attention to the scale!). Once corrected, they should make a final draft (each student does his/her own to be included in their portfolio), and they should write a paragraph explaining the process by which they created the graph. This can then be placed in their portfolio to be used later when solving systems of equations.