# Precalculus Reform at the University of Michigan: Reformed UM Precalculus

## Topics Covered in our Reformed Course

• Data analysis.
• Constructing linear models from real-world data sets.
• Functions.
• A basic "toolkit": constant, linear, quadratic, cubic, square root, absolute value, reciprocal, sine functions.
• Transformations, combinations, inverses, polynomial, rational, exponential, logarithmic.
• Re-expression to linearize data for analysis.
• Large-scale modeling (including applications of basic probability).
• Trigonometry: unit-circle, functional approach.
• Triangle applications done last.

## Characteristics of the Course

• Basic principles: the same as for our reformed calculus.
• The rule of four. Every topic should be studied geometrically, numerically, and algebraically, and communicated back to the instructor in a literate fashion.
• The way of Archimedes. Formal definitions and procedures evolve from the investigation of practical problems.
• Reformed text used: Contemporary Precalculus through Applications, by the mathematics faculty of the North Carolina School of Science and Mathematics, Janson, 1992.
• Less emphasis on manual algebraic manipulation, more on concepts.
• Emphasis on cooperative learning, in-class and with homework groups; de-emphasis on lecturing.

## Homework Assignments

• Both individual and group.
• Individual: typically 4-5 exercises daily, but nonrepetitive and nontrivial.
• Group: typically 3-4 exercises once or twice per week.
• Emphasis on applications and concepts.
• Writing skills and clarity of exposition stressed.

## Typical Group Homework Problem

The following table represents the measured amount g, in grams, of a certain chemical compound that can be dissolved in one liter of water at various temperatures T, measured in degrees Celsius.

```T     10   20   30   40   50   60   70   80   90
g   1.70 1.23 0.86 0.65 0.47 0.34 0.25 0.18 0.13
```
• In this relationship, which variable would it be natural to view as the independent variable and which as the dependent variable? Explain. (An aside: In practice, we would probably not add the command to "Explain." The students learn early in the course that this is a natural part of answering a question such as this.)
• Using median-median line regression, find a linear expression that models the number of grams of solute as a function of the temperature of the water.
• Describe the real-world physical meaning of the slope of the line given by your linear model.
• Describe the real-world physical meaning of each of the axis intercepts of the line given by your linear model.
• What are the natural domain and range of the function defining your linear model, taking into account physical constraints?
• Find an exponential model for the number of grams of solute as a function of the temperature of the water.
• Why would you expect the sign of the exponent in the equation that defines this model to be as it is?
• Let the above data set be called Data Set A. Suppose that another experimenter performs the same experiment and obtains other figures that will be called Data Set B. For low temperatures, the figures of Data Set B show that more of the compound can be dissolved in a liter of water than do the figures of Data Set A, but for high temperatures the opposite is true. If Data Set B were to be used instead of Data Set A to construct the exponential model, what would be the likely effect on each of the numbers appearing in the equation that defines the model?
• Which of the two models do you believe is better?

## Technology

• The text assumes technology is available to facilitate computations and graphing; the numbers and functions used are not always "nice," often arising from real-world data.
• TI-82 calculators are (essentially) required of all students; a few use other calculators at their own risk.
• Portable.
• Easy to learn to use.
• Have built-in features especially for the text being used.
• Programs can be transferred by cable link directly into calculators.
• The use of technology allows the exploration of topics once inaccessible in precalculus.
• For instance, students can learn to set up optimization problems, then find the solutions using the calculator. When they get to optimization problems in calculus, they already understand the concept and only have to learn the new methods. (Also, the graphical insight they get from solving the problems with the calculator can make them more alert to the need for knowing whether a place where a derivative is zero is a local maximum, a local minimum, or neither. In short, they may be less likely merely to set a derivative equal to zero and declare the problem finished.) An example follows.

## An Optimization Problem

Find the volume of the largest open box that can be formed from an 18.4" by 24.2" rectangle of cardboard by cutting congruent squares from each corner and folding up the flaps.

The students will find that

`V(x)=x(18.4-2x)(24.2-2x),`

where x is the side of each square and V(x) is the corresponding box volume. (Of course, finding this formula is the hard part of the exercise.) To finish solving using the calculator, they will also find it useful to know that the natural domain for this function is the closed interval [0,9.2]. Some experimenting (or some estimating of the size of the function) shows that the natural range will fit in the closed interval [0,1000]. Those familiar with the TI-82 will now recognize the following as the resulting graph window settings screen and the output from the calculator's graphical optimization routine.  The maximum volume is therefore about 686 cubic inches.

## "Prerequisite" Precalculus Knowledge

• We assume that the students have already seen, though not necessarily totally mastered, such topics as:
• The distance formula; laws of exponents and radicals; factoring of polynomials; simplification of rational algebraic expressions; equations of straight lines.
• Solution of equations, including those involving rational expressions and systems of linear equations; solution of inequalities, including absolute-value inequalities.
• Elementary properties of functions and graphs.
• Knowledge of trigonometry is not assumed; however, most students have some.
• Have our students really seen this material before? In most cases, they have. Here is the distribution of students in Math 105 in the fall, 1993 semester by the number of years of high school or college mathematics taken previously.
• 1 year: 0.7%
• 2 years: 1.4%
• 3 years: 15.9%
• 4 years: 77.5%
• 5 or more years: 4.3%
• Though the students have seen the material mentioned above, most have not mastered it, and are in Math 105 precisely because they failed a calculus placement test over these skills or by self-placement due to lack of comfort with the material.

## Gateway Testing

• Problem: Since our precalculus course does not stress manipulative algebraic skills, how can we guarantee that the students have these skills before being passed on to calculus?
• Solution: Gateway testing.
• Covers skills, not concepts.
• Comprehensive and rigorous.
• Pass/fail grading, but high score needed to pass.
• Usually offered outside class.
• May be repeatedly retried without penalty as long as it is eventually passed.
• New version of test available for each attempt.
• If a student is not at the skill level needed to pass the test, it is primarily the student's responsibility to reach that level, not the instructor's to spend valuable class time filling in holes in the students' knowledge that will vary from student to student.
• See the following article for more on our gateway testing program. We also have a WWW document available describing the UM gateway testing program, and another document describing the Lotus 123/TeX-based gateway testing software mentioned in the above article. This last-mentioned document includes a button you can push to have this software delivered into your disk directory.

## Homework Groups

• 3-4 students per group.
• Formed by instructors, not self-selected by students.
• Changed every 3-4 weeks.
• One or two group homework sets per week.
• Group homework is held to high standards of completeness, accuracy, and literacy.
• Group member roles (which change for each homework set).
• Recorder: keeps the "minutes".
• Scribe: writes up the final version of the homework.
• Clarifier: verbally summarizes and clarifies discussions during the group meetings.
• Manager: gets the group together, often provides refreshments.

## In-Class Cooperative Learning

• Students work on activities in pairs or larger groups, usually not the same groups as the homework groups.
• Activities are carefully chosen to get the students to interact, test out ideas on each other, reinforce individual understanding.
• Instructor has many roles, but not primarily that of a lecturer.
• Watch for general points of confusion; be prepared to interrupt and clarify.
• Summarize when groups are generally reaching some conclusion.
• Re-engage students not actively engaged with their group.

## Instructor Training and Support

• One-week training session (or professional development program) for those teaching our reformed precalculus or calculus for the first time.
• What to do when you have de-emphasized lecturing.
• Emphasis on concepts over manipulations.
• In-class cooperative learning.
• Homework groups.
• "Sick" groups and other problems with groups.
• The proper use of technology.
• Weekly meetings with instructors during the semester.

## Advantages of our Reformed Precalculus

• Students learn concepts better.
• Students have skills reinforced within the context of new material (so there is far less boredom).
• Students recognize applications as realistic and relevant.
• Technology allows realistic numbers and more complicated computations.
• Socialization valuable - mathematics not presented as "loner" activity.
• Emphasis on writing and other communication skills.

## Disadvantages of our Reformed Precalculus

• More work (and initially more stressful) for instructor.
• Requires instructor training.
• More difficult to implement with large class sizes.
• Technology absolutely required.
• More expensive for students.
• "Culture shock" for students - "I've never had a math course like this before. Will this really prepare me for calculus?"

## Questions? Want More Information? Return to the top of this document Go back to General Information on UM Precalculus Go back to Traditional UM Precalculus

Last modified Fri 12 May 1995 12:19 EDT

Bob Megginson
Department of Mathematics
University of Michigan
meggin@math.lsa.umich.edu