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[1] Equations Then, Functions Now
Since the beginning of the college algebra era (approximately 1960), the
primary emphasis in college algebra has been on solving equations. Thus
factoring, rules of signs, completing the square, etc. were important
techniques and a great deal of time was spent on trying to get students to
master them. The choice of equations (mostly first, second, and third degree
polynomials) was limited to those that could be solved by the "standard"
techniques. Thus a student would probably not be asked to solve the simple
looking equation:
The reform movements over the past fifteen years, aided by technology, have
stessed the multiple representation of functions-graphically, numerically,
symbolically, and in written form. With a graphing device (calculator,
computer), solving equations no longer requires mastering the techniques of
traditional college algebra. Furthermore, there are essentially no limitations
placed on the type of equation to be solved. Function values can be easily
displayed by tables and iterations. Students see functions in graphical and
symbolic form throughout the popular press, in business publications, and on
news broadcasts.
Today, in refocused college algebra courses, the emphasis has changed from
equations to functions. In particular, a major objective of these courses is
to have students experience developing functions through the modeling process.
For example, students collect data on vehicle breaking distance, plot the
data, fit a curve (function graph) to the scatter plot, and then use the
resulting function for predictive purposes. Another example is the use of
recursive sequences (functions) to model discrete change (i.e., credit car
payments) or to approximate continuous change (e.g., warming a cold can of
soda). There are huge advantages to this modeling - it helps connect the
mathematical ideas with the kinds of experiences students have in their
everyday lives; it gives the students a strong sense for the usefulness of the
mathematics; it serves to reinforce the behaverial characteristics of various
families of functions; and it serves as a unifying theme in the course.
Furthermore it provides students with the knowledge and understanding of
recursion - the mathematical language of the spreadsheet, which is the primary
technology tool of almost every other discipline. In addition, it is
fun! [2] Fundamental Skills (Elementary Algebra)
Developing and maintaining skill in applying elementary algebra operations is
important for several reasons including building self-confidence, easing
computations, and developing mathematical insight. Like most other skills,
repeated usage is necessary to maintain proficency. Because the lists of
elementary algebra skills expected of students often differs from instructor
to instructor, instructors are urged to talk with each other in order to form
a consensus. A next step is to provide students with frequent opportunities to
check their skills (e.g., five minute opening class activity, five minute
quiz, etc.). The following set of five questions is an example of an "opening
class" activity.
For each of the following, state if it is True or False and then give an
example that supports your answer. (That is, assign interger values to "a" and
"b" and then compute the value of each side of the equation.)
a.
b.
c.
d.
e.
Readers are invited to send their lists of fundamental skills to this
Newsletter along with suggestions for teaching fundamental skills.
[3] Class Activity: Generalizing the Concepts of Mean and Median to Two Dimensions
The mean (average) of a one-dimensional, numerical data set is the number
computed by adding all the data values together and dividing the resulting sum
by the number of data entries.
The median of a one-dimensional, numerical data set is the middle number when
the data is arranged in numerical order. If the set contains an even number of
entries, then the median is the average of the two center most
elements.
A fascinating question is how to extend these two concepts to a
two-dimensional set? In particular, how do you determine the mean and median
of the population of the conterminous United States (48 states plus the
District of Columbia)? The Census Bureau describes the Mean and Median Centers
as follows:
Mean center of population as: "The point at which an imaginary, flat,
weightless, and rigid map of the United States would balance if weights of
identical value were placed on it so that each weight represented the location
of one person."
Median center of population as "The intersection of two median lines,
a north-south line (a meridian of longitude) constructed so that half of the
Nation's population lives east and half lives west of it, and an east-west
line (a parallel of latitude) selected so that half of the Nation's population
lives north and half lives south of it."
The following tables give the Mean Center and the Median Center of the U.S.
population for selected years. (Source www.
census.gov/population/censusdata/popctr.pdf)
Table A: Mean Center
Table B: Median Center Activity a. Using the data in Tables A and B, draw a line plot of the Mean Center and Median Center on a map of the United States that shows latitude and longitude
lines. b. Write a paper interpreting the data and your line plots. Include (but do not be limited to) comments on: the overall trend; reasons for the overall trend;
predictions for 2000 and 2010. c. True or False: Is it possible for the Mean and Median Centers to move in opposite directions going north and south or going east and west. State your reasons. If it is possible for the centers to move in opposite directions, explain a scenario where this could
happen. d. Describe how you would determine the Geographic Center of the conterminous
United States. e. Explain the justification of the use of the cosine function in the following description for measuring longitude: For these distances, a degree of longitude at the equator was the unit of measurement. East-west distances along the equator could be measured in degrees, but any east-west degree distance north of the equator -- where all the United States is located -- had to be adjusted to recognize the conver- gence of meridians toward the poles. This adjustment required that each east- west distance, stated in degrees of longitude, be multiplied by the cosine of the latitude. This mathematical relationship is precise for a sphere and a very close approximation for the earth. (Source: www.census.gov/geo/www
/cenpop/calculate)
Follow-on class activity: students determine the Median
Center for their class. [4] Television SetsYvette Stepanian
Virginia Commonweal University
The following chart represents the number of television sets per 1000 people
in the Developed Countries between 1985 and 2003.
(Source: International Telecommunication Union and Earth Trends)
Plot the points on your calculator, where
a. Find the equation of the line that passes through the points representing the years 1990 and 2002. Clearly show your work. b. What does the slope of the linear model found in (a) represent here?
c. What is the value of the
your linear model? And what does it represent? d. According to the model found in (a), what should the production level have been in 1999? Show your work. e. According to the model found in (a) and using algebra, determine in which year there were 300 television sets per 1000 people. Show your work.
[5] Notices
* Supported by the National Science Foundation and the U.S. Military Academy. |
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