Fractions in bases 4, 5, and 6.  Do they terminate or repeat?  How to convert fractions to decimals in other bases.
Rational Number Project (RNP) Fraction circles - Lesson # 1, 2
Resource(s): 1, 2, 3, 4, 6, 9, 13

Day 2

How to convert fractions to decimals in other bases.
NCTM Illuminations – Equivalent Fractions Applet
Rational Number Project (RNP) Fraction circles - Lesson # 4, 6
Resource(s): 1, 2, 3, 4, 6, 9, 13

Day 3

How to convert fractions to decimals in other bases.
Rational Number Project (RNP) Fraction circles - Lesson # 9, 12
Resource(s): 1, 2, 3, 4, 6, 9, 13

Day 4

How to convert fractions to decimals in other bases.
Rational Number Project (RNP) Fraction circles - Lesson # 15 & 20.
Resource(s): 1, 2, 3, 4, 6, 9, 13

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
Resource(s): 1, 2, 3, 4, 6, 9, 13

Day 7

Making squares on the geoboard using irrational numbers
Resource(s): 1, 2, 3, 4, 6, 9, 13

Day 8

Now that you can make a square of area five, find a decimal to represent the square root of five. Bisection method
Resource(s): 1, 2, 3, 4, 6, 9, 13

Day 9

Continue with the bisection method and look at "divide and average" and the long division methods to approximate irrational numbers
Resource(s): 1, 2, 3, 4, 6, 9, 13

Day 10

Wrap-up irrational numbers.
Number systems – how they evolved or how they complete operations.
Resource(s): 1, 2, 3, 4, 6, 9, 13

Day 11

Integers - Positive and negative numbers – modeling operations and derivation of arithmetic rules; Modular arithmetic; Cayley tables – properties and inverses
Resource(s): 1, 2, 3, 4, 6, 9, 13

Day 12

Jordan curve theorem; Utility problem; Konigsberg bridge problem; Euler circuits and paths /Hamiltonian circuits and paths; Tournament matrix;
Resource(s): 1, 5, 7, 8, 9, 15

Day 13

Tournament matrix; Euler circuits/Hamiltonian circuits;
Resource(s): 1, 5, 7, 8, 9, 15

Day 14

Konigsberg bridge problem; Euler circuits and paths /Hamiltonian circuits and paths; Tournament matrix;
Resource(s): 1, 5, 6, 7, 8, 9, 13, 15

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.
Resource(s): 1, 6, 9, 13

Day 16

Scientific notation; basic notation and fundamental operations; Craig’s stories (number magnitude and estimation)
Resource(s): 1, 2, 3, 4, 9, 13

Day 17

Divisibility tests for the integers 2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,32
Classification, proof, and application.
Resource(s): 1, 5, 6, 7, 8, 9, 10. 11, 13, 15

Day 18

Divisibility tests from the perspective of Blocks, Flats, Longs, and Units
Resource(s): 1, 5, 6, 7, 8, 9, 10. 11, 13, 15

Day 19

Review closure, associative, zero, inverses (+ and x), commutative, distributive.
Resource(s): 1, 5, 6, 7, 8, 9, 10. 11, 13, 15

Day 20

The Division algorithm.
Resource(s): 1, 5, 6, 7, 8, 9, 10. 11, 13, 15

Day 21

Prime numbers – Locker problem; sieve of Eratosthenes; size and dimensions of a sieve; Fundamental theorem of arithmetic;
Resource(s): 1, 5, 6, 7, 8, 9, 10. 11, 13, 15

Day 22

Greatest common divisor, GCD's – four methods
Set definition, prime factorization, Euclidean algorithm, krazy methods
Resource(s): 1, 5, 6, 7, 8, 9, 10. 11, 13, 15

Interesting side note

Day 23

Wrap-up GCD's 
Least common multiple LCM's – four methods
Set definition, prime factorization, formula, krazy methods
Resource(s): 1, 5, 6, 7, 8, 9, 10. 11, 13, 15

Day 24

Wrap-up LCM

https://twitter.com/joann_sandford/status/1204433454273875968?s=20

Examin method for more than two numbers – GCD and LCM
Resource(s): 1, 5, 6, 7, 8, 9, 10. 11, 13, 15

Day 25

Cayley tables revisited -- Closure, Associative, Commutative, Identity, Inverses, and Distributive properties.
Resource(s): 1, 5, 6, 7, 8, 9, 10. 11, 13, 15

Day 26

Conclude Cayley tables -- Magic Math:
1)   first three digits of phone # times 80 || 2) add 1 || 3) multiply by 250 || 4) add the last four digits of your phone # || 5) add the last four digits of your phone # - AGAIN || 6) subtract 250 || 7) Divide by 2.  Do you recognize the answer?  Why does this work??
2)   Pick 2 4 digit numbers; make a sum chart; choose 4 values from chart so that no two share the same row or column; predict the sum -> it will be the sum of the digits in the first 2 numbers;
Resource(s): 1, 5, 6, 7, 8, 9, 10. 11, 13, 15

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;
Introduce Magic Squares;
Resource(s): 1, 5, 7, 8, 10, 11, 15

Day 30

Magic squares - conjectures and proof. 
Clever Counting hand out-- Combinations and permutations using intuition
Resource(s): 1, 5, 7, 8, 10, 11, 13, 15

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;
Adjacency matrices and matrix operations; Sales routes;
Resource(s): 1, 5, 7, 8, 10, 11, 15

Day 35

Sales routes; Use Excel to examine Fib Seq/ Lucas numbers, and golden ratio;sprouts (discrete yearbook)
Resource(s): 1, 5, 7, 8, 9, 10, 11, 13, 15

Day 36

Solve traveling salesman problems; develop Brute Force; Greedy algorithm; Nearest Neighbor; Repeated Nearest Neighbor algorithms
Resource(s): 1, 5, 7, 8, 10, 11, 15

Day 37

Solve traveling salesman problems; apply Brute Force; Greedy algorithm; Nearest Neighbor; Repeated Nearest Neighbor algorithms
Resource(s): 1, 5, 7, 8, 10, 11, 15

Day 38

Sorting algorithms; Model Bubble, Insertion, Selection, and Quicksort with cards
Resource(s): 1, 5, 7, 8, 10, 11, 14, 15

Day 39

Sorting algorithms; Analyze Bubble, Insertion, Selection, and Quicksort with arrays; Utilize Big O – notation to give estimate of time for algorithms.
Resource(s): 1, 5, 7, 8, 10, 11, 14, 15

Day 40 

Four four's activity – review and model instruction for order of operations and practice
Resource(s): 1, 5, 6, 7, 8, 9, 10. 11, 13, 15

Day 41 

Integers, scientific notation – small and large numbers, the division algorithm, prime numbers.
Resource(s): 1, 5, 6, 7, 8, 9, 10. 11, 13, 15

Day 42 

Greatest common divisor, least common multiple, fundamental theorem of arithmetic, famous number theory problems.
Resource(s): 1, 5, 6, 7, 8, 9, 10. 11, 13, 15

 Day 43

 Linear Diophantine equations; lattice points; solve ax + by = n; generate parametric solutions
Resource(s): 1, 5, 6, 7, 8, 9, 10. 11, 13, 15

 Day 44

Arithmetic sequences; geometric sequences; non-arithmetic sequences; recursive / iterative rules and explicit rules.
Resource(s): 1, 5, 7, 8, 9, 10, 11, 13, 15

 Day 45

Congruence theory using modular / clock arrays or tables.
Number puzzles and properties etc.
Resource(s): 1, 5, 7, 8, 9, 10, 11, 13, 15

 July 25

FINAL EXAM 11:30am – 3:00 pm

This course is taught with handouts from a variety of sources.
1.     Richgels, G.W., Rypkema, C., Frauenholtz, T., Sarles, G., Severson, A.R., Webb, D. Number Activities for Teachers, Bemidji State University.
2.     The Rational Number Project materials are available on-line
a.      http://www.cehd.umn.edu/rationalnumberproject/default.html
b.     http://www.cehd.umn.edu/rationalnumberproject/rnp2.html
http://www.cehd.umn.edu/rationalnumberproject/rnp1-09.html
3.     Carpenter, T.P., Fennema, E., Franke, M.L., Levi, L., Empson, S.B. (1999). Children’s Mathematics: Cognitively Guided Instruction. Heinemann (pub).
4.     Cramer, K.A., Monson, D. S., Wyberg, T., Leavitt, S., & Whitney, S. B. (2009). Models for initial decimal ideas, September 2009, Teaching Children Mathematics; NCTM (pub).
5.     Dossey, Otto, Spence, and Vanden Eynden (1997). Discrete Mathematics, Addison-Wesley (pub).
6.     Gallian, J. A. (2010). Contemporary Abstract Algebra, 7th ed.  Brooks/Cole Cengage Learning (pub). Chapter 1.
7.     Hausner. Discrete Mathematics. Saunders College publishing, 1992.
8.     Kenney, and Hirsch. Discrete Mathematics Across the Curriculum K-12. National Council of Teachers of Mathematics, 1991.
9.     Mathematics for Elementary Teachers a Contempory Approach (2011), Musser, Burger & Peterson, 9th ed.
10. Navigating through Discrete Mathematics in Grades K-5. National Council of Teachers of Mathematics, 2009.
11. Navigating through Discrete Mathematics in Grades 6-12. National Council of Teachers of Mathematics, 2008.
12. Principles and Standards for School Mathematics. National Council of Teachers of Mathematics, 2000.
13. Richard, T. Number Concepts for Elementary & Middle School Teachers, Bemidji State University (pub).
14. Sorting Algorithm. (n.d.) Retrieved from http://en.wikipedia.org/wiki/Sorting_algorithm
15. Wheeler, and Brawner. Discrete Mathematics for Teachers. Houghton Mifflin, 2005.

Technology:

A calculator. Use of a computer lab.

Department of Mathematics and Computer Science

8710.3320 MIDDLE LEVEL ENDORSEMENT LICENSE FOR TEACHERS OF MATHEMATICS

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:

In this syllabus you will find the word TEACH. This will mean to:

1. Launch:  This is where the teacher sets the context of the problem or activity being worked on this day.  This involves making sure the students clearly understand the mathematical context and the mathematical challenge of the day’s activities.
2. Explore:  This is the time where the students get to work in pairs, individually, or as a class to solve problems presented by the lesson.
3. Share: This occurs when most of the students have made sufficient progress toward solving the problem presented with today’s lesson.  It is during this phase that the students learn how others approached the problem and possible solution routes.  Helps students deepen their understanding of the mathematical ideas presented in the day’s lesson.
4. Summarize:  During this phase the teacher concludes the lesson by clearly stating what the main idea was in the lesson, being sure to clear up any confusion that may arise during the “share” segment.  Helps students focus their understanding of the mathematical ideas presented in the lesson.

Standard

K/A

Activity

Assessment

(2)  concepts of discrete mathematics:

(a) application of discrete models to problem situations using appropriate representations, including sequences, finite graphs and trees, matrices, and arrays;

K A

TEACH:
Introduce and study finite graphs and trees with the Konigsberg Bridge problem; develop vertex-edge graphs; identify and differentiate Euler and Hamiltonian circuits; use graphs to model round robin tournaments; use vertex-edge graphs to model communication/movement between adjacent positions; develop adjacency matrices and use matrix multiplication to determine n-move paths; use trees to model elimination tournaments, calculate number of games to win and total number of games played;
Days 34-37;

TEACH:
Sequences studied will include arithmetic, geometric, and other sequences from M3064 Notebook. Recursive and explicit formula’s will be used with the sequences. Also the Fibonacci and Lucas sequences will be studied.
Days 35, 44, 45

TEACH:
Use arrays to model sorting techniques.

Days 38,39

Assesment: test 2 and test 3.
-Students will classify finite vertex-edge graphs as Euler graphs, Euler circuits, Hamiltonian graphs, Hamiltonian circuits, or neither.
-Students will construct finite vertex-edge graphs that are a combination of Euler graphs, Euler circuits, Hamiltonian graphs, or Hamiltonian circuits or explain why it is impossible.
-Students will interpret tournament graphs and trees to identify the winner and assign places.
-Students will construct an adjacency matrix and find the number of paths of length n between two points and then specify the paths.

Assesment: test 5.
-Students will classify sequences as arithmetic, geometric, or neither.

Assesment: tests 4,5.
-Students will use arrays to model different sorting techniques from a starting order, step-by-step, to a specified ascending or descending order.

(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:
Study systematic counting techniques through the application of the multiplication principle. Introduce counting/multiplication principle to count letter arrangements of words, election of specific officers, election of representatives – develop permutation and combination formulae, nPr and nCr, to solve problems; Count number of types of five card hands that can be dealt from a standard fifty-two card deck; count number of t-topping pizzas from n toppings; connect to Pascal’s triangle; total number of pizzas from n toppings; view Pascal’s triangle as collection of binomial coefficients and combination notation;
Days 31-33

TEACH:
Explore, develop, analyze and compare algorithms used to solve traveling salesman problems; include brute force, cheapest link or greedy algorithm, nearest neighbor and repeated nearest neighbor algorithms to solve traveling salesman problems; apply systematic counting techniques to calculate the number of routes possible for a traveling salesman. Based upon this calculation, identify routes that have finite time requirements for solution and routes that practically can not be solved; apply nearest neighbor, cheapest link (greedy), repeated nearest neighbor algorithms to find possible solutions; estimate time to find optimal solution.
Days 36,37

Assesment: test 4, test 5;
-Students will apply systematic counting techniques to calculate the number of outcomes in permutation and combination problems.
-Students will apply systematic counting techniques to calculate the number of distinguishable arrangements of letters that can be made from words.

Assesment: test 5;
-Students will apply systematic counting techniques to determine the existence of a solution, the number of possible solutions, and the optimal solution for a traveling salesman problem.

(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:
Study the Brahman monk myth about the end of the world-64 golden disks; students will use pattern searching to develop a recursive / iterative algorithm for counting number of moves to solve an n-disk problem; develop an explicit formula for counting the number of moves; solve / prove the algorithms using mathematical induction; estimate the end of the earth based upon one move per second;
Day 29

TEACH:
Study organization of information and sorting problems; use decks of cards to model concretely sorting and tables to model semi-concretely; study selection, insertion, bubble, and quick sorts; estimate time for sorts; use big O notation to classify sorts; use big O estimates to calculate time difference for sorting n-items; choose appropriate sort for n items; Use case-by-case analysis to determine best sorting technique for data.

Days 38,39

TEACH:
Utilize organization of information to analyze or to do pattern searching on  sequences; use difference patterns to identify arithmetic, geometric or other sequences; use mathematical induction to establish prove the equality of recursive/iterative formulas and explicit formulas.
Day 44

Assesment: test 4
-Students will apply pattern searching, iteration and recursion, and mathematical induction to investigate, solve, and extend the results of the Brahman monk problem to solve a similar problem.

Assesment: tests 4,5;
-Students will use arrays to model different sorting algorithms.
-Students will be given specific parameters about the number of objects to be sorted and time estimates. They will then organize the information and do a case-by-case analysis to calculate the length of time to sort the objects. They will be asked to compare sorting algorithms with respect to time efficiency.

Assesment: test 5;
-Students will classify sequences as arithmetic, geometric, or neither.
-Students will be asked to formulate recursive and explicit formulae for arithmetic and geometric sequences.
-Students will be asked to verify the equality of iterative and recursive formula.

(d)  exploration, development, analysis, and comparison of algorithms designed to accomplish a task or solve a problem;

K A

TEACH:
Explore, develop, analyze and compare algorithms used to solve traveling salesman problems; include brute force, cheapest link or greedy algorithm, nearest neighbor and repeated nearest neighbor algorithms to solve traveling salesman problems; apply systematic counting techniques to calculate the number of routes possible for a traveling salesman. Based upon this calculation identify routes that have finite time requirements for solution and routes that practically can not be solved; apply nearest neighbor, cheapest link (greedy), repeated nearest neighbor algorithms to find possible solutions; estimate time to find optimal solution;
Days 35,36,37

TEACH:
Study sorting problems; use decks of cards to model concretely sorting and tables to model semi-concretely; study selection, insertion, bubble, and quick sorts; estimate time for sorts; use big O notation to classify sorts; use big O estimates to calculate time difference for sorting n-items; choose appropriate sort for n items; Use case-by-case analysis to determine best sorting technique for data.
Days 38,39

Assesment: test 5;
-Students will explore, develop, analyze, and compare algorithms designed to find solutions for a traveling salesman problem.

Assesment: test 4,test 5;
-Students will use arrays to model different sorting algorithms.
-Students will be given specific parameters about the number of objects to be sorted and time estimates. They will then calculate the length of time to sort the objects. They will be asked to compare sorting algorithms with respect to time efficiency.

(3)  concepts of number sense:

(a)  understand number systems; their properties; and relations, including whole numbers, integers, rational numbers, real numbers, and complex numbers;

K A

TEACH:
Examine the development of number systems or sets – counting, whole, integers, reals, complex and the operations each set completes; examine the properties each set possesses; determine sets that have specified properties;
Days 1-11

TEACH:
Study rotation and flip operations on a physical triangle and square; study the composition of functions/operations - rotate then rotate, rotate then flip, flip then rotate, flip then flip; find equivalent operations; perform the operations concretely and abstractly; build a Cayley table; look for identity operation(s), determine commutativity, associativity and inverses;
Days 11, 15

Assesment: test 1;
-Students will identify and give examples of different classes / sets of numbers, such as whole numbers, integers, rational numbers, real numbers, and complex numbers, and will determine properties and relations of these different classes / sets of numbers.

Assesment: test 2;
-Students will be asked to determine the properties and relations that a given set has based upon a Cayley table.

(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:
Build intuitions of numbers with contextual stories that require numbers of different magnitudes to make logical and mathematical sense. Use numbers so that students can use mental mathematics and make reasonable estimates to evaluate the accuracy of the completion of the stories.
Day 16

TEACH:
Build intuitions of numbers with contextual stories that require numbers of different magnitudes to make logical and mathematical sense. Utilize handouts from departmental book to connect standard notation, place value and scientific notation to build intuitions, mental math, estimation and reasonableness of answers. Review/introduce scientific notation and operations with numbers in scientific notation;

Day 16

Assessment: test 3;
-Students will use intuition, a sense of magnitude, mental mathematics, estimation, place value and reasonableness to match numbers to situations.

Assessment: test 3;
-Students will be asked to compute with numbers in scientific notation and to check that their answers are reasonable.

(c)  possess a sense for operations, application of properties of operations, and the estimation of results;

K A

TEACH:
Explore addition, subtraction, multiplication, division, fraction and decimal representation in bases 2-9; stress place value; study operational algorithms to understand why the operations work in base 10;
Days 1-13

Assessment: test 1;
Students will use their intuitive sense for operations, apply properties of operations, and estimation to fractions or decimals and to explain how / why an operation gives its results.

(d)  be able to translate among equivalent forms of numbers to facilitate problem solving; and

K A

TEACH:
Explore number representations in other bases; explore representation of rational numbers as decimals and percents; convert decimals, terminating and repeating, to rational numbers; define irrational numbers; be able to give examples of irrational numbers based upon definition; prove that the square root of a prime number is irrational;
Days 1-11

Assessment: tests 1,2;
-Students will convert a number from one form to another.
-Students will prove the square root of a specified prime is an irrational number.

(e)  be able to estimate quantities and evaluate the reasonableness of estimates;

K A

TEACH:
Explore the activity Clever Counting from the departmental handbook; make conjectures; using the multiplication principle, calculate the number of possibilities for an event and then convert the possibilities to time requirements; make reasonable estimates for various operations, to evaluate conjectures;
Days 30-33

Assessment: with test 3;
-Students will make reasonable assumptions about situations. They will take their assumptions to calculate estimates. They will use these estimates to answer questions about the situation.