Vision - Potential
Vision Within Every Instructor - Potential Within Every Student
Newsletter of the HBCU College Algebra Reform Consortium*
Number 70, September 2006
www.ContemporaryCollegeAlgebra.org

Contents:
[1] Exploratory Student - Learning How to Learn [2] A Bit of Geometry [3] In-class Small Group Activity [4] Queries [5] Notices

Starting with the April issue, the Vision-Potential Newsletter will be distributed electronically. In order to continue receiving the Newsletter, send your e-mail address to Don Small, don-small@usma.edu.

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[1] Exploratory Student - Learning How to Learn

The statement Educating students for the future, rather than training them for the past has a nice ring, but how can it be transformed into a syllabus when we don't know what the future needs will be? In the last few years, technology in the form of graphing calculators, spread sheets, and computer algebra systems has eliminated or lessened the need for mastering many hand computations that forty years ago were deemed fundamental. However, lists of fundamental topics, like the weather, is constantly changing. Factoring (nice) trinomials would have been on the list several years ago, but not today. Recursive sequences were not on the list years ago, but are today. Although topics will come and go, the following two guiding aspects of educating students for the future will take on increasing importance.

1. Developing students to become explor- atory learners. Teaching students how to question, to "what-if" situations, to be critical of results, to seek alternatives, to recognize mathematics in everyday events, to provide time for exploration/discovery are all important components in developing students to becoming exploratory learners. Of utmost importance, of course, is for the instructor to model this behavior. The instructor who injects an article from the morning paper into a lesson (e.g., increase in the cost of gasoline), asks follow-on questions, challenges the class with queries, shares her/his wonderment over something, etc., will enrich and broaden the intellectual horizons of the students, a critical component in becoming an educated person.

2. Learning How to Learn. The days of lifetime employment with a single company is a thing of the past, today changing jobs and/or professions (voluntarily or involuntarily) is the norm. For some there will be training programs to help them learn what is needed for a new position, but most workers will need to do this learning on their own. Thus an important objective of a formal education program is to prepare students to learn on their own.

Two aspects of the recommended pedagogy of the Contemporary College Algebra program are structured to address both of these guiding aspects of educating students for the future. One aspect changes the traditional role of homework from one of drill to primarily one of guided exploration in which the student studies the new material before it is discussed in class. The textbook acts as the guide. The written part of the assignment includes exercises over the new material as well as some over the past material. Students are expected to have questions on the new material and are expected to raise those questions at the start of the next class. Changing the culture surrounding homework and its purpose, along with instructor expectation of the student is difficult and requires perserverance on the part of the instructor.

The other aspect is to provide class time for small group activities. In general, these activities are related to the new material. For example an activity related to average, median, and mode (Sections 2.2-3) is to have each small group determine a list of ten numbers whose average is 7, median is 5, and mode is 2. The activity concludes with a report to the class from several of the groups giving their results and explaining their reasoning.

A suggestion for a 50 minute class is to divide the time into three parts:

Part One (approx. 15 minutes) is devoted

to student questions over the new mater-

ial and/or the homework exercises.

Part Two (approx. 25 minutes) consists

of a small group activity.

Part Three (approx. 10 minutes) is

devoted to introducing/motivating the

material for the next class.

This pedagogy provides students with a supportive environment for learning how to learn new material on their own.

[2] A Bit of Geometry

Kathy Bavelas

Gateway Community College

This small-group activity involves finding equations of lines and solving systems of equations involving medians, altitudes, and perpendicular bisectors of a triangle. The activity, which consists of several parts, could be spread over two or more classes. Parts of it could be assigned for homework. The conclusion is a nice and probably surprising result. (This activity is related to Sections 2.7 and 2.8 of the text Contemporary College Algebra: Data, Functions, Modeling.) For convenience in grading and to keep the fractions that naturally occur in this activity to reasonable ones, I recommend for part 2 that you provide the coordinates of the 3 vertices. I like students to deal with equations of horizontal and vertical lines in this activity so I purposely place one of the sides of the triangle on the or axis. Selecting even numbers for the coordinates will also keep the fractions that students have to use in this activity reasonable.

Part 1. Exploration

a. Draw a fairly large acute, scalene triangle.

b. Draw the 3 medians (carefully please).

They should meet in a point, label it M.

c. Draw the 3 altitudes (carefully please).

They should meet in a point, label it A.

d. Draw the 3 perpendicular, side bisectors

(carefully please). They should meet in

a point, label it P.

e. Make a conjecture concerning the points

M, A, and P.

Part 2. Choose a Triangle.

a. Choose an acute, scalene triangle. Graph

the triangle using graph paper. Scale your

axes so that your triangle will occupy

about ½ to ¾ of the center of the graph

paper.

b. Find an equation for each line that

contains a side of the triangle. Put

each equation in the slope-intercept

form.

Part 3. Median Lines.

a. Now find the coordinates of the midpoint

of each side and label them on your

triangle.

b. Draw the 3 median lines. The 3 medians

should meet in a point. Estimate the

coordinates of that point.

c. Determine an equation for each median.

d. Solve the system consisting of the 3

equations from part c. Label the solution

M. Your solution should be close to the

estimate you found in part b. If it isn't,

check all your work.

Part 4. Perpendicular Bisectors.

a. Draw the 3 perpendicular bisectors of

the sides of the triangle and determine

the equation of each one.

b. The 3 perpendicular bisectors should

meet in a point. Estimate the coord-

inates of the point of intersection.

c. Solve the system consisting of the 3

equations from part a. Label the

solution P. Your solution should be

close to the estimate you found in

part b. If it isn't, check all your work.

(Hint: The slopes of perpendicular lines

are negative reciprocals of each other.)

Part 5. Altitude Lines.

a. Draw the 3 altitude lines and determine

the equation of each one.

b. The 3 altitudes should meet in a point.

Estimate the coordinates of the point

of intersection.

c. Solve the system consisting of the 3

equations from part a. Label the

solution A. Your solution should be

close to the estimate you found in

part b. If it isn't, check all your work.

Part 6. Euler Line.

Verify that the points labeled M, P, and A are collinear. This line is called the Euler Line.

Follow-on Questions.

a. Where is the point A in a right triangle?

Explain your reasoning.

b. Where is the point P in a right triangle?

Explain your reasoning.

c. Does point M always lie between

points A and P? Explain your reasoning.

[3] In-class Small Group Activity

(This game-like activity is designed to accompany Section 2.4 "Variable Representation" in Contemporary College Algebra: Data, Functions, Models. The activity involves formulating a conjecture by considering several examples and then introducing variables in order to abstract the process for the purpose of proving [or disproving] the conjecture. In the process, students will be engaged in adding, multiplying, and dividing fractions.)

Problem.

"Pick any two positive integers, say 2 and 5. Add 1 to the second number and divide the result by the first number to produce a third number, (5+1)/2=3 Repeat this procedure by adding 1 to the third number and dividing by the second. Keep repeating this process until you notice something interesting. Now pick two other positive integers and perform the same process. What happens? How long does it take to happen?"

Activity:

Step One: Each small group is divided into pairs and then each pair forms a solution conjecture by experimenting with at least three pairs of positive integers.

Step Two: Each small group decides on one conjecture.

Step Three: Each group introduces variables for the two positive integers, say and or and , and then solves the problem. (Different groups could use different letters for its variables.)

Step Four: One or more groups explain to the class how their solution using variables verifies their conjecture.

Follow-on Questions:

a. Can the condition of positive integers

be replaced by just positive numbers?

Explain your reasoning.

b. Can the condition of positive integers

be replaced just numbers? Explain

your reasoning.

Problem number 20, page 212, Functioning in the Real World: A Precalculus Experience (Class Test Edition) by S.P. Gordon, F.S. Gordon, B.A. Fusaro, M. J. Siegel, A.C. Tucker.

[4] Queries

a. When a circle is deformed into an ellipse (by "squishing in" two opposite sides), the circumference does not change. Does the enclosed area change?

b. Consider a region enclosed by a large, flexible steel band (the steel can bend, but it cannot stretch or contract). Can the steel band be deformed so that the enclosed area is less (more) than the area of the original region? Explain your reasoning and illustrate with an example.

c. Select two points, say (3,2) and (7,5). Determine the coordinates of the midpoint of the line segment joining the two points. Verify that the determined point is in fact equidistant from the two given points. As a follow-on question, determine the point that is two-thirds the distance from (3,2) to (7,5).

d. (For Section 2.4) Six people had gathered to celebrate a special birthday. In the course of the conversation they discovered that except for the oldest person in the group each person was half the age of someone else in the group. What are their ages?

[5] Notices

1. Applications are now being accepted for the second cohort of schools to participate in the NSF funded program to refocus college algebra. Five schools will be selected to send a three person team to participate in the 2007-8 HBCU Retreat and Follow-On program. The two-year program includes a four day Retreat (June 2007), follow-up visits, presentations at the national Joint Mathematics Meetings, and an option to receive a $5,000 grant to help support refocusing the college algebra program. For further information, contact Don Small at don-small@usma.edu.

2. Laurette Foster and Don Small will present a minicourse on Contemporary College Algebra during the Joint Mathematics Meeting in New Orleans in January 2007.

3. Deadline for contributions to the October Newsletter is Monday, October 1, 2006. Opinion articles, suggestions for writing assignments, small group in-class activities, small group out-of-class projects, Queries, announcements, etc. are welcomed.

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