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
8th grade math up to calculus.
TOPICS
Middle School Math:
Algebra I/Algebra II
Surviving on a Deserted Island
Linear Programming and Consumer
& Producer Surplus
Math Objectives: Measures of Central
Tendency, Range, Quartiles, Box-andWhisker 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: 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: Plotting points in the first
quadrant, slope, direct and inverse
relationships
Math Objectives: graphing a cubic equation,
calculating 1st and 2nd derivatives,
economics connection between
derivative and marginal cost/revenue
Econ Objectives: Demand curves, Law of
Demand, Why quantity demanded
depends on price, Connections to the
Supply Activity and the Equilibrium
activity
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}
19
Minimum: 2
Lower Quartile: 5.5
Median: 11
14
11
Upper Quartile: 14
Maximum: 19
5.5
2
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:
Occupation
Bid Data
1
Nurse
40000, 9998, 35000, 30000, 10000, 60000
2
Engineer
20000, 25000, 30000, 10000, 100000
3
Carpenter
25000, 10000, 10000, 13000
4
Farmer
50000, 40000, 40000, 1, 15000
5
Navy Seal
50000, 100000, 100001, 45000
6
Fishing guide
20000, 15000, 20000, 20000, 50000
Enter this information on the top of Activity 9.2 and find the
Box and Whisker data for each of the 6 occupations.
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 CD
Ordered Pair
(Independent Variable)
(Dependent Variable)
(Dep. Variable, Indep. Variable)
32
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
(0, 32)
1
(1, 30)
2
3
4
5
6
7
8
9
10
11
12
13
14
15
DEMAND CURVE
CLOSURE: WRITING THE EQUATION
Activity 1.3 an be used as homework or as a closure activity.
However, it is very important to talk about the difference
between graphing in economics and in math and other sciences.
In mathematics, the slope of a line is typically viewed as:
𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒅𝒆𝒑𝒆𝒏𝒅𝒆𝒏𝒕 𝒗𝒂𝒓𝒊𝒂𝒃𝒍𝒆
𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒊𝒏𝒅𝒆𝒑𝒆𝒅𝒆𝒏𝒕 𝒗𝒂𝒓𝒊𝒂𝒃𝒍𝒆
Because economists put the independent variable on the
vertical axis, and the dependent variable on the horizontal axis,
we must refer to slope as:
𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒕𝒉𝒆 𝒗𝒂𝒓𝒊𝒂𝒃𝒍𝒆 𝒐𝒏 𝒕𝒉𝒆 𝒗𝒆𝒓𝒕𝒊𝒄𝒂𝒍 𝒂𝒙𝒊𝒔
𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒕𝒉𝒆 𝒗𝒂𝒓𝒊𝒂𝒃𝒍𝒆 𝒐𝒏 𝒕𝒉𝒆 𝒉𝒐𝒓𝒊𝒛𝒐𝒏𝒕𝒂𝒍 𝒂𝒙𝒊𝒔
So for the example we just completed, the slope would be:
𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒑𝒓𝒊𝒄𝒆
𝒄𝒉𝒂𝒏𝒈𝒆 𝒊𝒏 𝒒𝒖𝒂𝒏𝒕𝒊𝒕𝒚 𝒅𝒆𝒎𝒂𝒏𝒅𝒆𝒅
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.
A farmer has 10 acres to plant in wheat and barley. He has to plant at
least 7 acres. However, he has only $1200 to spend and each acre of
wheat costs $200 to plant and each acre of barley costs $100 to plant.
Moreover, the farmer has to get the planting done in 12 hours and it
takes an hour to plant an acre of wheat and 2 hours to plant an acre of
barley. If the profit is $500 per acre of wheat and $300 per acre of barley
how many acres of each should be planted to maximize profit?
Let w = the number of acres of wheat planted
Let b = the number of acres of barley planted
Constraint Functions:
𝒘≥𝟎
𝒃≥𝟎
𝒘 + 𝒃 ≤ 𝟏𝟎
𝒘+𝒃≥𝟕
𝟐𝟎𝟎𝒘 + 𝟏𝟎𝟎𝒃 ≤ 𝟏𝟐𝟎𝟎
𝒘 + 𝟐𝒃 ≤ 𝟏𝟐
Function to be maximized:
𝑷 𝒘, 𝒃 = 𝟓𝟎𝟎𝒘 + 𝟑𝟎𝟎𝒃
STEP 1: GRAPH THE CONSTRAINTS
Let’s graph the constraints together.
STEP 2: IDENTIFY THE VERTICES OF
THE FEASIBLE REGION
We can find the vertices by solving systems of equations.
STEP 3: PLUG THE VERTICES INTO THE
TWO VARIABLE LINEAR EQUATION TO
FIND THE MIN. AND/OR MAX.
Now, let’s plug the vertices found from the feasible region
into the profit equation to find the maximum and minimum
profit possibilities.
STEP 1: GRAPH THE CONSTRAINTS
𝒘≥𝟎
𝒃≥𝟎
𝒘 + 𝒃 ≤ 𝟏𝟎
𝒘+𝒃≥𝟕
𝟐𝟎𝟎𝒘 + 𝟏𝟎𝟎𝒃 ≤ 𝟏𝟐𝟎𝟎
𝒘 + 𝟐𝒃 ≤ 𝟏𝟐
Because both w and
b must be greater
than or equal to zero,
only the first
quadrant would
supply feasible
answers.
STEP 1: GRAPH THE CONSTRAINTS
𝒘≥𝟎
𝒃≥𝟎
𝒘 + 𝒃 ≤ 𝟏𝟎
𝒘+𝒃≥𝟕
𝟐𝟎𝟎𝒘 + 𝟏𝟎𝟎𝒃 ≤ 𝟏𝟐𝟎𝟎
𝒘 + 𝟐𝒃 ≤ 𝟏𝟐
Plot the constraint
function 𝒘 + 𝒃 ≤ 𝟏𝟎
By first graphing 𝒘 +
𝒃 = 𝟏𝟎 and then
testing a point to see
which region defined
by the line should be
shaded.
Plotting the line 𝒘 + 𝒃 = 𝟏𝟎
STEP 1: GRAPH THE CONSTRAINTS
𝒘≥𝟎
𝒃≥𝟎
𝒘 + 𝒃 ≤ 𝟏𝟎
𝒘+𝒃≥𝟕
𝟐𝟎𝟎𝒘 + 𝟏𝟎𝟎𝒃 ≤ 𝟏𝟐𝟎𝟎
𝒘 + 𝟐𝒃 ≤ 𝟏𝟐
Plot the constraint
function 𝒘 + 𝒃 ≤ 𝟏𝟎
Test point (0,0) makes
the inequality true: 𝟎 +
𝟎 ≤ 𝟏𝟎
So the region defined by
the line which contains
(0,0) is shaded.
Shading the region 𝒘 + 𝒃 ≤ 𝟏𝟎
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
𝒘≥𝟎
𝒃≥𝟎
𝒘 + 𝒃 ≤ 𝟏𝟎
𝒘+𝒃≥𝟕
𝟐𝟎𝟎𝒘 + 𝟏𝟎𝟎𝒃 ≤ 𝟏𝟐𝟎𝟎
𝒘 + 𝟐𝒃 ≤ 𝟏𝟐
Following the same
steps as used for the
first constraining
function, graph the
next inequality to
further constrain the
feasible region
Feasible region after graphing the
constraint fuction 𝒘 + 𝒃 ≥ 𝟕
STEP 1: GRAPH THE CONSTRAINTS
𝒘≥𝟎
𝒃≥𝟎
𝒘 + 𝒃 ≤ 𝟏𝟎
𝒘+𝒃≥𝟕
𝟐𝟎𝟎𝒘 + 𝟏𝟎𝟎𝒃 ≤ 𝟏𝟐𝟎𝟎
𝒘 + 𝟐𝒃 ≤ 𝟏𝟐
Following the same
steps as used for the
previous constraining
functions, graph the
next inequality to
further constrain the
feasible region
Feasible region after graphing the
constraint function
𝟐𝟎𝟎𝒘 + 𝟏𝟎𝟎𝒃 ≤ 𝟏𝟐𝟎𝟎
STEP 1: GRAPH THE CONSTRAINTS
𝒘≥𝟎
𝒃≥𝟎
𝒘 + 𝒃 ≤ 𝟏𝟎
𝒘+𝒃≥𝟕
𝟐𝟎𝟎𝒘 + 𝟏𝟎𝟎𝒃 ≤ 𝟏𝟐𝟎𝟎
𝒘 + 𝟐𝒃 ≤ 𝟏𝟐
Following the same
steps as used for the
previous constraining
functions, graph the
final inequality to
further constrain the
feasible region
Feasible region after graphing the
constraint function
𝒘 + 𝟐𝒃 ≤ 𝟏𝟐
STEP 2: LOCATE THE ORDERED PAIRS OF THE
VERTICES OF THE FEASIBLE REGION.
Now that the final feasible region has
been found, the vertices of this bounded
polygon must be found by solving
systems of equations.
The intersection of the orange and
purple lines can be found by solving the
system:
𝒘+𝒃=𝟕
𝟐𝟎𝟎𝒘 + 𝟏𝟎𝟎𝒃 = 𝟏𝟐𝟎𝟎
𝒘 = −𝒃 + 𝟕
𝟐𝟎𝟎 −𝒃 + 𝟕 + 𝟏𝟎𝟎𝒃 = 𝟏𝟐𝟎𝟎
−𝟐𝟎𝟎𝒃 + 𝟏𝟒𝟎𝟎 + 𝟏𝟎𝟎𝒃 = 𝟏𝟐𝟎𝟎
−𝟏𝟎𝟎𝒃 = −𝟐𝟎𝟎
𝒃=𝟐
And now, because 𝒘 = −𝒃 + 𝟕
𝒘=− 𝟐 +𝟕
𝒘=𝟓
The point must be (5,2)
STEP 2: LOCATE THE ORDERED PAIRS OF THE
VERTICES OF THE FEASIBLE REGION.
In the same way the other two
systems can be solved to find the
remaining vertices:
𝒘+𝒃=𝟕
𝒘 + 𝟐𝒃 = 𝟏𝟐
𝒃=𝟓
𝒘=𝟐
(2,5)
(4,4)
(5,2)
The 2nd point must be (2,5)
𝒘 + 𝟐𝒃 = 𝟏𝟐
𝟐𝟎𝟎𝒘 + 𝟏𝟎𝟎𝒃 = 𝟏𝟐𝟎𝟎
𝒃=𝟒
𝒘=𝟒
And the final point must be (4,4)
STEP 3: PLUG THE VERTICES INTO THE
TWO VARIABLE LINEAR EQUATION TO FIND
THE MIN. AND/OR MAX.
Now that the three vertices of the feasible region are known, let’s
plug each of them into the equation to calculate profit.
𝑷 𝒘, 𝒃 = 𝟓𝟎𝟎𝒘 + 𝟑𝟎𝟎𝒃
𝑷 𝟐, 𝟓 = 𝟓𝟎𝟎 𝟐 + 𝟑𝟎𝟎 𝟓
𝑷 𝟐, 𝟓 = 𝟏𝟎𝟎𝟎 + 𝟏𝟓𝟎𝟎
𝑷 𝟐, 𝟓 = 𝟐𝟓𝟎𝟎
𝑷 𝟓, 𝟐 = 𝟓𝟎𝟎 𝟓 + 𝟑𝟎𝟎 𝟐
𝑷 𝟓, 𝟐 = 𝟐𝟓𝟎𝟎 + 𝟔𝟎𝟎
𝑷 𝟓, 𝟐 = 𝟑𝟏𝟎𝟎
𝑷 𝟒, 𝟒 = 𝟓𝟎𝟎 𝟒 + 𝟑𝟎𝟎 𝟒
𝑷 𝟒, 𝟒 = 𝟐𝟎𝟎𝟎 + 𝟏𝟐𝟎𝟎
𝑷 𝟒, 𝟒 = 𝟑𝟐𝟎𝟎
The profit in planting 2
acres of wheat and 5 acres
of barley is $2500.
The profit in planting 5
acres of wheat and 2 acres
of barley is $3100.
The profit in planting 4
acres of wheat and 4 acres
of barley is $3200.
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
Let’s start by graphing the supply and demand curves for
Activity 5.1.
Please note that the equations are given in the forms:
𝑷 = 𝟏𝟓 − 𝟎. 𝟐𝑸𝒅
𝑷 = 𝟎. 𝟏 𝑸𝒔
Where P (price) is independent variable on the vertical axis,
and Q is the dependent variable on the horizontal axis. Does
that make these equation follow the form of
𝒚 = 𝒎𝒙 + 𝒃
or
𝒙=
𝟏
𝒚
𝒎
+ 𝒂 ???
After you’ve graphed both lines, draw a horizontal line
through the equilibrium point where the supply and demand
curves cross.
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
The original consumer surplus was ∆𝑬𝑭𝑯, after a
tax is introduced to the smaller triangle ∆𝑪𝑭𝑨.
The original producer surplus was ∆_____, after a tax
is introduced to the smaller triangle ∆_____.
The benefit to the government or outside entity is
represented by the rectangle ______.
What part of the original gains from trade
(represented by the largest triangle ∆𝑱𝑬𝑭) no
longer benefits anyone?
DEADWEIGHT LOSS
The small triangle ∆𝑪𝑬𝑮 is the deadweight loss. This is the
inefficiency of taxation. The area of the deadweight loss
represents the amount of money that benefits nobody
(consumer, producer, or government.) It arises because
buyers pay more than producers receive. This inefficiency
manifests as deadweight loss.
QUESTIONS
Please don’t hesitate to send an email
with an further math questions you
might have. I can be reached at:
robert.schmidt@tusd1.org