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.

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.

Additional Texts:

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

• Numbers and Algebra
• Parallels in Geometry

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

Grading Rubric

Resources related to mathematics education

Websites on math standards

• Ohio Department of Education Standards in Math and Model Curricula.
• National Common Core State Standards in English and Math: Adopted by 43 states (and counting), including Ohio.
• National Council of Teachers of Mathematics Standards and Focal Points.

Websites of elementary school texts from high performing countries

• Textbooks used in Singapore: singaporemath.com Primary Mathematics 1A, 1B (first grade) through 6A, 6B (sixth grade)
• Translations of textbooks used in Japan: http://www.globaledresources.com/ (Tokyo Shoseki’s Mathematics for Elementary School)

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/

Academic Misconduct Statement

It is the responsibility of the Committee on Academic Misconduct to investigate or establish procedures for the investigation of all reported cases of student academic misconduct. The term "academic misconduct" includes all forms of student academic misconduct wherever committed; illustrated by, but not limited to, cases of plagiarism and dishonest practices in connection with examinations. Instructors shall report all instances of alleged academic misconduct to the committee (Faculty Rule 3335-5-48.7). For additional information, see the Code of Student Conduct at http://studentlife.osu.edu/csc/