MATHEMATICS AND ECONOMICS: A WORKSHOP TO SHARPEN YOUR SKILLS

OVERVIEW Today we will be looking at three different Mathematics & Economics topics for your classrooms. In an effort to reach out to a wide variety of educators, we’ll be looking at topics designed for students with mathematical competencies ranging from 8 th grade math up to calculus.

TOPICS Middle School Math: Surviving on a Deserted Island Math Objectives: Measures of Central Tendency, Range, Quartiles, Box-and- Whisker Plots Econ Objectives: Operate within a specific budget to reach desired outcome, Make predictions about value of labor in the marketplace. Algebra I: The Slopes of Supply and Demand Math Objectives: Plotting points in the first quadrant, slope, direct and inverse relationships Econ Objectives: Demand curves, Law of Demand, Why quantity demanded depends on price, Connections to the Supply Activity and the Equilibrium activity Algebra I/Algebra II Linear Programming and Consumer & Producer Surplus Math objectives: Graphing linear inequalities, solving a system of equations, Areas of geometric shapes Econ Objectives: consumer surplus, producer surplus, efficiency, taxation, deadweight loss Optional Calculus Activity (time permitting) Math Objectives: graphing a cubic equation, calculating 1st and 2nd derivatives, economics connection between derivative and marginal cost/revenue Econ Objectives: Calculating Total Profit, Maximizing Total Profit, mathematical connections of marginal cost/revenue and first derivative of cost/revenue graphs

SURVIVING ON A DESERTED ISLAND Activity 9 From:Mathematics & Economics Connections for Life: Grades 6-8 We’re going to run this activity as you would with your own class. Please feel free to chime in at any time with questions.

WARM-UP Median: The element of a set that is the central value (the element in the middle) when listed in order from least to greatest Upper Quartile: The “new” median of just the values above the median of the entire set Lower Quartile: The “new” median of just the values below the median of the entire set When listed in order the Minimum, Lower Quartile, Median, Upper Quartile, and Maximum split the data set into 4 quartiles that each represent 25% of the entire data set. A graph of these quartiles on a number-line and displayed horizontally or vertically is called a box-and-whisker plot or just a box plot.

WARM-UP Given the set below, find all of the points required and create a Box Plot. {2, 4, 5, 6, 7, 8, 11, 13, 14, 14, 14, 15, 19}

WARM-UP Given the set below, find all of the points required and create a Box Plot. {2, 4, 5, 6, 7, 8, 11, 13, 14, 14, 14, 15, 19} Minimum: 2 Lower Quartile: 5.5 Median: 11 Upper Quartile: 14 Maximum:

If you were stranded on a deserted island, whom would you want there with you?

ACTIVITY 9.1 Read the instructions and complete Activity 9.1 Make sure that the sum of the 5 bids your team is making does not exceed $150,000. When you’ve finished your list, trade papers with another team.

9.1 Let’s create a list of the 6 most popular occupations, and the bids for each occupation: Enter this information on the top of Activity 9.2 and find the Box and Whisker data for each of the 6 occupations. OccupationBid Data 1 Nurse40000, 9998, 35000, 30000, 10000, Engineer20000, 25000, 30000, 10000, Carpenter 25000, 10000, 10000, Farmer 50000, 40000, 40000, 1, Navy Seal 50000, , , Fishing guide 20000, 15000, 20000, 20000, 50000

9.2 Which of the top occupations had the highest bid? Which of the top occupations had the lowest bid? Which of the top occupations had the largest number of bids? Which of the top occupations had the greatest range of bids? Which of the top 6 occupations had the single highest interquartile range? Which of the top occupations seems to have the most variability in the data? Do each of the 6 box plots look the same? How are they similar? How are they different?

9.3 (CLOSURE) Let’s find out which occupations your team wound up with. What skills seem to be highly valued? Why is that? Activity 9.3 could be distributed as homework or in-class closure.

SLOPES OF SUPPLY AND DEMAND Activity 1 From:Mathematics & Economics Connections for Life: Grades 9-12 For this example, we’ll complete Activity 1 and discuss Activities 2 & 3.

WARM-UP HS Book: pg 8 Document Camera

WARM-UP HS Book: pg 8 Document Camera

DEMAND How does the price of a certain CD change the number of CD’s that will be bought? Let’s do an activity to find out!

DEMAND SCHEDULE Demand Schedule Price of CD in $Quantity demanded of CDOrdered Pair (Independent Variable)(Dependent Variable)(Dep. Variable, Indep. Variable) 32 0 (0, 32) 30 1 (1, 30)

DEMAND CURVE

CLOSURE: WRITING THE EQUATION

WHAT DOES THE CHANGE IN AXES DO TO OUR EQUATIONS? Consider this graph from Activity 3. (pg 45 of the HS Book) It is used to show that the equilibrium between supply and demand occurs at the intersection of those two graphs. What do you notice about the equations of the graphs?

LINEAR PROGRAMMING AND CONSUMER & PRODUCER SURPLUS Mathematical Supplement & Activity 5 From:Mathematics & Economics Connections for Life: Grades 9-12 Now that we can graph in both a mathematical and economics setting, what can we use those graphs to find? Let’s look at two ideas that use the graphs of inequalities.

LINEAR PROGRAMMING Linear programming has nothing to do with computer programming. The use of the word “programming” here means “choosing a course of action.” Linear programming involves choosing a course of action when the mathematical model of the problem contains only linear functions. The maximization or minimization of some quantity is the objective in all linear programming problems. All LP problems have constraints that limit the degree to which the objective can be pursued. A feasible solution satisfies all the problem's constraints. An optimal solution is a feasible solution that results in the largest possible objective function value when maximizing (or smallest when minimizing). A graphical solution method can be used to solve a linear program with two variables.

LINEAR PROGRAMMING Steps for Solving a Linear Programming Question 1.Graph the constraints. 2.Locate the ordered pairs of the vertices of the feasible region. If the feasible region is bounded (or closed), it will have a minimum & a maximum. If the region is unbounded (or open), it will have only one (a minimum OR a maximum). 3. Plug the vertices into the two variable linear equation to find the min. and/or max.

STEP 1: GRAPH THE CONSTRAINTS Let’s graph the constraints together. We can find the vertices by solving systems of equations. Now, let’s plug the vertices found from the feasible region into the profit equation to find the maximum and minimum profit possibilities. STEP 2: IDENTIFY THE VERTICES OF THE FEASIBLE REGION STEP 3: PLUG THE VERTICES INTO THE TWO VARIABLE LINEAR EQUATION TO FIND THE MIN. AND/OR MAX.

STEP 1: GRAPH THE CONSTRAINTS

Now, the feasible region exists only for those points in both the green and red regions. This defines an even more constrained feasible region.

STEP 1: GRAPH THE CONSTRAINTS

STEP 2: LOCATE THE ORDERED PAIRS OF THE VERTICES OF THE FEASIBLE REGION.

(5,2) (2,5) (4,4)

STEP 3: PLUG THE VERTICES INTO THE TWO VARIABLE LINEAR EQUATION TO FIND THE MIN. AND/OR MAX. We have shown mathematically, that of all of the possible combinations of acres of wheat and barley to be planted, the famer will have a maximum profit of $3200 when planting 4 acres each of wheat and barley, and a minimum profit of $2500 when planting 2 acres of wheat and 5 acres of barley.

CONSUMER & PRODUCER SURPLUS Now let’s take a look at another use of graphing inequalities, but this time from an economics point of view. We’ll be looking at Activity 5 from the HS book: The Gains from Trade We will skip the Warm-up since we’ve just had practice graphing linear inequalities.

CONSUMER & PRODUCER SURPLUS

This shaded triangular region, above the horizontal line but below the demand curve is a representation of Consumer Surplus. It indicates the total amount of money that consumers were willing to pay for the product, but didn’t have to spend.

CONSUMER & PRODUCER SURPLUS Similarly, the shaded triangular region below the horizontal line but above the supply curve is a representation of Producer Surplus. It indicates the total amount of money that producers to earned that is more than the minimum amount they were willing to earn.

CONSUMER & PRODUCER SURPLUS Combined together, these two right triangles form the larger (non-right) triangle shown to the right. This represents total surplus or gains from trade. It is the combined benefits to both consumers and producers.

TAXATION What happens to the benefit to consumer and producer when the government (or some other outside entity) imposes a tax? Let’s look at Activity 5.2 together and see what happens to the consumer and producer surplus.

TAXATION

DEADWEIGHT LOSS

Please don’t hesitate to send an with an further math questions you might have. I can be reached at: QUESTIONS