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[1] Opportunity
The start of a new school year is a time for new beginnings,
particularly for first year students in a Contemporary College Algebra class.
For many of these students, their college algebra will be their terminal or
near terminal mathematics class. You, as instructor for this college gateway
course, have the opportunity (i.e., responsibility) to mold student attitudes
and to develop students to be exploratory learners. In addition, you
have the challenge to create and manage learning experiences for your students
that: * Prepare them for the quantitative work they will encounter in their academic careers and as a member of society; * Engender a positive and appreciative attitude toward mathematics;
* Develop problem-solving skills.
Although there is no definitive way to achieve these objectives, a common
thread that runs through instructor success stories is a student-centered
pedagogy that replaces lecturing with small-group activities and projects.
Here are a few of the suggestions that instructors have made for developing
positive attitudes and problem-solving skills:
1. Begin with "easy" activities and then build on success. E.g., give an
example of 10 numbers with a mean of 7. What-if the activity to require a mean
of 7 and a median of 5. What-if again to also require a mode of 4.
2. Give students the opportunity to present to the class their solution to an
activity and explain their reasoning.
3. As part of a homework assignment, ask students to bring to class real-life
examples of the topic being studied, e..g., "Applications of Linear
Equations." Provide class time for students to present and discuss their
examples.
4. Let students pick their own small-group project. The instructor needs to
clearly specify the objectives and the format for the project report.
5. Ask specific students to discuss the homework assignment - e.g., What is
the main topic? What questions did the reading evoke in your mind? How does
the new material relate to previous material? How can you use the new
material? etc.
One of the purposes of the Vision-Potential Newsletter is to provide
examples of small-group, in-class
activities. [2] "Warm-up" Exercises
This is an example of a set of exercises for starting a class. For each of the
statements a to e, state if it is True or False and then give an example that
supports your answer. (That is, assign integer values to "a" and "b" and then
compute the value of each side of the equation.)
a.
b.
c.
d.
e.
f. In the following multiplot, the function
i. Determine the equation for the function
ii. Describe how
[3] Writing Assignment
Taxes are a hot topic in this period of political debate. Should taxes be cut?
Should new, specialized taxes be imposed such as an excess profit tax or an
additional gas tax? Should the income tax be changed to a flat tax? These are
just a few of the questions that are being raised as we enter into a time of
presidential campaigning. "Fairness" is an issue that is often interpreted
differently depending on whether a person is favoring or opposing a given tax
proposal.
Suppose the issue at hand is to reduce the federal income tax rate by 5%. If
you were arguing for this issue based on the savings people would realize,
would you refer to the average savings people would receive or would
you refer to the median savings? Write a one-half page paper
supporting your choice of average or
median. [4] Mark-up or Discount Pricing?
Mark-up and Discount are the two primary ways a store manager determines the
list price of an item. For example, if the manager of your College Bookstore
marks-up your college algebra text by 30% it means that she increases the
wholesale price by 30% to get the list price (i.e., the list price is 130% of
the wholesale price). Whereas using the discount method, the store manager
determines a list price that when reduced by 30% gives the wholesale price.
Answer the following questions:
a. Do the mark-up and discount methods give the same list price?
b. If the manager wants to increase her profit, should she use the mark-up or
the discount method of pricing? Explain your reasoning.
c.. Explain how you determine the list price using the discount method.
d. Determine the list price on an item marked up 30% if its wholesale cost is
$90.
e. Determine the list price on an item discounted by 30% if its wholesale cost
is $90.
Ask your Bookstore manager which method of pricing she uses to price
textbooks. [5] ATV Registrations in Maine
Over the past 10-15 years, All Terrain Vehicle (ATV) riding has grown
tremendously as a recreational sport in Maine, complimenting the snowmobile
sport. The continuing growth is shown by the increase in ATV clubs from about
36 to over 130 within the past five years. The sport is a major contributor to
Maine's economy as reported by a 2005 study that estimated ATVs contributed
approximately $200 million to the Maine's economy. The following data shows
the number of ATV registrations in Maine for the years specified. Source:
Bangor Daily News,August 18, 2007.
Display this data in a scatter plot and then graphically fit a polynomial and
an exponential function to the data (without using any regression program). In
order to lessen round-off errors, let the independent variable be the number
of years since 1990 (i.e., shift the plot 1990 units to the left. Thus 1993 is
written as 3, 1995 is written as 5, etc. Determine which of your two functions
provides a better fit to the data (e.g., compare the errors for each of the
years). Explain your reasoning.
[6] Scientific Notation
Very large or very small numbers are usually expressed in scientific notation
rather than in decimal notation in order to avoid writing long decimal
expressions. For example, the mean distancebetween Earth and Pluto is
3,700,000,000 miles which is 3.7 E9 in scientific notation.
Notice the zeros in 500 represent multiplication by 10. Thus we can write
Examples:
The decimal number 8264.596 written in scientific notation is 8.264596 *
The decimal number 0.00000782 written in scientific notation is 7.82
*
The number (scientific notation) 2.8 E04 written in decimal notation is
28000.
Answer the following questions:
a. Avogadro's number, 6.023 *
b. How large (miles) is one light-year (the distance light travels in one
year)? Light travels at the rate of 186,000 miles per second. Assume that a
year has 365 days.
c. Proxima Centuri, the closest star to Earth, is approximately 4 light-years
away. Express the distance (miles) between Earth and Proxima Centuri in
scientific notation.
d. At 7:52 pm on September 7, 2007, the national debt was $
9,008,761,055,587.33. Express the debt in scientific notation.
e. Compute and express in scientific notation:
f. Compute and express in scientific notation:
[7] Running, Running, Running
With the exception of your own favorite track and field event, the running of
the mile is probably considered to be the premier event in track and field. In
the late 1940s and early 1950s, the hottest topic in track and field was when
would someone run a sub (below) 4-minute mile. Based on the 1913-1984 data for
the world records for the mile, in what year was the first sub 4-minute mile
run? Note the years listed are those from 1913.
Develop a linear and an exponential model of the world records for the mile. That is, draw a scatter plot showing the data and then fit a linear and an exponential curve to the scatter plot. The functions representing your curves are the models. Use your models to answer the following questions: a. Explain which model is "better." In what way is it better?
b. State the symbolic forms of your functions.
c. Superimpose the graphs of your models on your scatter plot.
d. Use your model to predict the world record in 2010, in 2030.
e. Explain how reliable your prediction is for 2030. Here is some additional information: on July 7, 1999, Hicham El Guerrouj of Morocco ran the mile in 205.88 seconds, a record that stands today (2007). Based on this additional information, how, if at all, would you alter your answers to parts e and f?
[8] Queries
a. Pick a number, add 3 to it, double the result, subtract 4 from the answer,
and then triple the result. If the result is 39, what number did you pick to
start? b. The radius of a wheel (including the tire) on a 998 Buick Regal is 12 inches. How fast (revolutions per minute) is the wheel turning when the car is moving at 60 mph?
[9] Notices
* Supported by the National Science Foundation and the U.S. Military Academy. |
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