Vision - Potential
Vision Within Every Instructor - Potential Within Every Student
Newsletter of the HBCU College Algebra Reform Consortium*
Number 78, October 2007
www.ContemporaryCollegeAlgebra.org

Contents:
[1] Activities for Starting Class [2] Painting a Bedroom [3] Identifying Functions [4] Life Insurance Rates [5] Positioning an Extension Ladder [6] Queries [7] Small Group Activity [8] Notices

[1] Activities for Starting Class

The start of a class presents an interesting, and sometimes frustrating, challenge to the instructor. Namely, how to manage the change in the student environment from one of socializing to one of academic learning. Often valuable class time is lost before the students become academically engaged in the lesson. Student centered pedagogy that engages students through small-group activities offers a way to replace the "start-up" challenge with a learning opportunity. The opportunity is to begin the class with a timed, small-group activity that concludes with two or more groups presenting their work to the class. The time limit aspect of the activity encourages students to start working immediately and the presentation aspect provides a means of accountability. In addition, this opportunity addresses the problem-solving and communication goals of the course.

In previous Newsletters, skill exercises were presented as examples of start-up activities. In the following two examples, some mathematical problem solving analysis is required to identify the skill work embedded in word problem settings.

1. A straight playground slide is to be built such that the top of the slide is 4 feet off the ground and the bottom of the slide rests on the ground. In addition the triangular area formed by the slide, the ground, and a vertical ladder to the top of the slide is 10 ft...

a. Determine the equation of the line

segment corresponding to a side of

the slide.

b. Determine the slope of the slide.

c. Interpret the meaning of the vertical

intercept of your linear equation

from Part a.

d. Determine the length of the slide.

2. A local Cab Company charges $1.00 on entering the cab plus $2.50 for each mile driven.

a. Develop a fee model that expresses the

cost of a trip as a function of the miles

driven.

b. Interpret the meaning of the slope in

your function equation in Part a.

c. Interpret the meaning of the vertical

intercept for your function equation.

d. Plot the fee function from Part a.

[2] Painting a Bedroom

Nancy wants to paint the walls of her bedroom which measures 11' by 14' by 8'. The room has two windows that measure (including trim) 40" wide by 64" high, two doorways that are 40" wide by 7' high, and an 8" baseboard that is not to be painted.

How many quarts of paint should Nancy buy? (One quart covers 100 square feet.)

[3] Identifying Functions

a. Table 1:

x	-2	- 3	-1	4	2
g(x)	9	11	7	-3	1

b. Table 2:

x	4	1	10	5	8
h(x)	5.2	2	6	5.4	5.8

c. Table 3:

x	-1	0	2	4	-2
k(x)	5.5	8	23	83	4.25

d. Table 4:

x	-1	0	2	5	6
k(x)	-3	2	6	3	-10

[4] Life Insurance Rates

Insure.com listed the following monthly rates for $1,000,000 life insurance policy. (Source: USA Today, September 9, 2007.)

Age		Monthly Rate
35		$21
40		$29
45		$47
50		$69
55		$105
60		$157
65		$256

Model the situation:

a. Create a scatter plot for the data.

b. Recognize the shape of the scatter plot.

Does the shape suggest a quadratic

polynomial or an exponential model?

Explain.

c. Fit a curve to the scatter plot. (The

equation of the curve defines a

function which is the model.)

d. Use your model to predict the monthly

rate for a person whose age is 70.

[5] Positioning an Extension Ladder

With respect to safety, what is the proper elevation angle for an extension ladder leaning against the side of a house? A brochure of the Lynn Ladder and Scaffolding Company provides the recommendations listed in the following table. The height is the vertical distance in feet from the ground to the bearing point of the ladder against the house. The base distance is the horizontal distance in feet of the resting point of the ladder from a vertical line passing through the bearing point of the ladder (i.e., horizontal distance of the foot of the ladder from the house).

Height Base Distance 9.5 2.5 13.5 3.5 17.5 4.5 21.5 5.5 25 6.5 28 7 31 8

.

a. Create a scatter plot of the data. (Which

variable is the independent variable?)

b. Fit a curve to the scatter plot.

c. Determine the slope of the ladder using

the equation of the curve found in Part b.

d. Determine the angle of elevation of the

ladder from the slope in part c.

e. Explain why the answers in parts c and d

are reasonable. If the answers are not

reasonable, redo your analysis.

[6] Queries

You are interested in designing the largest area that can be enclosed with 100 feet of fencing. For each of the following, be prepared to explain your reasoning and to justify that your answer is in fact a maximum.

a. How large a rectangular area can be

enclosed with a 100 foot fence?

b. How large a circular area can be

enclosed with a 100 foot fence?

c. How large a triangular area can be

enclosed with a 100 foot fence?

[7] Small Group Activity

For this business related activity, divide the class into pairs of students, have them work the following problem, then call on different pairs to describe what they did and to explain their reasoning. Ask how many pairs used a graphical approach? Call on different pairs to present their follow-on problem. This activity could be used to lead into a discussion of the reasonableness of a demand curve being decreasing and a supply curve being increasing (both curves representing functions of price). Another discussion topic emanating from this activity is the interpretation of the intersection point of the supply and demand curves as an equilibrium point. What does equilibrium mean? What happens when the price is changed from the equilibrium price? For example, is the demand higher or lower than the supply when the price is less than the equilibrium price?

Problem. One summer day, Brenda decides to sell apples from her Dad's apple tree to the people walking along her street. She decides that the price of an apple should depend on how many apples she plans to sell. (The more apples she sells, the higher she has to climb in the tree to pick them and thus the amount of her work per apple increases.) She also realizes that as the price per apple increases, fewer people will buy apples. Therefore her supply function is increasing and her demand function is decreasing. Suppose her supply function is . and her demand function is . where, in both functions, . denotes the price of an apple. Thus if Brenda thought she could only charge 5 cents per apple, she would just pick 5 apples (the low hanging fruit); while at 15 cents per apple, she would be willing to pick 45 apples. On the demand side Brenda expects that at 5 cents apiece, there is a demand for 30 apples, but at 15 cents per apple the demand is only for 10 apples.

Your tasks are to:

a. Form a multiplot of the supply and

demand functions.

b. Interpret the vertical asymptotes of

your two plots.

c. Determine the price at which Brenda

would "sell out" while satisfying the

demand.

d. Determine how much Brenda would

make at the price in Part c.

Follow-on Problem. Brenda's grandfather offers to subsidize the price of apples by 5 cents per apple. Assume Brenda adjusts her supply function to reflect her grandfather's subsidy and keeps her original demand function. Now:

e. Form a new multiplot of the supply

and demand functions.

f. How does your new multiplot compare

with the multiplot in Part a?

g. Determine the price that Brenda would

sell out while satisfying the demand.

h. Determine how much Brenda would

make at the price in Part g plus her

grandfather's subsidy.

i. Develop a different follow-on problem

.

[8] Notices

1. The sixth edition of the text Contemporary College Algebra: Data, Functions, Modeling is now in preparation. Comments and suggestions for improving the text are welcome. Please send them via e-mail to Don Small

   <don-small@usma.edu>.
   
   

2. The 2008 Joint Mathematics Meetings will be held in San Diego, CA, January 7-9, 2008. Two special sessions of interest are:
   

   Poster Session: Monday, January 7 from 2:00 to 4:00 PM
   

   "Sharing a Residue" session, Monday, January 7 from 5:00 to 7:00 PM
   
   

3. The next issue of the Vision-Potential Newsletter will appear in November 2007. Deadline for contributions to the November Newsletter is Thursday, November 1, 2007. Opinion articles, suggestions for writing assignments, small group in-class activities, small group out-of-class projects, Queries, announcements, etc. are welcomed. Please send material to Don Small, <don-small@usma.edu>.
   
   

4. Subscribe to this Newsletter
   
   

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