AMATYC 2007 Brief Calculus: Mary Pearce and Sharon Welker mdpearce@waketech.edu
, sfwelker@waketech.edu
AMATYC Conference, Minneapolis, MN November 1, 2007
Presenters: Mary D. Pearce and Sharon F. Welker
Wake Technical Community College Raleigh, NC www.waketech.edu/~mdpearce (presentations)
Brief Calculus students learn many concepts, including Riemann Sums, Lorenz Curves and the Gini Index. These and other calculus topics can be modified to engage students in active learning using spreadsheet technology and Internet activities. Updating strategies for assessing student learning and collecting data related to learning outcomes helps the instructor maintain a strong course.
Session Goals Participants will review specific lessons that can be used in a Brief
Calculus (or Business Calculus) class incorporating spreadsheets and real-world applications. Participants will see how such applications incorporate strategies from
Beyond Crossroads.
This presentation will illustrate how faculty can use the Beyond Crossroads (BC) implementation recommendations to update and revise specific mathematical lessons.
The three standards to be addressed are (1) Student Learning and the Learning
Environment, (2) Instruction, and (3) Assessment of Student Learning.
BC
(1) Implementation Standard: Student Learning and the Learning Environment
Mathematics faculty and their institutions will create an environment that optimizes the learning of mathematics for all students.
The following lessons touch on a variety of learning style characteristics and strategies for students, listed on page 21. “Relate new mathematics topics to previous knowledge.
Discuss mathematics ideas with others. Practice problems using computer software.
Consult other sources for specific real-world examples of mathematics concepts and procedures.” A very important expectation is stated on page 23, students will “recognize that problem solving is an essential skill to be developed or improved in any mathematics course.”
(2) Implementation Standard: Instruction
Mathematics faculty will use a variety of teaching strategies that reflect the results of research to enhance student learning.
The following lessons agree with characteristics of Student-Centered Classrooms, given on page 52. “Knowledge is constructed by students gathering and synthesizing information. Teaching and assessment are intertwined. Instructors and students learn together.”
(3) Implementation Standard: Assessment of Student Learning
Mathematics faculty will use results from the ongoing assessment of student learning of mathematics to improve curriculum, materials and teaching methods.
The following lessons incorporate the Assessment Implementation Cycle, page 30, focused on the classroom level, using a variety of assessment tools.
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AMATYC 2007 Brief Calculus: Mary Pearce and Sharon Welker mdpearce@waketech.edu
, sfwelker@waketech.edu
Lesson Plan: Riemann Sums
Objectives:

Students will understand the purpose of Riemann Sums

Students will use Riemann Sums to find the area under a curve

Students will use Excel to compute Riemann Sums
Motivational Problem:
The model R
 
268 .
12 d
2 
1482 .
5 d describes the average box office receipts per day for the movie The Pink Panther , where d represents the days since release.
Revenue
$2,500.00
$2,000.00
$1,500.00
$1,000.00
$500.00
$0.00
0 1 2 3 4 5 6
Instruction:
Class begins with a discussion of the questions: What do you think the area underneath this curve from 0 days to 5 days represents? How could you compute this area if all you had was a ruler?
Next the students are guided through a slider sheet in Excel that shows the area divided into left-handed, right-handed, or mid-point centered rectangles with questions asking the students to add up the areas.
After the students understand how to add up the areas of the rectangle the instructor guides them through setting up a template to calculate the area using any number of rectangles.
The fourth sheet is a slider that helps the students see that the more rectangles that are used the closer the sum becomes to the “actual” area.
The final sheet is an additional real world example using data from the Space Shuttle.
Assessment:
Students complete an out-of-class assignment in Excel that practices both Riemann Sums and the Fundamental Theorem of Calculus.
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AMATYC 2007 Brief Calculus: Mary Pearce and Sharon Welker mdpearce@waketech.edu
, sfwelker@waketech.edu
Supplements:
Riemann Sums
Riemann Sums provide a procedure to estimate the area under a curve using a series of rectangles. The graph below shows the area using six rectangles using the left-sided endpoint.
The graph below shows the area using fifteen rectangles using the left-sided endpoint.
The more rectangles used the better the approximation of the area under the curve.
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AMATYC 2007 Brief Calculus: Mary Pearce and Sharon Welker mdpearce@waketech.edu
, sfwelker@waketech.edu
Summations of this type, i n 

1 f ( x i
)
 x , are called Riemann Sums.
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AMATYC 2007 Brief Calculus: Mary Pearce and Sharon Welker mdpearce@waketech.edu
, sfwelker@waketech.edu
Spreadsheet Template: Computing the Area Under a Curve
The template below will allow you to compute the area under any curve using Excel:
A B C D E F G H
1
2
3
4
5 a a b b n n dx
=(B2-A2)/C2 ith rect
1
2
…
… xi
=A2 f f(xi)
(F2)
Ai
=G2*$D$2
=F2+$D$2 f (E3) =G3*$D$2
…
…
…
…
…
…
For the above template: a.
The a and b are the left and right endpoints, respectively, of the total area. b.
The n is the number of rectangles used. c.
The dx , or ∆ x , is the width of each rectangle, and is computed by ( b – a )/ n . d.
The ith rect represents which rectangle you are computing. e.
The xi represents the x coordinate for that rectangle. f.
The f(xi) is the function evaluated at the corresponding xi . g.
The Ai is the area for that rectangle.
Alternatives: a.
If using the Right endpoint for the rectangle height, then the formula in F2 will be “=A2 + D2”. b.
If using the Midpoint for the rectangle height, then F2 will be “=A2 + D2/2”.
Use the above template to complete the table below for the model for daily receipts of the movie The Pink Panther :
Number of
Rectangles, n
3
Approximate
Area under f ( x )
6
25
50
100
500
1.
Enter the values for a , b and n .
2.
Enter the formulas for dx , xi and Ai .
3.
Enter the given function f ( x ) into the f(xi) column, using the xi values for x .
4.
Select the four cells in row 3 (E3:H3) and drag down until the value of ith rect is equal to the value of n .
5.
Sum up column H for the Total Area .
As the number of rectangles increases, what happens to the value of the approximate area?
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AMATYC 2007 Brief Calculus: Mary Pearce and Sharon Welker mdpearce@waketech.edu
, sfwelker@waketech.edu
Out-of-class Assignment:
1.
The managers of an employee health plan for a firm have determined that the rate of change of the balance B in millions of dollars is given by the formula dB

10 e
0 .
1 t 
8 , where t is the years since 1985. Use Riemann Sums with 10 dt rectangles to determine how much the balance increased from 1989 to 1995.
2.
A small firm makes quilts and has determined that the rate of change of its profit P in hundreds of dollars is given by the formula dP dn
 
0 .
1 n
2 
2 .
03 n

9 .
87 , where n is the number of quilts made in a week. Use Riemann Sums with 50 rectangles to determine how much the profit increases if production is increased from 4 to 9 quilts per week.
3.
A small firm makes quilts and has determined that the rate of change of its profit P in hundreds of dollars is given by the formula dP dn
 
0 .
1 n
2 
2 .
03 n

9 .
87 , where n is the number of quilts made in a week. Use the Fundamental Theorem of Calculus to determine the profit if production is increased from 4 to 9 quilts per week.
4.
An average-size man with a parachute will fall at a speed of dS dt

9 .
81 t

5 .
11 , where t is the number of seconds since jumping and S is the number of meters. How far does the man fall between 5 and 10 seconds?
5.
The managers of an employee health plan for a firm have determined that the rate of change of the balance B in millions of dollars is given by the formula dB

10 e
0 .
1 t 
8 , where t is the years since 1985. Use the Fundamental Theorem of dt
Calculus to determine the balance from 1989 to 1995. (We have not done this antiderivative rule, but see if you can figure out what the appropriate antiderivative should be.) Your final answer should be close to Problem 1.
In-class Assessments:

Collect one or more of the above problems

Critical Thinking Quiz or Activity: first person creates a problem, pass this to the second person who solves the problem, then pass to a third person who will critique the problem and the solution ((is the question too hard/too easy, do you think the solution is correct, should they show more steps, etc)

Test questions will be used to assess student learning
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AMATYC 2007 Brief Calculus: Mary Pearce and Sharon Welker mdpearce@waketech.edu
, sfwelker@waketech.edu
Activity for Riemann Sums
First Person’s Name__________________________________________
A.
Make-up a problem to be solved with the use of Riemann Sums
Second Person’s Name__________________________________________
B.
Work the Riemann Sum problem above in Excel and record your answer below.
Third Person’s Name__________________________________________
C.
Critique the above two tasks using complete sentences. Comment on the problem in
A (for example, do you think this would be a good test question/bad test question, is it toohard/tooeasy, etc.). Grade the answer in B.
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AMATYC 2007 Brief Calculus: Mary Pearce and Sharon Welker mdpearce@waketech.edu
, sfwelker@waketech.edu
Lesson Plan: Lorenz Curves and the Gini Index
Objectives:

Students will find real-world connections for this specific calculus topic

Students will create Lorenz curves

Students will find and interpret the Gini index
Motivational Problem:
Students will be given the following pre-class homework assignment.
Directions : The purpose of this assignment is for students to explore the internet or nontextbook print material to develop an understanding of a Lorenz Curve and the Gini
Index. Answer the questions, and search for the terms given below and find 2 different types of sources that help you understand the concepts.
I. What do you think : are the rich getting richer and the poor getting poorer? What is the trend of economic equality in the US?
II. Find data to support your intuition…or not.
Look up the following topics that will be studied in our text section 7.4.
(1) Lorenz Curve (2) Gini Index
--Explain what it is.
--Explain where it is used.
--Cite the reference for each.
Instruction:
Class begins with discussion and interactive responses, based on the assignment. This is an immediate assessment of student engagement. Instructors guide the discussion to ensure that the information is correct.
Instructor leads the class in specific examples on a guided lecture spreadsheet.
Incorporate on-going assessment by asking questions for interactive responses.
Students work with others to complete the practice problems.
Assessment:
Students complete examples and homework problems, and use a writing assignment to reflect on the learning activity
8

AMATYC 2007 Brief Calculus: Mary Pearce and Sharon Welker mdpearce@waketech.edu
, sfwelker@waketech.edu
Supplements:
This could be prepared in Excel or Power Point slides, or class handout for guided lecture notes
Lorenz Curves and the Gini Index
How is income distributed among the citizens of a country?
How equitable is this distribution?
Define a Lorenz Curve as a function y = L(x) where x represents the percentage of the population of a country (starting with the poorest and working up in income) y represents percentage of total income that the percentage x receives
Example:
If the poorest 25% of the population received 8% of the total income of a country, then
L(0.25) = 0.08
What properties must a Lorenz curve have?
What is the Domain?
What is the Range?
What 2 points are on all Lorenz curves?
What shape do you anticipate the graph will have?
What do you notice about the curves?
How can we visualize the discrepancy between a perfect distribution, and reality?
Would you argue for a situation that has more, or less, area between the curves?
What mathematical insight can we use to compare 2 values?
Define the Gini Index to be the ratio of the discrepancy to the ideal, so
9

AMATYC 2007 Brief Calculus: Mary Pearce and Sharon Welker mdpearce@waketech.edu
, sfwelker@waketech.edu
1

0
( x

L ( x ) dx
1

0 xdx

2 *
1

0
( x

L ( x )) dx
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AMATYC 2007 Brief Calculus: Mary Pearce and Sharon Welker mdpearce@waketech.edu
, sfwelker@waketech.edu
In-class Learning Activity, to be completed as an Out-of-Class Assignment
Students complete the following in pairs or small groups. This can be provided as a handout, or be posted as an Excel document.
1.
A developing country’s income distribution is described by the function
L ( x )

19
20 x
2 
1
20 x , where x is the percentage of the population. a.
Compute L (0.2) and L (0.8). Interpret your results. b.
What percent of the wealth does the upper 30% control? c.
Find the Gini Index.
2.
In a certain country 5% of the wealth is controlled by the lower 20% of the population and 37.5% of the wealth is controlled by the lower 60 % of the population. Find the Lorenz curve for the country. a.
Compute L (0.3) and L (0.7). Interpret your results. b.
What percent of the wealth is controlled by the upper 20% of the population? c.
Find the Gini Index.
3.
In a study conducted by a certain country’s Economic Development board, it was found that the Lorentz curve for the distribution of income of stockbrokers was described by the following function
S ( x )

11
12 x
2 
1
12 x and that of high school teachers by the following function
T ( x )

5
6 x 2 
1
6 x , where x is the population of each profession. Which profession has a more equitable income distribution? Defend your answer.
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AMATYC 2007 Brief Calculus: Mary Pearce and Sharon Welker mdpearce@waketech.edu
, sfwelker@waketech.edu
4.
Answer our original motivating problem: Are the rich getting richer and the poor getting poorer?
1977 Percent of Income Earned by the
Bottom…..
20
3.6
40
12.5
60
27.5
80
50.7
95
78.3
Percent of population
Percent of Income
1993 Percent of Income Earned by the
Bottom…..
Percent of population
Percent of Income
2001 Percent of Income Earned by the
Bottom…..
Percent of population
Percent of Income
20
3.6
20
3.5
40
12.6
40
12.2
60
27.7
60
26.8
80
51.2
80
49.8
95
79
95
77.6
Find the Gini Index for the years 1977, 1993, and 2001, using the data provided by the U.S. Census Bureau for household income in the United States for the given year.
Show supporting work
Compare the values. Are incomes becoming less equally distributed or more equally distributed? Explain.
Assessment:

At the end of class, students complete a minute paper. (1) What were the key concepts covered today? (2) Which learning activities helped you understand the concepts?

In addition to completing class examples, standard homework problems are assigned.

Test questions will be used to assess student learning
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