The following is a math problem written in the ways it might have been written in the years indicated.

1960s:     A peasant sells a bag of potatoes for $10. His costs amount to 4/5 of
                his selling price.  What is his profit?

1970s:     A farmer sells a bag of potatoes for $10.  His costs amount to 4/5 of his
                selling price, that is, $8.  What is his profit?

1970s (new math):  A farmer exchanges a set, P, of potatoes for set, M, of
                money.  The cardinality of the set M is equal to 10, and each element of
                M is worth $1.  Draw ten big dots representing the elements of M.  The
                set C of production costs is composed of two big dots less than the set
                M.  Represent C as a subset of M and give the answer to the question:
                What is the cardinality of the set of profits?

1980s:     A farmer sells a bag of potatoes for $10.  His production costs are $8,
                and his profit is $2.  Underline the word "potatoes" and discuss with
                your classmates the work that farmers do.

1990s (new-new math):  A farmer sells a bag of potatoes for $10.  His or her production
                costs are 0.80 of his or her revenue.  On your calculator, graph revenue vs.
                costs.  Run the POTATO program to determine the profit.  Discuss the
                result with students in your group.  Write a brief essay that analyzes
                this example in the real world of economics.  Discuss the positive and
                negative aspects of farming as a career choice.  Alternate assessment:
                Make a drawing of a farmer and cut it out.  Make a representation of a
                potato out of paper.  Make a garden in the classroom window-box and grow
                a  potato plant.  Go on the internet and find out the futures price for potatoes
                for next month.  Compare your results with other students and then place
                everything you have done on the problem into your portfolio.

                 How you might expect to see this problem presented in Mr. Hansen's class.

                A farmer sells a bag of potatoes for $10.

                a. How many bags would he have to sell to get more than $62.50?
                b. How much money would he get if he sold 22 bags?
                c. How much money would the farmer get if he sold b number of bags?
                d. What are the two values that are varying in this problem?
                e. What is the function?
                f. What type of function does it seem to be?

                   Assume that the farmer hires you to sell the potatoes for him at the
                   Farmers' Market.  He agrees to pay you $1.60 for each bag of potatoes
                   that you sell, plus an additional $15 for your expenses for the day.

                 g. Define a variable for the number of bags of potatoes you sell.
                 h. Write an expression for the amount of money the farmer will pay you at
                     the end of the day.
                 i. Write an equation, for t (the total amount you will earn) in terms of the
                     variable you defined.
                 j. Graph the equation on paper.  Be sure to include all the required parts of
                     your graph (title, labels, intervals marked, etc.)  Make sure your graph indicates
                     whether this data is continuous, or discrete.  Identify the domain and range
                     of your graph, and indicate any limits that are appropriate.
                 k. Use your graph to estimate about how much you will earn if you sell 16 bags.
                 l.  Use your graph to determine how many bags of potatoes you will need to
                      sell to earn enough money to fix your car ($75.)
                 m. What is the slope of the line indicated by your data?
                 n. What does the number you get for the slope represent in this problem?
                 o. What does the y-intercept represent in the problem?
                 p.  Assume that you are selling the potatoes for $10 per bag.  Write an
                       expression that represents the amount of money the farmer will have at the
                       end of the day, after he pays you.
                 q.  Write the inequalities that represent the linear programming for the following:
                       What is the number of bags of potatoes that you must sell in order for you to
                       earn at least $85.50?  Simultaneously graph all of the inequalities you have found
                       and shade the region that represents the possible solutions.  What is the range of
                       dollars you will earn if you sell between
                       11 and 24 bags of potatoes?

                 ...        etc.
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