|
|
| Home | Textbook | Newsletter | Discussion Forum | Contact |
 Textbook
|
|
[1] Refocusing College Algebra
The national movement to refocus college algebra from being a pre-precalculus
course to one that provides a basis for addressing the quantitative needs
students will encounter in society, the work place, and in their other courses
is rapidly expanding. Within the past few years all of the major mathematical
associations have published curriculum guidelines that contain specific
recommendations for refocusing college algebra. These guidelines call for
introducing elementary data analysis, modeling, use of technology, increased
emphasis on communication and on small group work. With respect to pedagogy,
the recommendations call for a change from instructor centered to student
centered instruction. The modeling theme in refocused courses requires a deeper understanding of functions than is usually found in traditional courses. Many small group activities involve using graphing calculators or spreadsheets to fit functions to scatter plots. Success in these projects require students to be able to recognize the shapes of the basic functions and to understand their properties. Modeling with recursive sequences (difference equations, discrete dynamical systems) is another activity that distinguishes refocused courses from the traditional courses. These activities provide students with a feel for the power of mathematics in modeling the real world; it serves to reinforce the behavioral characteristics of various families of functions; it links the mathematics to what is done in many of the other disciplines; and it provides the students with the knowledge and understanding of recursion - the mathematical language of the spreadsheet, which is the primary technological tool of almost every quantitative discipline. The encouragement and support for refocusing college algebra has expanded beyond curriculum guidelines. Consider the following:
(1) Next month (November 2006) the Mathematical Association of America (MAA)
will hold a small working conference on "Algebra: Gateway to a Technological
Future;" (2) In 2007, the Center for the Study of Mathematics Curriculums is
planning a national conference on algebra, one portion of which will be on
college algebra; (3) The American Association of State Colleges and
Universities is developing an initiative to focus on success in mathematics
which will include an emphasis on college algebra; (4) The Mayor's office of
the City of San Antonio has established a government/industry/academic task
force to improve the college algrebra courses and increase student success in
them; (5) The National Science Foundation (NSF) is funding an MAA program to
renew college algebra at eleven colleges and universities; (6) The NSF is
funding a three year program at the U.S. Military Academy to assist eleven
Historically Black Colleges and Universities to refocus their college algebra
programs.
Leaders in the Contemporary College Algebra program have been instrumental in
deveoping the national movement to refocus college algebra. [2] Graphing "Sense"
Graphing programs on calculators or computers offer wonderful opportunities
for graphical exploration. However, they also present a danger as they allow
students to plot with very little "thinking input." As an instructor whose
teaching career began long before the development of graphing calculators, I
am impressed with the ability to show the effects of changing a parameter by
just moving a slider bar. However, many of my students see it as just a
game. Helping students develop a graphing sense involves hand sketching and
written explanations of the effects of changing parameter values as well as
using technology for explorations. The following activity is directed toward
the hand sketching aspect of graphing.
In each exercise, students working in pairs are asked to sketch (without
technology) a multiplot of the two given functions and then compare the two
functions taking into account: domain, long-term behavior, dominance of one
function over the other, intersections, asymptotes, etc.
a.
b.
c.
d.
e.
f.
g.
[3] Algebraic Manipulations
Several students have difficulty with "basic" algebraic manipulations even
though they may have spent considerable time "drilling" on them. These
students would be better served if part of their drill time was replaced with
activities directed toward conceptual understanding. Having students analyze
expressions for correctness, as the following illustrates, is one such
possibility.
For each of the following, state if it is True or False and then give two
numerical examples that support your answer. (For instance, assign integer
values to "a" and "b" and then
compute the value of each side of the equation.)
a.
b.
c.
d.
e.
[4] Three In-class Activities1. Paper Folding. When a single sheet of paper is folded in half, the result is two sheets. A second folding of the two sheets produces four sheets. Make a table with three columns - the first column shows the number of folds from 0 to 5, the second column shows the number of sheets, and the third column expresses the number of sheets in exponential form. Model this folding phenomenon by expressing the number of sheets as a function of the number of folds. Now create a thickness function by multiplying your sheet function by the thickness of one sheet. (How can you determine the thickness of a sheet since a single sheet is too thin to measure accurately?) a. Estimate the thickness after you have made 10 folds? 20 folds? 50 folds? b. Compute the thickness in feet after 10 folds, 20 folds, and then compute the thickness in miles for 50 folds. c. How many (theoretical) folds would be required for the stack of sheets to reach
the moon, the sun?
2. Sailboat Sail. A Marine Sail company makes and sells sails at a cost of $30
per square foot of finished sail. How much would a triangular sail with
dimensions 33 feet by 27 feet by 21 feet cost? (Does the sail form a right
triangle? How do you know?) Hints a. Draw a sketch of the sail and denote the lengths of the sides of the triangle. b. Expand your sketch by enclosing it in a right triangle. Label the unknown portions of the sides with variables. c. Use an inverse cosine function to
determine the size of an angle.
3. Assume the Newton dormitory on your campus has three floors and is
approximately 70 feet tall whereas the Euler dormitory has nine floors and is
approximately170 feet tall. Both buildings have flat roofs and the height
measurements include the basements. Develop a model that predicts the height
of a dormitory building as a function of the number of floors it
contains. [5] Shopping for Lower Gasoline Prices
The volatility of gasoline prices often means that prices across a region are
not always uniform. It is not unusual for gas stations within ten miles of
each other to charge different amounts for the same type of gas. The variation
in price encourages some people to "shop around" to get the lowest price. How
does one determine if traveling to a station for a lower price actually saves
money? For example, suppose Sam gets 24 mpg with his car and usually buys 12
gallons when he "fills up." Also suppose that Sam lives near a gas station
that charges $2.70 per gallon and he knows of another station 10 miles away
that charges $2.60 per gallon. Ignoring the time involved and the wear on the
car, would Sam save money buying gas at the station 10 miles away? Explain
your reasoning.
Follow up question: Ignoring time and wear on the car, how far could Sam drive
to buy gas at $2.55 and still save money? [6] Cardiovascular Fitness(This activity/project requires an instrument, such as LabPro, to measure and record a heart rate.)
One measure of cardiovascular fitness is how quickly the heart rate (HR) drops
during the first three minutes after exercise and also how long it takes the
heart rate to return to "normal," what is called the sedentary heart rate. The
time from the end of the exercise until the normal heart rate is obtained is
called the recovery time. Contact your physical education department for their standard or guideline data on: Exercise HR, 3 minute HR, Sedentary HR, and Recovery Time. Compare the HR of one or more students against this standard by doing the following: a. Measure the student's normal heart rate. b. Have the student conduct a vigorous exercise for five minutes and then collect heart rate data for the next three minutes. c. Have groups of students model the heart rate data. d. Use the heart rate model to predict the recovery time. e. Compare the results to the standard or guideline results.
Complete the activity/project by having each group critique the experiment and
their model. How reliable is the result? What are the strengths and weaknesses
of the model? What assumptions were made in developing the model?
etc.
An alternative to collecting data is to use the following data from a similar
activity that was carried out in a class at West Point this fall. The question
posed to the class was: Are incoming cadets (Plebes) in better cardiovascular
shape than senior cadets? The Physical Education department provided the following data averaged over the senior class.
A Plebe was recruited and his heart rate data was obtained during class. After measuring his sedentary heart rate (75 bpm), he was sent to run up and down a flight of stairs for five minutes. When he returned to class, the following data was obtained using LabPro and a computer.
Time
(min)
Heart Rate (bpm) 0.134 181.8319472 0.268 173.6367575 0.402 172.2158703 0.536 163.8150485 0.67 160.4477429 0.804 153.8698733 0.938 150.0037194 1.072 146.3272227 1.206 142.5035335 1.34 139.6956319 1.474 135.6851702 1.608 132.3432303 1.742 130.7893923 1.876 128.1471051 2.01 125.609584 2.144 124.3781393 2.278 122.4645295 2.412 118.9704246 2.546 125.0094219 2.68 122.4645295 2.814 118.8089375 2.948 117.6384269 3.082 117.061887
3.216
116.0002203 [7] Notices
* Supported by the National Science Foundation and the U.S. Military Academy. |
|
| Home | Textbook | Newsletter | Discussion Forum | Contact |
