Welcome to Math for Teachers!

[picture of Niles]

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 1136, Measurement and 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/SP15/1136.html.

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Excellent math teachers have been far too scarce in American education. This course and its prequel, Math 1135, give you the keys to how and why elementary mathematics works. We will develop a thorough understanding of the ideas, how they are related to eachother, and how they are related to more advanced ideas. This is the basic knowledge which underpins truly great teaching!

Course Objectives

The course consists of fundamental topics in geometry and measurement. This includes the concepts of length, area, volume, angles, units of measurement, precision and error. Algebraic expressions and functions, primarily in the linear case, are introduced to express geometric relationships.

The basic properties of two and three dimensional geometric shapes and their relationships are a central part of the course. Special emphasis is put on geometric reasoning through problem solving, including unknown angle, length, area, and volume. The course also covers topics on transformations in the plane, symmetries, congruence, and similarity. Some geometric constructions and basic geometric proofs are included. The skills developed throughtout the course are applied at the end in a brief introduction to probability.


  • Informal and formal proofs with angles
  • Geometric constructions
  • Angle sum in triangles, polygons
  • Pythagorean Theorem statement and proof
  • Planar shapes
  • Linear equations and graphs
  • Algebra and linear equations
  • Measurement of length, area, volume
  • Units of measurement
  • Perimeter, area and volume of 2D and 3D shapes
  • Plane transformations: translations, rotations, dilations
  • Introduction to coordinate geometry
  • Introduction to probability

Section Information

During Spring 2015, we have two sections of Math 1136; these are numbered 30976 and 30977. The syllabus and content for the sections will be the same but the class meeting times and locations are different. To distinguish them, each section also has a codename after one of the standard web colors.

Class meets at the following place and times following the OSU Registrar's academic calendar:

Section 30976 (Indigo)

Hopewell 112
Mondays, Wednesdays 12:45 – 2:05
and Fridays 12:45 – 2:50.

Section 30977 (Peru)

Hopewell 124
Mondays, Wednesdays 3:55—5:15
and Fridays 3:05 – 5:10.

Course Calendar (+)

An approximate outline, updated regularly, is available here.

Homework Assignments(+)

Assignment information will be posted here as the semester unfolds. A number such as A.B.c refers to problem or practice exercise c at the end of section A.B.

You only need to turn in the Problems; the Practice Exercises are suggestions for your study, and may be discussed in class. Remember to type your homework and bring a printed copy to class!

The homework template I mentioned in class is available here. A little minicourse on latex is available here, and here's a list of beautiful examples of latex typesetting (or related *tex typesetting). Let Niles know if you're interested in more!

Past Homework

Homework 1 due Friday, 01/16

Practice Exercises (do not turn in): 10.2.4, 10.4.5
Problems (type, print, and turn in): 10.1.4, 10.1.8, 10.2.4, 10.2.10
Bonus problem: Carefully explain part 4 of Activity 10D, and its relationship to part 2.

Homework 2 due Wednesday, 01/21

Practice Exercises (do not turn in): 10.5.4, 10.5.8, 9.1.1, 9.1.5
Problems (type, print, and turn in): 10.4.1, 10.5.3, 10.5.11, 10.5.19

Homework 3 due Monday, 01/26

Practice Exercises (do not turn in): 9.1.5, 9.2.5
Problems (type, print, and turn in): 10.5.11, 9.1.4, 9.1.9, 9.2.9

Homework 4 due Friday, 01/30

Practice Exercises (do not turn in): 9.3.1, 9.3.4, 9.4.2, 9.5.3
Problems (type, print, and turn in): 9.3.3, 9.3.5, 9.3.7, 9.4.5
Additional problem: Use the parallel postulate (and triangle congruence) to explain why the diagonals in a rhombus bisect the interior angles. Go on to explain why diagonals in a rhombus are perpendicular bisectors.

Homework 5 due Monday, 02/02

Problems (type, print, and turn in): 9.5.8, 9.5.10, 9.5.16, 9.6.1, 9.6.5
Additional problem: moved to HW 6.
Additional problem 2: Consider the sequence formed by ones digits of powers of 2. What is the pattern, and what is the ones digit of 2100?

Homework 6 due Friday, 02/06

Problems (type, print, and turn in): 9.6.7, 9.6.13, 9.7.2, 9.7.5, 9.7.10, 9.7.12
Additional problem: Explain why a quadrilateral having opposite angles equal must be a parallelogram. Then use this to explain why a quadrilateral having opposite sides equal must be a parallelogram.

Homework 7 due Friday, 02/13

Type and turn in at least four problems from each group on the review sheet.

Extra Credit due Monday, 02/16

Find a general expression for the terms in a quadratic sequence whose first terms are
\[ c, (c + b), (c + b + a), \ldots \] It might be easiest to first try this for specific values of \(c\), \(b\), and \(a\). Email Niles if you'd like a hint.

Extra Credit due Monday, 02/16

Follow the steps in this article to make a cool Islamic art pattern with compass and straightedge!

Homework 8 due Friday, 02/20

Problems: 11.1.4, 11.3.4, 11.4.5, 11.4.9, 11.4.21

Homework 9 due Monday, 02/23

Problems: 12.1.4, 12.2.4, 12.3.2, 12.3.7, 12.4.9, 12.5.3

Homework 10 due Friday, 02/27

Problems: 14.5.6, 14.5.16, 12.6.2, 12.6.4
Additional Problem: Use a ruler to carefully estimate the square constant (permieter/diagonal).

Homework 11 due Monday, 03/02

Problems: 12.7.3, 12.8.4, 12.8.12, 12.9.1, 12.9.6

Homework 12 due Friday, 03/06

Problems: 13.1.3, 13.1.4, 13.1.8, 13.2.9, 13.2.14

Homework 13 due Monday, 03/09

Problems: 13.3.11, 13.3.22, 13.4.1, 14.1.4, 14.1.5
Additional problem: Draw a pattern which, when cut out and folded up, makes an oblique rectangular prism.

Homework 14 due Friday, 03/13

Type up and turn in 12 problems of your choosing from chapters 11 and 12.

Homework 15 due Monday, 03/23

Problems 14.1.8 — 14.1.13.
You can use this link to download graph paper.

Writing Assignment due Monday, 03/23

See guidelines and suggested topics in class email or here. I've opened a dropbox in our Carmen course for you turn your papers in. Please submit your document in pdf and use a filename that begins with your last name.

Here is a video illustrating Archimedes Method for finiding the volume of the sphere.

Homework 16 due Friday, 03/27

1. Draw objects that have the following combinations of symmetries:

  • horizontal and vertical reflection symmetry
  • rotation symmetry, but no reflection symmetry
  • both reflection and rotation symmetry
  • translation and reflection symmetry, but no rotation symmetry
  • glide-reflection symmetry and rotation symmetry

2. Use geogebra.org or another tool of your choosing to do activity 14 F. Print a nice-looking pattern that you create in this way.

3. Problem 14.2.8

Homework 17 due Monday, 03/30

  • Why doesn't ASS give a triangle congruence? See the text, or Activity 14I, "triangle T"
  • Describe what happens when you combine a reflection and a rotation (for example, on an octagon)
  • Construct a square and an octagon as in Activity 14M.
  • Look up the Newark Earthworks, and appreciate the Octagon!

Homework 18 due Friday, 04/03

Problems: 14.3.6, 14.3.7, 14.4.7

Homework 19 due Monday, 04/06

Problems: 14.5.19, 14.5.21, 14.5.24, 14.6.4, 14.6.7

Homework 20 due Friday, 04/10

Turn in your work on the review sheet problems.

Writing assignment due Monday, 04/13

Revisit your writing assignments and make sure that you've clearly attributed content from your references. See email for more details.

Extra Credit due Monday, 04/13

An equilateral triangle has 6 symmetries: 3 reflections and 3 rotations (including the "identity", which rotates by 0 or 360 degrees). Make a "multiplication table" which shows what you get by combining each pair. You will need to be very careful about the order that you combine them in, as combining symmetries is not necessarily commutative.

A square has 8 symmetries total. Make a "multiplication table" which shows what you get by combining each pair. For this to be feasible, you will want to see how understanding just a few combinations can allow you to deduce the others. This will allow you to notice and use patterns in the multiplication table to complete it.

After you have thought about this for a while, you should take a look at the Wikipedia article about the dihedral group. If you look before you've really thought about this, it may be too overwhelming!

Homework 21 due Friday, 04/17

Problems 15.1.7, 15.1.9
One of 15.2.2, 15.2.3, 15.2.4
One of 15.2.5, 15.2.6, 15.2.7
Problems 15.3.7, 15.3.9, 15.3.13

Homework 22 due Monday, 04/20

Problems 16.1.2, 16.1.5, 16.2.3, 16.3.6
Additional problem: What is the exact theoretical probability that 5 out of 10 coin flips will be heads?

Homework 23 due Friday, 04/24

Problems 16.3.17, 16.3.18, 16.4.12, 16.4.15

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Basic information


Niles Johnson
Office: Hopewell 189
Office Hours: Mondays and Wednesdays 2:15 – 3:45 (in ECC); Thursdays by appointment


Mathematics for Elementary Teachers with Activities, 4th Edition by Sybilla Beckmann. ISBN 0-321-82572-1. The 4th edition (Monkeys on the cover) is substantially improved over earlier editions. Moreover, it is available in electronic or loose-leaf formats which are cheaper and more portable than the bound version.

Exam Schedule

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

Midterms (3):

Friday Feb. 13, Friday March 13, Friday April 10.


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


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.

Math Learning Center

The Math Learning Center (Warner 214) is a great resource, and I encourage you make use of it. The staff there has experience with our course and would love to help you!

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: 12%
  • Homework: 5%
  • Writing Assignments: 4%
  • Quizzes: 14%
  • Midterm exams: 15% each, for a total of 45%
  • Final exam: 20%

The two lowest quiz and homework grades will be dropped. If your final exam score is higher than one of your midterms, then that grade will replace the lowest midterm grade.


Thoughout this semester we will be focused on the how and why of elementary 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.

Grading Rubric

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.

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

Websites of elementary school texts from high performing countries

Other websites of interest

  • Project INTERMATH, which focuses on building teachers' mathematical content knowledge through mathematical investigations that are supported by technology.
  • Problems and tasks that Sybilla Beckmann wrote for her 6th graders during the 2004/2005 school year organized according to the grade 6 Georgia Performance Standards.
  • 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:


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.

Disability Statement

Students with disabilities that have been certified by the Office for Disability Services will be appropriately accommodated, and should inform the instructor as soon as possible of their needs. The Office for Disability Services is located in 150 Pomerene Hall, 1760 Neil Avenue; telephone (614) 292-3307 and VRS (614) 429-1334; webpage 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. For additional information, see the Code of Student Conduct:

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