Teacher Education Program Mission Statement:
BSU prepares teachers through inquisitive, involved, reflective practice. The framework outlining our program sets a standard that is rigorous, exemplary and innovative. The curricular structure is research based and organized around the Standards of Effective Practice. Graduates are proficient, collaborative, technologically literate and environmentally aware teachers, who work effectively in various settings with diverse learners.
Math 3064
Number Concepts for Teachers
Bemidji State University
Spring Semester 2020
Math 3064/5064 – section 01 – Number Concepts for Teachers
Meets: M W 2:00 – 3:50 am in DH 113
Instructor: Todd Frauenholtz
Office: S 207G Phone: 755-2817
E-mail: tfrauenholtz@bemidjistate.edu
Website: http://faculty.bemidjistate.edu/tfrauenholtz
Office hours: by arrangement
Math help center: HS 232
Professional Education Department Mission Statement:
“The Bemidji State University Professional Education program is preparing today's teachers for tomorrow, through effective, inquisitive, and reflective practice. Our students are proficient, self-reliant, and thoughtful practitioners, developed in a viable and growing program, who can teach effectively in various settings with diverse learners."
Course Description -- Number Concepts for Middle School Teachers (4 credits)
This course helps meet the BOT rule with respect to number sense. Provides a background in special number concepts that are pertinent to middle school mathematics. Topics include elementary algebra, properties of integers, prime and composite numbers, divisors, GCDs, LCMs, the number of divisors, the sum of divisors, the Euclidean Algorithm, famous unsolved problems, finite mathematical systems, modular arithmetic and congruencies, and sequences. Emphasis given to problem solving techniques as they relate to number concepts and algebraic representation.
Class participation and quizzes: Quizzes will be given approximately every week and generally unannounced. Students must be present at the beginning of the class session to take a quiz. Make-ups will not be allowed for missed quizzes but your lowest quiz score will be dropped from your grade. All cell phones must be turned off during class.
Homework: Doing the homework will prepare you for the quizzes and exams. You are responsible for understanding how to do each problem; I recommend study groups of three to four. Students will also be expected to complete projects and papers for this course. These will count as homework in the grade.
April 1st will be BSU’s Annual Student Scholarship and Creative Achievement Conference. Class will not meet this day; students who choose to attend the conference may earn extra-credit points.
Exams: There are two exams planned – one midterm and the final exam. Make-up exams will be given only under special circumstances and need to be discussed with me beforehand. The final exam is scheduled for Tuesday, May 5th, from 1:00 pm – 3:00 pm.
Grades: Grades for this course will be based upon quizzes, tests, homework and a comprehensive final exam; the quizzes may be unannounced. Items for both will come from the assigned homework and in-class activities. The following grading scale will be used to determine grades:
A -- 90 – 100 %
B -- 80 – 89 %
C -- 70 – 79 %
D -- 60 – 69 %
F -- Below 60 %
Tests, quizzes, and assignments will be used to calculate the final grade. The instructor reserves the right to adjust the grading scale if necessary.
Working through the assigned problems is essential to learning mathematics. Showing your work is the only way to receive partial credit and hence is very important.
Academic integrity: Students are expected to practice the highest standards of ethics, honesty, and integrity in all of their academic work. Any form of academic dishonesty (e.g., plagiarism, cheating, misrepresentation) may result in disciplinary action. Possible disciplinary actions may include failure for part or all of a course, as well as suspension from the University.
Upon request, this document and others distributed in this course can be made available in alternate formats. Please contact the instructor, Todd Frauenholtz, at 755-2817 or Chris Hoffman in the Office for Students with Disabilities at 755-3883 for assistance.
Tentative Daily Course Outline
Day 1 |
Fractions in bases 4, 5, and 6. Do they terminate or repeat? How to convert fractions to decimals in other bases. |
Day 2 |
How to convert fractions to decimals in other bases. |
Day 3 |
How to convert fractions to decimals in other bases. |
Day 4 |
How to convert fractions to decimals in other bases. Check out a game of Fraction Tracks |
Day 5 |
Fraction circles and repeating to terminating decimals Decimal operations |
Day 6 |
Irrational numbers on the geoboard |
Day 7 |
Making squares on the geoboard using irrational numbers |
Day 8 |
Now that you can make a square of area five, find a decimal to represent the square root of five. Bisection method |
Day 9 |
Continue with the bisection method and look at "divide and average" and the long division methods to approximate irrational numbers |
Day 10 |
Wrap-up irrational numbers. |
Day 11 |
Integers - Positive and negative numbers – modeling operations and derivation of arithmetic rules; Modular arithmetic; Cayley tables – properties and inverses |
Day 12 |
Jordan curve theorem; Utility problem; Konigsberg bridge problem; Euler circuits and paths /Hamiltonian circuits and paths; Tournament matrix; |
Day 13 |
Tournament matrix; Euler circuits/Hamiltonian circuits; |
Day 14 |
Konigsberg bridge problem; Euler circuits and paths /Hamiltonian circuits and paths; Tournament matrix; |
Day 15 |
Rotations and flips of a triangle; Composition of functions; Cayley table – properties and inverses; develop same material for rotations and flips of a square. |
Day 16 |
Scientific notation; basic notation and fundamental operations; Craig’s stories (number magnitude and estimation) |
Day 17 |
Divisibility tests for the integers 2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,32 |
Day 18 |
Divisibility tests from the perspective of Blocks, Flats, Longs, and Units |
Day 19 |
Review closure, associative, zero, inverses (+ and x), commutative, distributive. |
Day 20 |
The Division algorithm. |
Day 21 |
Prime numbers – Locker problem; sieve of Eratosthenes; size and dimensions of a sieve; Fundamental theorem of arithmetic; |
Day 22 |
Greatest common divisor, GCD's – four methods
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Day 23 |
Wrap-up GCD's |
Day 24 |
Wrap-up LCM https://twitter.com/joann_sandford/status/1204433454273875968?s=20 |
Day 25 |
Cayley tables revisited -- Closure, Associative, Commutative, Identity, Inverses, and Distributive properties. |
Day 26 |
Conclude Cayley tables -- Magic Math: |
Day 29 |
The myth of Brahman monks - Tower of Hanoi and the 64 gold disks; develop recursive algorithm to solve problem; develop explicit formula for calculating number of moves until end of earth; prove explicit and recursive formulae with Mathematical Induction; |
Day 30 |
Magic squares - conjectures and proof. |
Day 31 |
Clever Counting -- Combinations and permutations using intuition Resource(s): 1, 5, 7, 8, 10, 11, 13, 15 |
Day 32 |
Combinations and permutations – develop and apply nPr and nCr formulae. Resource(s): 1, 5, 7, 8, 10, 11, 13, 15 |
Day 33 |
Counting -- permutations, paths, combinations, codes, five card hands, pizza problems ... examine Pascal’s triangle as a table of binary coefficients; Resource(s): 1, 5, 7, 8, 10, 11, 13, 15 |
Day 34 |
Jordan curve theorem; Konigsberg bridges; Euler circuits and paths; Hamiltonian circuits and paths; other applications of vertex edge graphs; |
Day 35 |
Sales routes; Use Excel to examine Fib Seq/ Lucas numbers, and golden ratio;sprouts (discrete yearbook) |
Day 36 |
Solve traveling salesman problems; develop Brute Force; Greedy algorithm; Nearest Neighbor; Repeated Nearest Neighbor algorithms |
Day 37 |
Solve traveling salesman problems; apply Brute Force; Greedy algorithm; Nearest Neighbor; Repeated Nearest Neighbor algorithms |
Day 38 |
Sorting algorithms; Model Bubble, Insertion, Selection, and Quicksort with cards |
Day 39 |
Sorting algorithms; Analyze Bubble, Insertion, Selection, and Quicksort with arrays; Utilize Big O – notation to give estimate of time for algorithms. |
Day 40 |
Four four's activity – review and model instruction for order of operations and practice |
Day 41 |
Integers, scientific notation – small and large numbers, the division algorithm, prime numbers. |
Day 42 |
Greatest common divisor, least common multiple, fundamental theorem of arithmetic, famous number theory problems. |
Day 43 |
Linear Diophantine equations; lattice points; solve ax + by = n; generate parametric solutions |
Day 44 |
Arithmetic sequences; geometric sequences; non-arithmetic sequences; recursive / iterative rules and explicit rules. |
Day 45 |
Congruence theory using modular / clock arrays or tables. |
July 25 |
FINAL EXAM 11:30am – 3:00 pm |
Prerequisites
MATH 1011 or consent of instructor.
Required Text
No text required – course taught with handouts
Resources: |
This course is taught with handouts from a variety of sources. |
Technology: |
A calculator. Use of a computer lab. |
Board of Teaching Standards
8710.3320 MIDDLE LEVEL ENDORSEMENT LICENSE FOR TEACHERS OF MATHEMATICS.
Department of Mathematics and Computer Science
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8710.3320 MIDDLE LEVEL ENDORSEMENT LICENSE FOR TEACHERS OF MATHEMATICS |
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C.A teacher with a middle level endorsement for teaching mathematics in grades 5 through 8 must demonstrate knowledge of fundamental concepts of mathematics and the connections among them. The teacher must know and apply: |
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In this syllabus you will find the word TEACH. This will mean to:
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Standard |
K/A |
Activity |
Assessment |
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(2) concepts of discrete mathematics: |
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(a) application of discrete models to problem situations using appropriate representations, including sequences, finite graphs and trees, matrices, and arrays; |
K A |
TEACH:
TEACH: Days 38,39 |
Assesment: test 2 and test 3.
Assesment: test 5.
Assesment: tests 4,5. |
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(b) application of systematic counting techniques in problem situations to include determining the existence of a solution, the number of possible solutions, and the optimal solution; |
K A |
TEACH: TEACH: |
Assesment: test 4, test 5;
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(c) application of discrete mathematics strategies including pattern searching; organization of information; sorting; case-by-case analysis; iteration and recursion; and mathematical induction to investigate, solve, and extend problems; and |
K A |
TEACH: TEACH: Days 38,39
TEACH: |
Assesment: test 4
Assesment: test 5; |
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(d) exploration, development, analysis, and comparison of algorithms designed to accomplish a task or solve a problem; |
K A |
TEACH: TEACH: |
Assesment: test 5;
Assesment: test 4,test 5; |
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(3) concepts of number sense: |
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(a) understand number systems; their properties; and relations, including whole numbers, integers, rational numbers, real numbers, and complex numbers; |
K A |
TEACH: TEACH: |
Assesment: test 1;
Assesment: test 2; |
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(b) possess an intuitive sense of numbers including a sense of magnitude, mental mathematics, estimation, place value, and a sense of reasonableness of results; |
K A |
TEACH: TEACH: Day 16 |
Assessment: test 3;
Assessment: test 3; |
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(c) possess a sense for operations, application of properties of operations, and the estimation of results; |
K A |
TEACH: |
Assessment: test 1; |
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(d) be able to translate among equivalent forms of numbers to facilitate problem solving; and |
K A |
TEACH: |
Assessment: tests 1,2; |
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(e) be able to estimate quantities and evaluate the reasonableness of estimates; |
K A |
TEACH: |
Assessment: with test 3; |
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Technology Requirements and Expectations
Students will use internet browsers to access information and answer questions posed in class. Students may use calculators, spreadsheets, and data programs such as Excel, Tinkerplots, Fathom 2, or Minitab to answer problems. Written assignments for class will be composed using a word processor such as Microsoft Word.
Teaching Methodology
Polya’s problem solving steps
1. Understand the problem
Lesson Sequencing
Intuitions Þ Concrete Û Semi-Concrete Û Abstract
Dr. Glen Richgels’ Teaching/Learning Principles
1. Teach the way students learn
2. Use group work, heterogeneous, 3-4, change monthly
3. Communication student Û student
4. Communication student Û teacher
5. Multiple solution paths
6. Use contextual settings / problem solving
7. Assessment
Instructional practices modeled after principles from Principles and Standards for School Mathematics and the Cognitively Guided Instruction research project from the University of Wisconsin-Madison (WCER).
University Policies and Procedures
http://www.bemidjistate.edu/students/handbook/policies/
Academic Integrity
BSU students are expected to practice the highest standards of ethics, honesty and integrity in all of their academic work. Any form of academic dishonesty (e.g., plagiarism, cheating and misrepresentation) may result in disciplinary action. Possible disciplinary actions may include failure for part of all of a course as well as suspension from the University.
Students with Special Needs
Upon request this document can be made available in alternate formats. Please contact Kathi Hagen at Disabilities Services at (218) 755-3883 for assistance or the AUC Office at 262-6753 or (800) 369-4970.
Student Rights and Responsibilities
Student Code of Ethics
http://www.bemidjistate.edu/academics/catalog/10catalog/GradCatalog/Frontpages/sectionIV/rights.html
Student Academic Rights and Responsibilities
http://www.bemidjistate.edu/students/handbook/policies/academic_integrity/rights_responsibilities.cfm
Instructor Rights and Responsibilities
- I work with all students and expect success from all students. It is my expectation for those students who attend class regularly and complete assignments that they will earn an A or B.
- I am available for help whenever I am in my office. I encourage students to do homework at a table outside of my office so that I can help them whenever they have difficulties. Help is also available through email and at my home, if prior arrangements have been made.
- I will try to give grade status reports at least every three weeks.
Course Policies
Attendance: Daily attendance is expected
Participation: Class participation and group work is expected
updated 1/8/2020
by Todd
Frauenholtz