Vision - Potential
Vision Within Every Instructor - Potential Within Every Student
Newsletter of the HBCU College Algebra Reform Consortium*
Number 64, October 2005
www.ContemporaryCollegeAlgebra.org

Contents:
[1] A Comparison Study of the Contemporary College Algebra with Traditional College Algebra [2] Fun Project: Collisions of Air Hockey Pucks [3] Activity: Identifying Graphs [4] In-class Activity -- Lines & Half-planes [5] Notices

[1] A Comparison Study of the Contemporary College Algebra with Traditional College Algebra

The results cited in this article are taken from Amie Ellington's comparison study of the Contemporary College Algebra (CCA) and traditional college algebra. The full study appears in the September issue of Primus under the title "A Modeling-Based College Algebra Course and It's Effect on Student Achievement." The study was based on eight CCA sections (pilot) and eleven traditional sections (control) taught at Virginia Commonwealth University (VCU) in the 2004 Fall semester. Each section had 35 students. The students were not preassigned and did not know the difference between the sections when they enrolled.

The course grade results were:

Grade	CCA		Traditional
A	16.90%		12.75%
B	30.99%		17.11%
C	23.94%		19.84%
D	12.32%		12.75%
F	10.22%		17.21%
W	5.63%		20.34%

Table 1

The reduction in the withdrawal rate from 20.34% in the traditional sections to 5.63% in the CCA sections is particularly striking. Furthermore 89.6% of the students in the CCA sections took the final exam (5.63% withdrew and 4.77% of the remaining did not take the final exam). In comparison only 71.33% of the students in the traditional sections took the final exam (20.34% withdrew and 8.33% of the remaining did not take the final exam). Thus not only did the students in the CCA sections earn higher grades, considerably more of them stayed in the course through the final exam than was the case in the traditional sections.

In order to address the question: How well do students completing the CCA course do on skill type questions?, the final exam contained a common portion that all students, CCA and traditional, took. The common portion consisted of ten skill questions and three application questions. All the instructors involved agreed on the questions and this portion of the exam was graded by one person who was not involved with any of the nineteen sections. The grade results were:

	CCA	Traditional
	A,B,C	A,B,C
All Questions	57.14%	41.05%
Skill Questions	53.78%	35.09%
Application Questions	61.29%	47.02%

Table 2

These results clearly demonstrate that students develop the appropriate algebraic skills in courses that de-emphasize skill work in favor of modeling.

The study included an analysis of how well the students did on their follow-on mathematics course during the the Spring semester 2005. VCU students passing college algebra have two options for their follow-on course---precalculus or business calculus. The precalculus is a very traditional, skill oriented course while the business calculus is applications oriented. Table 3 shows that students in the CCA sections did better than those in the traditional sections in the business calculus, but not as well in the precalculus course. The percentages of students passing (A,B,C) were:

Courses	CCA	Traditional
Business Calculus	72.60%	69.57%
Precalculus	56.38%	70.42%

Table 3

A slightly higher percentage of students completing the CCA course (69%) took a follow-on mathematics compared to those in the traditional sections (63%).

Table 4 gives the percentage comparison measured over two semesters, the students in the CCA sections (37.3%) realized a significantly greater success rate then those in the traditional sections (28.3%). This was a result of the higher passing rate and greater percentage of students taking a follow-on mathematics course. The results were:

Follow-on Course	CCA	Traditional
Business Calculus or Precalculus	37.3%	28.3%
Business Calculus	18.7%	8.1%
Precalculus	18.7%	20.2%

Table 4

(The complete article is posted under Instructor Resources on our webpage.

[2] Fun Project: Collisions of Air Hockey Pucks

(Frank Wattenberg, U.S. Military Academy, suggested this project.) The project involves using systems of equations to model the effects when two air hockey pucks collide head-on, taking into account their velocities and weights. The particular questions to be answered are:

1. Suppose a moving puck hits a stationary puck of the same weight head-on. What will happen?

2. Suppose a moving puck hits a stationary puck of heavier (lighter) weight head-on. What will happen?

3. Suppose two pucks of the same weight and traveling at the same velocity collide head-on. What will happen?

4. Suppose two pucks of the same weight and traveling at different velocities collide head-on. What will happen?

It is recommended that this project be introduced and the questions discussed in class before students work in their groups. The discussion should be based on the students intuition and experience. A physical demonstration using different colored tennis balls for pucks would help generate enthusiasm and student interest.

This project provides a nice opportunity to invite a physics instructor to speak to your students about the importance of conservation laws in science. (In Section 2.8, the Law of Conservation of Mass was used to balance chemical equations.) In particular, the introduction to the project needs to include the conservation laws of energy and momentum.

Law of Conservation of Energy: the total kinetic energy of the two pucks is the same before and after the collision. (The kinetic energy of an object with weight and velocity is )

Law of Conservation of Momentum: the total momentum of the two pucks is the same before and after the collision. (The momentum of an object with weight and velocity is .)

The students should be encouraged to define variables representing the weight and velocity of each puck before collision and the weight and velocity of each puck after collision. Constructing an equation for each of the two conservation laws gives a system of two equations to solve. (For instance, let denote weight, denote the velocity of one puck and the velocity of the other other puck. Also let subscript one denote time immediately before collision and subscript two denote time immediately after collision. Then for Question 1 the Law of Conservation of Momentum states:

or

An equation involving and can similarly be obtained from the Law of Conservation of Energy. The solution of the resulting system of equations leads to two outcomes. It is important that the students interpret each of the outcomes to see if they are reasonable. (Recall the interpretation leg in the Problem Solving/Modeling Process diagram.)

[3] Activity: Identifying Graphs

For this activity, divide the students into pairs. The instructor sketches a graph on the board and then each pair of students do the following:

a. Identify the basic shape of the graph.

b. Develop a function representing the function

represented by the graph.

c. Create a "story" corresponding to the

graph and their function.

After an appropriate amount of time, the instructor calls on different pairs to share what they have done and to explain their reasoning. Here are some examples:

1a. Graph:

1b. Curve is decreasing and concave up-

ward. There is an intercept on the

vertical axis, but possibly not on the

horizontal axis. The horizontal axis is

an asymptote. An approximate func-

tion is (The intercept on

the vertical axis rules out a function

of the form Why?)

1c. Interpretation. Water draining out of

a bathtub---vertical axis represents

the depth of the water in inches over

the drain and the horizontal axis

represents time in minutes.

2a. Graph:

2b. The curve suggests a periodic func-

tion, say sine or cosine. Since the

curve passes through the origin,

would be a reasonable

approximation.

2c. Interpretation. The number of hours

of possible sunlight in Buffalo, New

York. Hours are measured on the

vertical axis and days of the year on

the horizontal axis

3a. Graph:

3b. Curve is increasing, concave upward

at first and then changes to concave

downward. There appears to be two

horizontal asymptotes, the horizontal

axis and where the cure appears to

flatten out. A logistic curve would be

a reasonable approximation,

(A reasonable app-

roach would be to approximate with a

two-part function. The first part could

be a quadratic function or an exponent-

ial function and the second part could

be a negative quadratic shifted up and

to the right or the negative of a nega-

tive exponential shifted up and to the

right.)

3c. Interpretation. The amount of fish

mass in an aquarium.

[4] In-class Activity -- Lines & Half-planes

Ask the class to explain what is meant by saying "A straight line divides the plane into two half planes." When this is understood, divide the class into pairs.

1. Ask each pair to determine the equation of a line that does not include the point (2,3). (Different pairs will probably have different lines.) Then ask each pair to use an inequality to analytically describe the half-plane that contains the point (2,3). Ask one of the pairs to explain to the class what they did and their reasoning.

2. Ask each pair to determine a second line that does not contain the point (2, 3) and is not parallel to their first line. Using inequalities, analytically describe the "half-plane" that contains the point (2,3). Ask one of the pairs to explain to the class what they did and their reasoning.
   
   
   

[5] Notices

1. Past issues of the Vision - Potential Newsletter are available on our website: www//ContemporaryCollegeAlgebra.org.

2. Faculty Development Workshop for Contemporary College Algebra: Nov. 3-4, 2005, Houston, TX. Contact Laurette Foster (lbfoster@pvamu.edu).

3. REQUEST: The November issue of our Newsletter will focus on final exam questions. Please send in samples of your final exams questions so that they can be shared through the Newsletter.

4. Deadline for contributions to the November Newsletter is November 1, 2005. In addition to final exam questions, opinion articles, suggestions for writing assignments, small group in-class activities, small group out-of-class projects, Queries, announcements, etc. are welcomed.

5. Subscribe to this Newsletter
   
   

* Supported by the National Science Foundation and the U.S. Military Academy.