# Math 135: Algebraic Reasoning for Teaching Mathematics (Instructor: Steffen Lempp)

#### SCHOLARSHIP:

The Brookhill Foundation now supports Math 135 students as well as students taking the math/science minor for elementary and special education majors by a scholarship!

#### CLASS POLICIES:

Handout stating class policies

#### HOMEWORK:

Homework will be assigned on Wednesdays at the beginning of lecture and due the following Wednesday at the beginning of class.

There will be no credit for late homework except in case of illness or family emergency.

• HW # 1 DUE 1/26: Primary 5A, p. 25, # 1, 4, 7, 10 (using bar diagrams) and p. 60, # 2, 4, 6, 8 (in two ways, using both methods on page 58); Lecture Notes, p. 4, # 1.1
• HW # 2 DUE 2/4: Lecture Notes, p. 8, # 1.3, 1.4; NEM 1, p. 25, # 5-7 and p. 27, # 1, 5; Primary 6A, p. 29, # 3, 6 and p. 33, # 2, 6 and p. 38, # 4, 6 and p. 60, # 7, 10 and p. 68, # 8, 10 (all using bar diagrams!)
• HW # 3 DUE 2/9: Primary 6A, p. 82, # 2, 4, 7 (using pictures as in text); NEM 1, p. 141, # 2dfh, 4dfh and p. 147, # 2dfh, 4dfh (In the last two exercises, explain each step carefully by a rule or a definition!)
• HW # 4 DUE 2/16: Lecture Notes, p. 23, # 2.1; NEM 1, p. 139, # 4dg, 5, 7, 10, 13 and p. 174, # 10, 13 and p. 181, # 17, 21; NEM 2, p. 117, # 4, 6, 8, 9 (GST = general sales tax)
• HW # 5 DUE 2/23: Lecture Notes, p. 33, #2.2 and p. 35, #2.4 and p. 41, # 2.5; NEM 1, p. 158, # 3dhl and p. 161, # 12, 24 and p. 166, # 4, 10, 17, 20 (with the method of this section, i.e., using only one variable!); NEM 3A, p. 77, # 1df, 3abc, 4bd, 6bd, 8
• HW # 6 DUE 3/2: NEM 2, p. 53, # 4, 6, 10 (literal = linear) and p. 78, # 17, 37 and p. 141, # 6, 12 (by elimination method) and p. 148, # 10, 20 and p. 151, # 11, 18, 20, 21
• HW # 7 DUE 3/9: Lecture Notes, p. 61, # 3.1-3.2; NEM 2, p. 159, # 1 and p. 163, # 3dfhj and p. 164, # 4dhl, 5-7
• HW # 8 DUE 3/23: NEM 3A, p. 47, # 4, 18, 22 and p. 50, # 5, 10, 12, 14 and p. 55, #3-5 (Typo for # 22 on p. 49: Second inequality should read "2x+1 < x+2")
• HW # 9 DUE 3/30: NEM 2, p. 122, # 3, 5, 9, 10; NEM 3A, p. 102, # 2, 10, 12 and p. 106, # 1bfj, 2 and p. 109, # 2, 6, 8, 12, 14
• HW # 10 DUE 4/6: NEM 2, p. 28, # 3dhl, 4bdh and p. 31, # 2bdf and p. 33, # 6dhlpt; NEM 3A, p. 114, # 6, 10, 16, 17; also solve:
1. |.5x+4| > 2.5
2. |4(x-5)| ≤ 3
3. |x-1| < 2x+5
• HW # 11 DUE 4/13: NEM 2, p. 65, # 1hlpt and p. 66, # 3, 7, 11, 14; NEM 3A, p. 31, # 1dh, 2dh and p. 39, # 12, 15, 17, 20; Lecture Notes p. 98, # 5.1 and p. 99, # 5.2 (Note: Solve all NEM 2 quadratic equations by factorization!)
• HW # 12 DUE 4/20: NEM 2, p. 133, # 3 and p. 135, # 1, 2, 4, 5; NEM 3A, p. 34, # 1bf, 2lptz, 4bd and p. 41, # 2, 4 and p. 129, # 5 and p. 138, # 1, 5; also solve:
1. 2x2 < 5x-1
2. -x2+5 ≥ 2x
• HW # 13 DUE 4/27: Lecture Notes, p. 125, # 6.1, Add'l Math, p. 78, # 1d, 2df, 4df and p. 81, # 1dg, 3bd, 5 and p. 83, #1c, 2f, 3c (note that in the Singapore math books, "index" = "exponent", and "surd" = "algebraic expression involving roots")
• HW # 14 DUE 5/6: Add'l Math, p. 87, # 1e, 2b, 3di and p. 91, # 2cd, 4cf and p. 97, # 2dgh, 6, 8, 10 and p. 100, # 2, 5, 7 and p. 103, # 2ace, 4, 9, 13

#### PROJECT TOPICS

Each project requires a group of two students to compare how the topic is introduced in one of our Singapore math schoolbooks and one American math schoolbook of your choice. Each student should pick a partner and a joint topic. (Pick an American schoolbook of your choice from CIMC, e.g., the one you yourself used in your own school, or one that is currently used in a school you are familiar with. The more different that book is from our Singapore math schoolbooks, the more interesting the project will be.)

The comparison I am looking for is how the mathematical concepts are introduced. I'm not necessarily looking for a value judgment from you, which one you like better, but mainly for an investigation of how the presentation of the topic differs from a mathematical point of view. E.g., is the concept defined differently? Are the examples by which it is introduced very different? Etc.?

Your project will consist of a short (joint) paper (of 2-3 pages), with copies of the relevant "other" schoolbook pages attached, and a short presentation (10-15 minutes) in class about this comparison of topics. You can use an overhead projector or copied handouts if you prefer. (I'll help you get an overhead projector if you need one.) Presentations will start in mid-April. I suggest that each group meet with me briefly a few days before your presentation for feedback.

Here is a list of possible topics (with reference to treatment in my Lecture Notes):

1. How are the properties of arithmetic (commutative law, etc.) first presented? (Cf. Proposition 1.2. Presentation by NC and KH on April 29.)
2. How is the slope of a line defined? (How is it motivated that one can define the slope as one single number? (Cf. section 2.4. Presentation by CM and JR on May 2.)
3. How is solving two linear equations in two unknowns presented? (Cf. section 2.5. Presnetation by SS and KVD on May 4.)

#### COPIED PAGES FROM OTHER SINGAPORE PRIMARY MATH BOOKS:

For copyright reasons, the links below are password protected. Please get your user name and password from me by @math.wisc.edu">email if you have forgotten it. Use of these links will be monitored, and unauthorized use is prohibited.

#### EXAMS:

Makeup exams will be scheduled only with the instructor's consent, and only in case of illness or family emergency or conflict with another required class. In the latter case, please let me know as soon as possible!
Calculators of any kind are discouraged for any part of this course and will not be allowed during exams.
• Midterm Exam 1: Monday, February 28, 7:15-8:15 p.m., B139 Van Vleck Hall
• Midterm Exam 2: Monday, April 4, 7:00-8:00 p.m., B139 Van Vleck Hall
• Final exam: Tuesday, May 10, 10:05-12:05, B139 Van Vleck Hall

#### PURPOSE OF THIS COURSE:

This course is the first of three courses of the math component of the Mathematics-Science Dual Minor for all Elementary Education and Special Education majors wishing to enhance their content preparation in mathematics and science. This minor is particularly suitable for those Elementary Education majors seeking Middle Childhood-Early Adolescence certification and intending to teach mathematics and science in middle school.

The other two math courses for this minor, Math 136 and Math 138, will focus on precalculus and early calculus; and on probability, statistics and combinatorics, respectively.

Note that in spring semester 2011 only, Math 135 can also be taken in place of Math 132 by students already admitted into the Elementary Education or Special Education program.

Math 135 focuses on the mathematical content needed to teach pre-algebra and algebra in upper-level elementary and middle school.

The core instructional goals of Math 135 are:

• problem-solving;
• making mathematically grounded arguments about the strengths and weaknesses of a range of solution strategies (including standard techniques)
• examining the rationale behind middle-school students' mathematical work and how it connects to prior mathematical understanding and future mathematical concepts
• flexible use of multiple representations such as graphs, tables, and equations (including different forms)
• using functions to model real-world phenomena
• modeling real-world problems ("word problems") as mathematical problems and then interpreting the mathematical solution in the real-world context
• symbolic proficiency (solving equations and inequalities, simplifying expressions, factoring, etc.)
The mathematical content topics of Math 135 include:
• Review: basic properties of the real numbers
• Linear Functions: proportional relationships, linear equations and systems of linear equations, linear inequalities
• Quadratic Functions: different forms of quadratic equations, factoring quadratic polynomials, completing the square and quadratic formula, graphing quadratic functions, quadratic inequalities, brief discussion of polynomial functions
• Exponential Functions: understanding the difference between additive and multiplicative growth, exponential rules, exponential growth and decay, brief discussion of logarithmic functions

Prepared by Steffen Lempp (@math.wisc.edu">lemppmath.wisc.edu)