Properties of the Real Numbers I

Spring 1998

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Instructor: Dr. Art Duval

Please feel free to come by my office any time during scheduled office hours. You are welcome to come at other times, but in that case you might want to make an appointment, just to make sure that I will be there then. You can make an appointment simply by talking to me before or after class, by calling me at my office or at home, or by sending e-mail.

You may also ask any questions directly via phone or email. If I'm not in when you call, please leave a message on the voice-mail or answering machine with your name, number, and a good time for me to call you back. I will try to repond to your phone or e-mail message as soon as possible.

I will also be available after class in the clasroom for a while most days.


The problems in the course are intended to acquaint you with the following concepts in roughly the following order: Counting and Whole Numbers, Integers, Base Representations, Number Theory, Geometric Congruence and Similarity, Ratio, Proportions, Rational Numbers, Decimal Representations, Irrational Numbers, Data Analysis, and Symmetry Transformations. In addition to these mathematical concepts, you will become familiar with the following computer software: ClarisWorks (word processor), Divide & Conquer, Geometer's Sketchpad, and Function Probe.

Guiding Philosophic Principles

The purpose of this course is for you to become involved with a wide variety of situations and contexts which give rise to mathematical concepts essential for K-8 teaching. You will be expected to engage in a dialogue between grounded activity and systematic inquiry. Grounded activities will include situations arising from physical activity with strings, sticks, blocks, cardboard, marbles, dice, coins, cut paper, models, photographs, and anything else under the sun. Systematic inquiry will involve the fullest possible use of the tools that are commonly available in our culture, including both physical tools, such as rulers, measuring cups, scales, stopwatches, thermometers, projectors, calculators, and computers; and linguistic tools, such as words, numbers, symbols, drawings, diagrams, tables, graphs, along with computer software which can be considered as both a physical and a linguistic tool.

Mathematics is a dialogue between grounded activity and systematic inquiry. This dialogue is a deep and essential element of what it means to be human, and all humans are engaged in some form of this dialogue. The refinement of one's expression of this dialogue is achieved through a broader set of physical experiences, and clearer communication tested in the context of social interaction.

Course Expectations

Working in small groups with your classmates, or alone, you will engage each problem or situation presented in class, and will attempt to describe and explain the results of that engagement in a written report. You must write your report yourself (a private oral exam is always possible, in case of irregularities). A description, experiment, explanation or proof is anything which is both meaningful to you and convinces other people of the validity of your thoughts, words and activities. Reports may incorporate any available media including written words, symbols, pictures, diagrams, models, tables, graphs, videos, computer discs, etc.

There will be an initial due date (approximately weekly), by which time some work (however partial) must be turned in. If you do not turn in a report by the initial due date, you will not be able to resubmit it; please see me as soon as possible if some emergency prevents you from attending class and turning in your report

Reports will be returned as soon as possible with questions and comments, after which you may respond, revise, amend, and then resubmit the project. Resubmitted solutions will again be returned with comments and may again be revised and resubmitted. After the initial reports are returned, selected students will be chosen to present their work in class. After such presentations others may still continue to resubmit those same projects in their own personal way making full or partial use of what has been presented. Originality and diversity of expression will always be encouraged.

Each project will be returned with one of three marks:

"check minus"
Some engagement with the project, but substantial questions remain.
A well reasoned explanation, but some questions remain.
"check plus"
A complete and thorough explanation, no further questions.
If you eventually achieve a "chek" on 10 of the projects, you will get a grade of C or better. If you eventually achieve a "check plus" on 11 of the projects, you will get an A.

Remember to pace yourself in a reasonable way. It is virtually impossible to turn in well-written resubmissions if you leave them all until the end of the semester, so the following deadlines will apply:

Your final grade will depend on a portfolio of all your submissions for the semester; therefore, keep all your submissions (even after turning in subsequent resubmissions for the same project). Your portfolio will be due on Wednesday, May 6.


Because of the experimental nature of this class, on-time attendance is mandatory at all times. If you have more than four unexcused absences, you will be dropped from the class with an F. I will usually excuse an absence if you tell me about it in advance, or, in cases of emergencies, as soon as possible afterwards.


All students will be required to purchase several computer discs, a ruler, a compass, scissors, and other incidental supplies as required by the problems. I recommend the purchase of a Texas Instruments Math Explorer Calculator, since that is what is widely available in most schools. A textbook is not required, but many of the topics that will emerge in this course are discussed in the book A Problem Solving Approach to Mathematics for Elementary School Teachers, by Billstein, Libeskind, and Lott, 6th ed. (1997), Addison Wesley. Students may wish to consult this book at their discretion. Students who work together might easily share a textbook. The math department is working on making this book available in the UTEP bookstore.

Final Word

Mathematics is not a march down any particular road, but rather a walk in a garden with many branching paths that circle and wind back onto themselves. Visitors stroll in many different ways, pausing to look down at a single small flower, or gaze out over an enchanting vista, or perhaps even to water, weed or plant (for a garden is a human creation constructed within the constraints of life). Each new return brings its own special set of views and experiences. - David Dennis