# Welcome to Math for Teachers!

NOTE: This is the syllabus for a PREVIOUS VERSION of the course. See nilesjohnson.net/teaching.html for current courses.

This is the course homepage and syllabus for Math 2137, Algebra and Coordinate Geometry for Teachers. Here you will find our course calendar, current assignments, basic course information, and links to additional content of interest. This syllabus may need to be updated as the course progresses, and you will always find the current version at http://www.nilesjohnson.net/teaching/AU16/2137.html.

This boy is using geometry to understand algebra and algebra to understand geometry.

The topics of middle-grade math bring adventure and excitement! Together, Math 2137 and 2138 will hone your skills of mathematical explanation and explore the profound relationships between algebra and geometry.

## Course Objectives

This course integrates the various types of numbers introduced in the previous course to present them as members of a single (real) number system. The notion that new numbers are discovered as solutions to equations is promoted, and motivated by connecting various equations with mathematical models.

Matrices are introduced and used as linear transformations, mainly in the plane. The complex numbers are introduced as general solutions to quadratic equations and the relationship between complex arithmetic and transformations in the plane is explored.

The course finishes with several weeks of geometry content for middle grade teachers, including more material on proofs, triangle congruence, and non-Euclidean geometry. The main example is "Taxicab geometry", based on the ell_1 norm.

### Topics

• Polynomial arithmetic as “base-x” and binomial theorem
• Real number system
• Polynomial equations and their roots
• Exponential and logarithm functions
• Complex numbers
• Matrices
• Complex arithmetic and linear transformations in the plane
• Geometry proofs
• Taxicab geometry

### Learning Goals

• Understand polynomial arithmetic from the perspective of place value.
• Unified perspective on the real number system, including situations modeled by different numbers, and numbers as solutions to equations.
• Familiarity with complex numbers and matrices from algebraic and geometric points of view.
• Awareness of non-Eulidean geometries and the importance of the parallel postulate.
• Ability to create and evaluate geometric proofs.
• Identify major historical developments in algebra and number systems including contributions of significant figures and diverse cultures.

## Section Information (#25274)

Hopewell 119
Mondays, Wednesdays 3:55—5:15

## Course Calendars

Our class meets according to the OSU Registrar's academic calendar.

Our class will generally follow this calendar of topics; this may be adjusted as the course goes on.

#### Homework 9

Due Monday, 11/28:

1. Read all of Lang chapter 6.
2. Which classes of mappings does the identity transformation belong to?
3. 6.1.1: Describe the fixed points of various mappings.
4. 6.4.10, 6.4.11: Composition of isometries, congruences, and permutations.
5. 6.6.7: SAS triangle congruence.

#### Homework 8

Due Monday, 11/07:

1. Consider the 2 x 2 matrices A and B below. Let I be the identity matrix, also below. $A = \begin{bmatrix}3 & 1\\2 & 4\end{bmatrix} \quad B = \begin{bmatrix}p & q\\r & s\end{bmatrix} \quad I = \begin{bmatrix}1 & 0\\0 & 1\end{bmatrix}$ Explain how the equation AB = I can be viewed as a system of four linear equations with four variables. Solve this system of equations.
2. Solve exercises 2.1.4, 2.1.9(b) both directly, and with the determinant formula. Explain how the two methods are related.
3. Consider a square with corners labeled (in counter-clockwise order) 1, 2, 3, 4. Let r be the permutation (1 2 3 4) and let s be the permutaiton (2 4). Note that composition of permutations is not generally commutative, but it is associative. Write all the different permutations you can get by combining r and s. (There should be 8 distinct possibilities; combining any two yields one of these 8, and you can show all of these on an 8 x 8 grid like a multiplication table.) Compare with exercises 9 -- 12 in section 6.4.

#### Homework 7

Due Monday, 10/31:
17.1.4, 7.1.7, 17.2.1, 17.3.5, 17.4.3.

#### Homework 6

Prepare for student presentations. A list of main topics is due Monday, 10/17. Presentations will be the following week: 10/24 and 10/26.

Subjects are:

• Complex numbers (Ch. 15)
• Hyperbolas (Sec. 12.4, 12.5)
• Permutations (Ch. 14; mostly sec. 14.3)

#### Homework 5

Due Monday, 10/17:
13.1.8, 13.2.1, 13.2.3, 13.2.8(a), 13.2.8(d), [two problems of your choosing from 13.3], 13.4.6, 13.5.5.
Additional 1: Let a be a real number greater than 1, and let f be a function. Discuss the differences between the graphs of f(x), a * f(x), and f(a*x). Give one or two illuminating examples.
Additional 2: Let f(x) = 2x^3 - 3x^2 - x + 4. Use the method outlined in class to rewrite f as a function of (x-c) for c = 3 and c = -1.

#### Homework 4

Due Monday, 09/26:
1. Find the center and radius for the circle described by $x^2 + 4x + y^2 - 2y - 4 = 0$ Hint: complete the squares
2. Find 3 Pythagorean triples that we have not ever discussed in class, and are not multiples of ones discussed in class.
3. Find 3 rational points on the circle from part 1. Hint: use translation and dilation from the unit circle.
4. Consider a parallelogram with side lengths 3 and 5 and with short diagonal 4. What is the length of the long diagonal?
10.2.1, 10.2.5

#### Homework 3

Due Monday, 09/12:
Prepare class presentations! Plan for 20 -- 30 minutes. In addition to your presentation, choose a few helpful homework problems to assign to the other students.

#### Homework 2

Due Wednesday, 09/07:
3.2.1, 3.2.3, 3.4.4, 3.4.8, 3.4.13, 3.4.19, 3.4.27, 3.4.29
Additional Problem: Show that $$\mathbb{Q}(\sqrt{5})$$ satisfies the additivity and multiplicativity axioms for number systems (the most interesting one is multiplicative inverses).
Also: Prepare a 20 minute presentation on the sections you volunteered for in class! Either 8.3/8.4, Ch. 4, or 9.1/9.2.

#### Homework 1

Due Monday, 08/31:
1.6.12, 1.6.14, 1.6.16, 2.1.2, 2.2.3, 2.2.10
Additional Problem: Use the Pythagorean theorem to define distance in 3-space. Find the distance between the points P = (1, 2, 4) and Q = (-1, 3, -2).

## Basic information

### Instructor

Niles Johnson
Office: Hopewell 184
Office Hours: Mondays and Wednesdays 10:30 – 12:00; Thursdays by appointment

### Textbook

Basic Mathematics by Serge Lang. ISBN-13 978-0387967875.

The style of this book is different from typical math textbooks, and is much closer to the way mathematicians communicate with eachother. Lang invites the student to experience and learn mathematics at a level which is deep, but not complex. Our focus will be Parts III and IV, but it is illuminating to compare the first two parts with Beckmann's text.

Bart Snapp has developed free texts for middle grade teachers, and they can be a helpful supplement to Lang.

### Exam Schedule

We will have two in-class midterms and a final exam. All students must take the exams at the scheduled times indicated here.

#### Midterms (2):

Wednesday, Sept. 28; Wednesday, Nov. 09

#### Final:

Scheduled by the registrar. See campus schedule for exam times.

### Attendance

Active participation in class is an essential part of this course, and so attendance every day is required. Please let Niles know as soon as possible if an illness or other commitment will prevent you from attending. Homework is due at the beginning of each class, and late homework cannot be accepted unless prior arrangements have been made. See full late policy below.

### Calculators and Mathematical Software

There are a variety of modern tools which support mathematics learning and application. We'll use several of them in this course, but no technology will be used on the quizzes or exams. In particular, calculators and cell phones will not be permitted. The only materials you'll need are writing instruments and your mind.

## Assessment

Your final grade will be based on written homework, written and oral in-class participation, quizzes, midterms, and a final exam. The precise breakdown is as follows:

• In-class participation: 20%
• Homework: 20%
• Midterm exams: 20% each, for a total of 40%
• Final exam: 20%

### Participation

Thoughout this semester we will be focused on the how and why of basic mathematics. This means that you will be responsible both for knowing the content and for knowing how to explain the content. We will practice this in a variety of ways, and much of this practice will take place in class.

Teaching mathematics requires listening carefully to students, assessing their ideas, and responding in ways that make sense. Our class participation is designed to practice these essential skills. You will have opportunities on a daily basis to listen to your fellow classmates explain ideas and ask questions. You will be asked to respond with your ideas and with additional questions. Together we will see how and why mathematics works! Participating in this course includes all of the following:

• Show interest in mathematical ideas.
• Show interest in different ways of approaching mathematical ideas.
• Listen carefully to different ways of solving a problem.
• Carefully evaluate a proposed method of solution.
• State whether you agree or (respectfully) disagree with a statement.
• Show interest in learning with and from others.

### Homework and Quizzes

Homework and Quizzes will be scored using the rubric below. Scores are based both on mathematical correctness and quality of explanations. Homework should be typed and presented as you would an essay. If necessary, you may draw diagrams by hand on separate pages. Quizzes will be hand-written during the first 10 or 15 minutes of class.

The homework is intended to give you time to develop your explanations and understanding of the content. You are encouraged to work with your classmates on this, but you must write your own explanations. Homework is designed to help you learn, and not as an assessment tool. Therefore grades for homework will be recorded on a complete/incomplete basis (scores of 3 and higher are complete). The numerical scores are given only for your information.

The weekly quizzes are opportunities to evaluate your current grasp of the material. They will be very short and based on previous homework problems or class activities. Quiz grades will be recored as scored.

### Late Work Policy

If you miss class on the day homework is due, or forget to bring a printed copy of your homework, you may send me a copy by email the next day. I will accept up to three homeworks this way.

If you must miss class on the date of a quiz or exam, please do the following:

• Contact me before the time of the assessment to let me know you will be away.
• Provide a reason and objective documentation (e.g., a note from your health care provider).
• Make up the assessment within two days of your return to campus, and before the next class if possible.
• Miss no more than two assessments this way.

If you do each of the items above, you will have no penalty. For each item that you do not complete there will be a 10% reduction in your final score.

The descriptions on the rubric below are meant as a guide to help answer the question What constitutes a good explanation of mathematics? This is a subtle and challenging question—one that is worthy of considerable time and energy. As you write your explanations, consider the distinction between Procedure and Conceptual Meaning. A complete explanation addresses both of these, but the conceptual meaning is essential. Explanations that address procedure only are a disservice to students and do not support their future learning. They are also much less interesting!

Quizzes and exams will follow this scale; Niles will use these for comments on homework too, although homework grades are recorded on a complete/incomplete basis. Homework will be considered complete if the average score is R (3) or higher.

G++
Exemplary
Excellent work that exceeds the assignment guidelines.
G+
Good plus (10)
Correct procedure with an explanation that effectively addresses relevant definitions and concepts.
G
Good (9)
Correct procedure with nearly complete explanation.
Em/G
Emerging/Good (8)
Well-developed but incomplete explanation.
Procedural errors are minor or nonexistant.
Em+
Emerging plus (7)
Emerging explanation that shows understanding.
Procedural errors are minor or nonexistant.
Em
Emerging (6)
Explanation that mentions core definitions or conceptual meaning relevant to the question.
Possibly some non-minor procedural errors.
R/Em
Relevant/Emerging (5)
Work that has merit but also has significant shortcomings in the procedure and/or explanation.
R
Relevant effort (3)
Work that shows relevant effort but is seriously flawed.
NC
No credit (0)
No work submitted, or no relevant effort shown.

## Resources related to mathematics education

### Other websites of interest

• Project INTERMATH, which focuses on building teachers' mathematical content knowledge through mathematical investigations that are supported by technology.
• Purplemath has lessons and practice problems for elementary mathematics.
• Khan Academy has math videos for children and resources for parents or teachers.
• Report by the National Council on Teacher Quality on mathematics in the U.S.: No Common Denominator. The report includes a great appendix on content that mathematics teachers must know, and ranks Beckmann’s textbook highest overall among elementary mathematics content textbooks.

## GEC Information

This Mathematics course can be used, depending on your degree program, to satisfy the Quantitative and Logical Skills category of the General Education Requirement (GEC). The goals and learning objectives for this category are:

#### Goals:

Courses in quantitative and logical skills develop logical reasoning, including the ability to identify valid arguments, use mathematical models and draw conclusions based on quantitative data.

#### Learning objectives:

Students comprehend mathematical concepts and methods adequate to construct valid arguments and understand inductive and deductive reasoning, scientific inference and general problem solving.

## Support Services

All students are encouraged to take advantage of campus support services. These include tutoring, academic advising, counseling, and the math learning center! A complete list is available at http://newark.osu.edu/students/support-services.html.

## Accommodation Statement

If you need accommodations due to a disability, you must first register with the Office for Disability Services (ODS) at 226 Warner Center, ext. 441. After you receive your authorized accommodation from ODS, you should show me your access plan and discuss your needs with me. Ideally, we should meet within the first week of class.

http://newark.osu.edu/students/student-life/disability-services.html

http://www.ods.ohio-state.edu/