# MAT 506 Foundations of the Rational Numbers (3 units)

Instructor: Stan Yoshinobu
Office: NSM A-117
Office hours: 4 hours per week TBA
Phone: 310-243-2367
Email: syoshinobu@csudh.edu

### Course Description

This course covers the theory and applications of Rational numbers. Topics central to the course are Number systems, representation of numbers, equivalence classes, rationality and irrationality, properties of the rational number system, applications of the central ideas to proportional reasoning, and developing intuitive models for the standard rules and algorithms. The course meets for 3 hours of lecture per week.

### Prerequisites

MAT 543 or concurrent enrollment. Students must have graduate standing and one year of full time secondary teaching.

### Objectives

After completing MAT 506 the student will be able to

State definitions of basic concepts

Understand the rational number notation and equivalence classes

Recognize irrational numbers and prove that some numbers are irrational

Construct proofs of mathematical assertions

Construct intuitive, concrete explanations and models of standard rational number rules and algorithms

Engage in problem solving using foundations of Rational numbers and proportional reasoning

Be knowledgeable of the current research and theories of proportional reasoning instruction

Create short units of instruction that address common misconceptions of Rational numbers and proportional reasoning

### Expected outcomes

Students should be able to demonstrate through written assignments, tests, and oral presentations that they have achieved the objectives of MAT 506.

### Method of Evaluating Outcomes

Evaluations are based on homework, class participation, short tests, scheduled presentations and group projects that cover students' understanding of topics.

Students' grades are based on homework, class participation, short tests, and scheduled presentations and group work. The instructor determines the relative weights of these factors.

### Text

No single text will be assigned, as no single book exists that is suitable for the course. Library research by students and class handouts will be used instead.

#### Course Schedule

• Week 1: Common misconceptions about fractions and proportional reasoning group discussion

• Week 2: Delving into theory. Building the Rational numbers from scratch, number systems, representing numbers and equivalence classes

• Week 3: Continuing the development of the theory of Rational numbers. Equivalence classes

• Week 4: Incompleteness of the Rational Numbers: Irrationality and Rationality

• Week 5: Arbitrarily close: The density of the Rational numbers in the real number system

• Week 6: Developing concrete models for the addition and subtraction of fractions

• Week 7: Developing concrete models for multiplication and division of fractions

• Week 8: Creating number sense with rational numbers and choosing group project topics

• Week 9: Problem solving using proportional reasoning and the structure of the Rational numbers

• Week 10: Problem solving using proportional reasoning and the structure of the Rational numbers

• Week 11: Problem solving using proportional reasoning and the structure of the Rational numbers

• Week 12: Teaching and learning issues in proportional reasoning

• Week 13: Teaching and learning issues in proportional reasoning. Discuss math Ed papers about proportional reasoning.

• Week 14: Group Presentations

• Week 15: Group Presentations

### Attendance Requirements

Students are required to attend all scheduled classes.

### Policy on Due Dates and Make-Up Work

Due dates of assignments and projects will be announced in class. No late work will be accepted.

### Schedule of Examinations

The instructor sets all test dates. Students will be notified well in advance. The final group project is due on the last day of class.