Contemporary College Algebra
By Don Small, U.S. Military Academy, West Point, NY 10996

Overview

Contemporary College Algebra is a course designed to educate students for the future rather than to train them for the past. The course, developed in collaboration with faculty in several disciplines as well as with people in the workplace, provides a strong base for quantitative literacy programs.

The primary goal of the course is to empower students to become exploratory learners, not to master a list of algebraic rules. Some of the means that are used to establish an exploratory environment for the students include:

1. Queries for engaging students in questioning and exploring the material being presented.
   
   
2. Exercises that explicitly ask students to: explore, ask what-if type questions, make up examples, further investigate worked examples, or iterate for the purposed of recognizing a pattern and developing a sense for the behavior of the solution.
   
   
3. Graphically fit a curve to a data set.

Other goals of the course include:

1. Improve communication skills: reading, writing, listening, presenting.
   
   
2. Small group work: in-class activities and out-of-class group projects. In-class activities culminate in student presentations to the class and out-of-class projects culminate in a written report.
   
   
3. Use technology: every student is expected to have daily access to a graphing calculator and/or computer.
   
   
4. Modeling: to empower students to use mathematics to quantify real-life situations.
   
   
5. Build confidence: develop personal confidence as a problem solver. Develop confidence in the iterative process: try something, note the errors, modify previous attempt to lessen the errors, and try again.
   
   
6. Enjoy applying mathematics to meaningful situations.
   
   

Concepts and techniques are introduced and motivated by real-life situations. Computational techniques are introduced in response to the need to solve real-life situations. For example, the quadratic formula is introduced in order to solve motion problems involving quadratic equations. The course emphasizes the importance of understanding elementary data analysis, the ability to extract a function relation from a data set, and the ability to mathematically model real-life situations in different disciplines.

Graphical analysis and problem solving in the modeling sense pervade the text. It is fundamentally important that students understand and experience the three stage problem-solving/modeling process applied to real-life problems. Because these problems usually involve dependency on many factors, the first stage of the process, creating a model, involves making assumptions in order to make the problem tractable. The second stage is the analysis-computation stage that leads to a mathematical solution. The results of this stage are often accomplished through the use of technology. The third stage is the interpretation of the mathematical solution in the setting of the original problem. If the solution is not applicable or meaningful, the process is repeated after modifying the assumptions. This process is illustrated by the following diagram.

A nice example of this process is developing a vehicle stopping distance model for a road network serving residential and business areas as well as highway travel. The first stage is accomplished by accepting the assumptions and data presented in the 1995 Texas Drivers Handbook (dry, paved roads).

Students are expected to form a scatter plot of the data and then fit a curve to the plot (second stage). An acceptable curve is given by the regression equation with . The third stage, interpretation, raises serious questions about the applicability of the model particularly for slow speeds. According to the solution, the stopping distance for zero speed is 46 feet! Because speed limits of 5, 10, and 15 miles per hour are common near schools, hospitals, and in congested areas, the model must be applicable to these speeds. Therefore changes in the assumptions and/or model need to be made and the process repeated. Two possible modifications to the assumptions would be to include (0,0) as a data point or to require that the regression curve pass through the origin. Requiring the latter yields the regression equation . The value is 0.9977, only slightly less than for the previous model.

Features of the course include:

Overview: The analysis of data is the starting point for most of the topics. Instructors are encouraged to begin the course by having students fill out a chart of individual characteristics (e.g., height, weight, shoe size, eye color, male/female, hair color, etc.) This data chart is then reproduced and distributed to the students as their "Class Data." Questions involving the Class Data are asked throughout the semester. For example, compile a class profile based on the Class Data.

Data and Variables: Study how to read and display data---table, pie chart, scatter and line plots, and bar charts. Learn the meanings, use, and methods to compute the three principle summary measures of a set---average (mean), median, mode. The understanding that data is information about a variable, introduces the meaning of variable and its use as a mathematical pronoun. The exploration of relations between variables leads to the study of straight lines. Applications of linear equations lead naturally to systems of linear equations, linear inequalities, and their applications in linear programming.

Functions. The concept of function is one of the most important concepts in mathematics. The concept is introduced informally through discussions of academic grades, modeling water level in a well, and warming a can of soda. Definitions of function and related terms are then clearly presented and illustrated. Graphically extracting function relations from data sets introduces the shapes of the basic functions---power, radical, exponential, logarithmic, and periodic (sine, cosine). The skill to graphically fit a curve to a data plot is enhanced by studying the function transformations of shifting and scaling. The development of linear regression for lines passing through the origin is used to illustrate the regression concept. However, students are expected to use regression programs in their calculators when computing a regression function. The algebra of functions is developed graphically, symbolically, and numerically. The ability to display data and to graphically approximate numerical solutions of equations and zeros of functions is an important thread throughout the course.

Modeling. The concluding chapter in the text for the course contains several sections, each of which is devoted to modeling real-life situations in a particular discipline (e.g., business, life sciences, economics, the arts, etc.). The principle techniques used are graphical approximations and recursive sequences (e.g., New Situation = Old Situation + Change). An example of how this chapter is used as a capstone for the course is given by Carrington Stewart of Texas Southern University. He divides his class into seven groups and assigns a particular section to each group. The group's responsibility is then to learn the material in their section well enough to present it to the class.

Fun Projects

These are small group, out-of-class projects. Two or three Fun Projects are assigned each semester. Each project involves an inquiry component as well as a written report. The following three Fun Projects illustrate the scope of these projects.

• Recommended Daily Amount of Sodium

  On a plane ride to Montana, a stewardess gave Don a can of Welch's orange Juice. The nutrition label on the can listed 15 mg of sodium representing 1% of the recommended daily value (DV) based on a 2,000 calorie/day diet. The person in the next seat had a can of Canada Dry Ginger Ale which listed 90 mg of sodium representing 4% of the DV. Can both of these labels be correct? Can you tell the DV for a 2,000 calorie/day diet from the information on a nutrition label? How accurate is the information on nutrition labels? In particular, would it make a difference if the weight or the percentage figure were rounded off to a full integer?

  Your tasks are:

  1. Record the weight (mg) and the percentage amounts listed for sodium on at least ten other varieties of soda.

  2. For each variety, determine the allowable range of DV assuming that both the weight and the percentage figures were rounded to full integers.

  3. Are there any contradictions in the data that you collected? That is, could the sodium information on all of the labels be correct.

  4. What information concerning the DV for sodium in a 2,000 calorie/day diet can you extract from your data?

  5. What is the official Food and Drug Administration DV for sodium? (Hint: Research the FDA website.)

  6. Include in your written project report, a one page essay on nutrition labels. Base your essay on the FDA article "Scouting for Sodium" that originally appeared in the September 1994 FDA Consumer and was later revised and reprinted in September 1995. The article can be found at http://www.fda.gov/fdac/foodlabel/sodium.html.

• The Optimal Dimensions of a Soda Can

  The article "The Aluminum Beverage Can" in the September 1994 issue of Scientific American estimated that 100 billion drink cans are produced every year. With this number of cans, a sizeable increase in profits can be realized by making a small reduction in the amount of material that is used in producing a drink can. Today's 12-ounce beverage can weighs approximately 0.48 ounces compared to the 0.66 ounces when the cans were first introduced in the 1960s. The savings from a further reduction of one percent are approximately $20 million dollars per year. In this Fun Project, you are asked to optimize the dimensions of a 12-ounce aluminum beverage can. Assume that the can is a closed cylinder. That is, ignore the neck, domed base, and pull-tab of an actual can. Also assume that the idealized can holds 12 ounces (1 ounce = 1.8047 cubic inches).

  Your tasks are:

  1. Write a letter to a soft drink company asking for the approximate number of 12-ounce aluminum beverage cans the industry produced during the previous year. Include a copy of your letter and the response received in your project report.

  2. Carefully measure the diameter and height of a 12-ounce aluminum soda can.

  3. Assume that the material of the can has uniform thickness (bottom, top, sides). Let the thickness be one unit. Express the volume of material of a 12-ounce aluminum soda can as a function of the radius of the can and then plot the function.

  4. Using the plot from Task 3, graphically determine the dimensions (height and radius) that minimize the volume of material of the can.

  5. Compare your results against the dimensions of an actual soda can. If the results are considerably different, explain why the soft drink company would not use your dimensions in order to save money.

  6. Change the assumption in Task 3 to be that the thickness of the top is 3 times the thickness of the sides and bottom (to allow for the pull-tab). Now repeat Task 3. Compare your results against those of an actual can. Comment on the comparison.

  Teacher notes:

  1. If the students have access to a physics lab, encourage them to measure the thickness of the bottom, sides, and top of a soda can.

  2. This project offers an interdisciplinary cooperation opportunity with the English Department. For example, invite an English instructor to speak to your class about writing a business letter.

  
  
  

• Oxygen Levels in the Naraguagus River

  Environmentalists have grown concerned over the algae build-up in the Naraguagus River. A scientist assigned to look into the issue asked that the oxygen levels at a designated location in the river be monitored for four successive days. Due to a mix-up in communications, the field worker assigned to do the monitoring understood that he was to take oxygen readings in the river at four different times on the same day. He reported the following data. (The oxygen level is reported as the number of milligrams of oxygen for every 1,000 grams of water.)

  Time	Oxygen Level
  5:00 am	8
  11:00 am	10
  5:00 pm	17
  midnight	10

  The scientist, disappointed by having data for only one day rather than four days, has turned to your group for help. She comments that the day-to-day changes in the weather during the four-day period were minimal and asks your group to develop several models of the oxygen level in the water as a function of the time of day for the four-day period. She then wants you to determine which model is best and to describe your reasoning in clear statements.

  Your Tasks are:

  1. Research oxygen levels in a river.

     1. a. List five factors that affect oxygen levels in a river.

     2. Determine the most important factor (explain your reasoning).

     3. Describe how oxygen gets into the water.

     4. Explain why oxygen levels may differ at different times of the day.

  2. Plot the data.

  3. Graphically fit a curve to your data plot. List the major characteristics you would like the curve to have.

  4. Using the regression capability of your graphing calculator, determine the following regression models: (a) Linear (b) Quadratic (c) Cubic (d.) Quartic (e) Sine

  5. Determine which of the five regression models gives the best fit for the four-day period. Describe your reasoning for each of the five regression models.

  6. Superimpose the plot of your best model on the plot of the data.

  7. Use your best model to predict the oxygen level at 2:00 PM on the third day.

  
  
  

Summary

Anecdotal evidence strongly suggests that student-centered pedagogy combined with emphasis on modeling real life situations has a positive impact on student attitudes. Several instructors have commented on "how alive the class is" and on the depth of the questions students ask. The use of technology has enabled students to model and analyze situations, which they had previously been unable to do because of the algebraic manipulations involved. The emphasis on graphical analysis, rather than just symbolic, has bolstered student confidence. Group work, particularly projects, has also contributed to student confidence and self-satisfaction. Student pride is clearly evident in the project reports.

Contact Person: Don Small, Dept. of Math. Sci., U.S. Military Academy, West Point, NY 10996 [don-small@usma.edu] (845) 938-2227

The text for the course, Contemporary College Algebra: Data, Functions, Modeling, 5th edition, by Don Small (McGraw-Hill College Custom Series) was developed through the HBCU Consortium for College Algebra Reform. A monthly newsletter Vision-Potential helps network the instructors teaching the course.