Effective: Late Spring 8-Week, 2018/2019

MATH 155: *Algebraic Reasoning For Elementary And Middle School Teachers

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  Course Description

This course introduces some basic concepts of number theory and modern algebra that underlie elementary and middle grade arithmetic and algebra, with a focus on collaborative learning and technology.

Prerequisite: MATH 102 and MATH 150 (or higher)

Proctored Exams: Midterm and Final



  • Blanton, M., Levi, L., Crites, T., and Dougherty, B. . (2011). Developing Essential Understanding of Algebraic Thinking: Grades 3-5.NCTM.  
    • [ISBN-978-0873536684]
  • Lloyd, G., Herbel-Eisenmann, B., and Star, J.R.. (2011). Developing Essential Understanding of Expressions, Equations & Functions; Grades 6-8 .NCTM.  
    • [ISBN-978-0873536707]
  • Bassarear, Tom. (2011). Mathematics for Elementary School Teachers - Manipulative Kit (2nd). Houghton Mifflin Harcourt.  
    • [ISBN-978-0-618-19093-5]
  • TI-84 Graphing Calculator (may be purchased through MBS). 
    • Note: You may use this calculator while taking the proctored exams.


  • Desmos Online Grapher http://www.desmos.com:  


Microsoft Excel will be needed. Any version of this software is acceptable. Microsoft Office 365 may be obtained free of charge through Technology Services.

MBS Information

Textbooks for the course may be ordered from MBS Direct. You can order

For additional information about the bookstore, visit http://www.mbsbooks.com.

  Course Overview

Algebra is an area of mathematics that permits us to describe and model the real world. You will explore the big ideas of algebraic thinking that are currently taught in elementary and middle schools in the United States. You will complete activities that build conceptual understanding between and among these big ideas. Topics covered include properties of real numbers, mental math, equal vs equivalent, growing patterns, functions, linear and non-linear models, recursive functions, linear inequalities and systems, and simplifying, combining, evaluating and factoring quadratic expressions and equations.

  Technology Requirements

Participation in this course will require the basic technology for all online classes at Columbia College:
  • A computer with reliable Internet access
  • A web browser
  • Acrobat Reader
  • Microsoft Office or another word processor such as Open Office

You can find more details about standard technical requirements for our courses on our site.

  Course Objectives

  • To progress from a procedural/computational understanding of mathematics to a broad understanding encompassing logical reasoning, generalization, abstraction, and formal proof.
  • To use technology (calculator and computer) as a learning and teaching tool for mathematics.
  • To learn the algorithmic approach to problem solving.
  • To display an understanding of the nature of rigorous proof. To write elementary proofs, especially proofs by induction and basic number theory proof.

  Measurable Learning Outcomes

  • Know the basic properties of the real numbers including commutativity, associativity, identity, distributivity.
  • Use the basic properties of the real numbers to determine equivalent algebraic equations and solve algebraic equations.
  • Use equations to model problem solving situations
  • Understand and use a variable to generalize a pattern, to represent a fixed but unknown number, to represent a quantity varies in relation to another quantity and that a variable can be a discrete or continuous quantity.
  • Use quantitative reasoning to generalize relationships
  • Use functional thinking to generalize relationships between covarying quantities and to express those relationships in words, symbols, tables, or graphs and reason with those relationships to analyze function behavior.
  • Use functional thinking to generalize relationships between covarying quantities and to express those relationships in words, symbols, tables, or graphs and reason with those relationships to analyze function behavior.
  • Compare and contrast the concepts of equality or equivalence.
  • Compare and contrast the concepts of equality or equivalence.
  • Understand that an inequality can describe a relationship between equalities and solve these inequalities.
  • Understand and describe recursive relationships
  • Classify functions based on the rate at which the variables change and the situations that they model
  • Solve equations using symbolic, graphical and numerical methods.


Grading Scale

Grade Points Percent
A 720-800 90-100%
B 640-719 80-89%
C 560-639 70-79%
D 480-559 60-69%
F 0-479 0-59%

Grade Weights

Assignment Category Points Percent
Discussion (9) 210 26%
Homework (8) 200 25%
Quizzes (6) 90 11%
Midterm Exam (1) 150 19%
Final Exam (1) 150 19%
Total 800 100%

  Schedule of Due Dates

Week 1

Assignment Points Due
Introduction Discussion 10 Thursday/Saturday
Discussion 1 25
Homework 1 25 Friday
Quiz 1 15 Sunday

Week 2

Assignment Points Due
Discussion 2 25 Wednesday/Saturday
Homework 2 25 Friday
Quiz 2 15 Sunday
Proctor Information N/A

Week 3

Assignment Points Due
Discussion 3 25 Wednesday/Saturday
Homework 3 25 Friday
Quiz 3 15 Sunday

Week 4

Assignment Points Due
Discussion 4 25 Wednesday/Saturday
Homework 4 25 Friday
Midterm Exam 150 Sunday

Week 5

Assignment Points Due
Discussion 5 25 Wednesday/Sunday
Homework 5 25 Friday
Quiz 4 15 Sunday

Week 6

Assignment Points Due
Discussion 6 25 Wednesday/Saturday
Homework 6 25 Friday
Quiz 5 15 Sunday

Week 7

Assignment Points Due
Discussion 7 25 Wednesday/Saturday
Homework 7 25 Friday
Quiz 6 15 Sunday

Week 8

Assignment Points Due
Discussion 8 25 Wednesday/Saturday
Homework 8 25 Friday
Final Exam 150 Saturday
Total Points: 800

  Assignment Overview


Each week you will participate in an online discussion worth 25 points each. In Week 1, you will have have an additional 10 point introductory discussion. You should respond fully to the initial discussion question and post at least two responses. Initial posts are due by 11:59 pm Wednesday, except for Week 1 when they are due on Thursday. Responses are due by 11:59 pm on Saturdays.

Your posts should be supported with examples with evidence of synthesis of readings and/or outside sources. Formal paragraph structure with college-level grammar and spelling is expected. Additional grading criteria will be available in the course.


Weekly Homework is required to allow you to practice mathematical skills. Each homework assignment is worth 25 points and due by 11:59 PM on Fridays. The Homework worksheets will be available in the Dropbox, and should be completed, saved, and resubmitted to their respective Dropbox.


There are six quizzes containing 5 short, objective-style questions. You will have 30 minutes to complete the quizzes and up to two attempts. The highest score will be recorded. Quizzes open on Monday and are due by 11:59 pm Sundays. They are worth 15 points.


The course has a Midterm and a Final Exam. Both must be proctored. They will contain M/C, T/F, or short answer questions. You will have two hours to complete the exams and just one attempt. They are available beginning on Monday of their respective weeks. The Midterm is due by 11:59 pm Sunday of week 4, the Final is due by 11:59 Saturday of week 8. Each exam is worth 150 points.

  • The Midterm covers material from weeks 1-4.
  • The Final covers material from weeks 5-8

  Course Outline

Click on each week to view details about the activities scheduled for that week.

  •  Developing Essential Understanding of Algebraic Thinking: pp. 1-24
  • Lecture: Properties of Real Numbers and Arithmetic
Introduction Discussion
Tell us a little about yourself. Items you might share are your interest in math, your career ambitions or family. Let's get to know each other! (Reminder: Two responses are expected in this and all discussions.)
Discussion 1
 The concepts of even and odd numbers seem to be simple to us, but for children this may not always be the case.

Watch the video clip at this University of Michigan site and respond.  You can use these questions to help, but feel free to respond in other ways.

1.  Were you surprised by the students' various interpretations of odd and even numbers?

2. What do you think is the importance of allowing the students to share and defend their thinking?

3. What role is the teacher playing in the classroom?

4. Is this how you remember learning about even and odd numbers and or what you have seen in the classrooms?

5. Why do you think the teacher is letting the kids develop the definition?  (i.e. They have a working definition of even number)

Homework 1
Homework 1 covers sum and products charts, classifying real numbers, and properties of real numbers
Quiz 1
Quiz 1 covers the week 1 readings.
  • Developing Essential Understanding of Algebraic Thinking: pp. 25-31
  • Lecture: Expressions, Equality, and Equivalence
Discussion 2
View the algebraic puzzles (Word download) that were posted on my Facebook page, prompting quite a discussion between me and my friends. (Note: puzzles will be available in the course.) Many answers were found and many paths were taken. People got excited about it even if they were not "math" people. 
  • Share these images with at least 10 people and then report your findings in the discussion for this week. What were their thought processes and reactions? Were they able to solve them? 
  • Are you able to solve the puzzles? Discuss what the equal sign is representing in these puzzles.
Homework 2
Homework 2 covers balancing scale and equivalent expressions problems.
Quiz 2
Quiz 2 covers week 2 readings.
Proctor Information
Submit your proctor form to the appropriate Dropbox folder by the end of the week. Remember to “Save” the form before placing it in Dropbox. See the Content area for more information.
  • Developing Essential Understanding of Algebraic Thinking: pp. 32-38
  • Lecture: Growing Patterns
Discussion 3

In the article found on the Ontario Ministry of Education site, Paying Attention to Algebra, on pages 19-21, an activity and student work is shown.  Consider the use of representations, tables, and graphs in helping students understand the roles of variables and to eventually write, use, and solve equations. As you do you may wish to respond to all of these prompts:

  • Do you believe these types of experiences are important in helping children develop their algebraic reasoning skills?
  • Did you have experiences such as this when developing your algebraic reasoning?
  • Do you wish to provide such experiences for your students?
  • What does your text suggest (pg. 30-38)?
Homework 3
Homework 3 covers counting dot patterns.
Quiz 3
Quiz 3 covers week 3 readings.
  • Developing Essential Understanding of Expressions, Equations & Functions: pp. 44-56
  • Lecture: Function Machines
Discussion 4
 The 5 Big Ideas are summarized on pages 12- 13 of your text (Developing Essential Understanding of Algebraic Thinking) and are further delineated in the remaining portion of the book.  We have completed many activities so far in class and I attempted to align these activities to these big ideas.  
  • Choose an activity that we have done that you feel aligns to a big idea and explain why you think it aligns.
Homework 4
Homework 4 covers body ratios.
Midterm Exam
The Midterm Exam is proctored and covers material learned in weeks 1-4. See the course for additional information.
  • Developing Essential Understanding of Expressions, Equations & Functions: pp. 57-66
  • Lecture: Linear and Non-linear Models
Discussion 5
  Watch this video about Bottles and Bungee Barbie. It is a high school class working on a lesson in modeling.
  • What did you learn by watching this video as compared to participating in similar lessons? 
  • Do you think you could do a similar lesson (revised perhaps) with upper elementary and middle school students?
Homework 5
Homework 5 covers body ratios.
Quiz 4
Quiz 4 covers week 5 readings.
Discussion 6
This week we have been modeling with recursive functions. In past weeks we have modeled with both linear and non-linear functions that are called explicit functions. 
  • Describe the differences between these two types of functions (explicit and recursive).
  • Sometimes functions are more easily described using an explicit function; give an example of such a function.
  • In other cases, functions can be more are more easily described using recursive notation. Give an example of such a function.
Homework 6
Complete Homework 6.
Quiz 5
Quiz 5 covers week 6 readings.
Course Evaluation
Please evaluate the course. You will have an opportunity to evaluate the course near the end of the session. A link sent to your CougarMail will allow you to access the evaluation. Please note that these evaluations are provided so that I can improve the course, find out what students perceive to be its strengths and weaknesses, and in general assess the success of the course. Please do take the time to fill this out.
  • Developing Essential Understanding of Expressions, Equations & Functions: pp. 38-43
  • Lecture: Linear Inequalities and Systems
Discussion 7
  Systems of linear inequalities are similar, yet different, to systems of linear equations. Compare and contrast:
  • How the systems are written
  • How the systems are solved
  • How the systems look graphically
  • How the solutions are written
Homework 7
Homework 7 covers linear programming.
Quiz 6
Quiz 6 covers week 7 readings.
  • Developing Essential Understanding of Expressions, Equations & Functions: Introduction, pp. 12-24, 32-36, 67-79
  • Lecture: Algebra Tiles
Discussion 8
 The 5 Big Ideas are summarized on pages 7-11 of your text (Developing Essential Understanding of Expressions, Equations & Functions) and are further delineated in the remaining portion of the book.  We have completed many activities so far in class and I attempted to align these activities to these big ideas.
  • Choose an activity that we have done that you feel aligns to a big idea and explain why you think it aligns.
Homework 8
Complete Homework 8.
Final Exam
The Final Exam must be proctored and covers material from weeks 5-8. See the course for additional information.

  Course Policies

Student Conduct

All Columbia College students, whether enrolled in a land-based or online course, are responsible for behaving in a manner consistent with Columbia College's Student Conduct Code and Acceptable Use Policy. Students violating these policies will be referred to the office of Student Affairs and/or the office of Academic Affairs for possible disciplinary action. The Student Code of Conduct and the Computer Use Policy for students can be found in the Columbia College Student Handbook. The Handbook is available online; you can also obtain a copy by calling the Student Affairs office (Campus Life) at 573-875-7400. The teacher maintains the right to manage a positive learning environment, and all students must adhere to the conventions of online etiquette.


Your grade will be based in large part on the originality of your ideas and your written presentation of these ideas. Presenting the words, ideas, or expression of another in any form as your own is plagiarism. Students who fail to properly give credit for information contained in their written work (papers, journals, exams, etc.) are violating the intellectual property rights of the original author. For proper citation of the original authors, you should reference the appropriate publication manual for your degree program or course (APA, MLA, etc.). Violations are taken seriously in higher education and may result in a failing grade on the assignment, a grade of "F" for the course, or dismissal from the College.

Collaboration conducted between students without prior permission from the instructor is considered plagiarism and will be treated as such. Spouses and roommates taking the same course should be particularly careful.

All required papers may be submitted for textual similarity review to Turnitin.com for the detection of plagiarism. All submitted papers may be included in the Turnitin.com reference database for the purpose of detecting plagiarism. This service is subject to the Terms and Conditions of Use posted on the Turnitin.com site.


There will be no discrimination on the basis of sex, race, color, national origin, sexual orientation, religion, ideology, political affiliation, veteran status, age, physical handicap, or marital status.

Student Accessibility Resources

Students with documented disabilities who may need academic services for this course are required to register with the office of Student Accessibility Resources. Until the student has been cleared through this office, accommodations do not have to be granted. If you are a student who has a documented disability, it is important for you to read the entire syllabus as soon as possible. The structure or the content of the course may make an accommodation not feasible. Student Accessibility Resources is located in Student Affairs in AHSC 215 and can be reached by phone at (573) 875-7626 or email at sar@ccis.edu.

Online Participation

You are expected to read the assigned texts and participate in the discussions and other course activities each week. Assignments should be posted by the due dates stated on the grading schedule in your syllabus. If an emergency arises that prevents you from participating in class, please let your instructor know as soon as possible.

Attendance Policy

Attendance for a week will be counted as having submitted any assigned activity for which points are earned. Attendance for the week is based upon the date work is submitted. A class week is defined as the period of time between Monday and Sunday (except for week 8, when the work and the course will end on Saturday at midnight.) The course and system deadlines are based on the Central Time Zone.

Cougar Email

All students are provided a CougarMail account when they enroll in classes at Columbia College. You are responsible for monitoring email from that account for important messages from the College and from your instructor. You may forward your Cougar email account to another account; however, the College cannot be held responsible for breaches in security or service interruptions with other email providers.

Students should use email for private messages to the instructor and other students. The class discussions are for public messages so the class members can each see what others have to say about any given topic and respond.

Late Assignment Policy

An online class requires regular participation and a commitment to your instructor and your classmates to regularly engage in the reading, discussion and writing assignments. Although most of the online communication for this course is asynchronous, you must be able to commit to the schedule of work for the class for the next eight weeks. You must keep up with the schedule of reading and writing to successfully complete the class.

No late discussion posts will be accepted. However late initial posts may receive partial credit provided they are posted before the end of the week deadlines (Saturdays).

Late exams are not accepted without prior approval. Approval is only given under significant extenuating circumstances and must be requested before the due date.

I do not accept late weekly quizzes.

I do accept late dropbox assignments, but will deduct 5% for each day they are late.

Course Evaluation

You will have an opportunity to evaluate the course near the end of the session. A link will be sent to your CougarMail that will allow you to access the evaluation. Be assured that the evaluations are anonymous and that your instructor will not be able to see them until after final grades are submitted.

Proctor Policy

Students taking courses that require proctored exams must submit their completed proctor request forms to their instructors by the end of the second week of the session. Proctors located at Columbia College campuses are automatically approved. The use of ProctorU services is also automatically approved. The instructor of each course will consider any other choice of proctor for approval or denial. Additional proctor choices the instructor will consider include: public librarians, high school or college instructors, high school or college counseling services, commanding officers, education service officers, and other proctoring services. Personal friends, family members, athletic coaches and direct supervisors are not acceptable.

  Additional Resources

Orientation for New Students

This course is offered online, using course management software provided by Desire2Learn and Columbia College. The course user guide provides details about taking an online course at Columbia College. You may also want to visit the course demonstration to view a sample course before this one opens.

Technical Support

If you have problems accessing the course or posting your assignments, contact your instructor, the Columbia College Helpdesk, or the D2L Helpdesk for assistance. Contact information is also available within the online course environment.

Online Tutoring

Smarthinking is a free online tutoring service available to all Columbia College students. Smarthinking provides real-time online tutoring and homework help for Math, English, and Writing. Smarthinking also provides access to live tutorials in writing and math, as well as a full range of study resources, including writing manuals, sample problems, and study skills manuals. You can access the service from wherever you have a connection to the Internet. I encourage you to take advantage of this free service provided by the college.

Access Smarthinking through CougarTrack under Students -> Academics -> Academic Resources.