Bemidji State University
M3064/ Number Concepts for Middle
School Teachers (4 credits)
Summer 2013, MTWRF
8-11:30 AM, ARCC T229
Instructor: Dr. Glen
Richgels
Email: --
grichgels@bemidjistate.edu
Office Phone:
755-2824
Professional Education Department Mission Statement:
ŌThe 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 new 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.
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. 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 3. Carpenter, T.P., Fennema,
E., Franke, M.L., Levi, L., Empson,
S.B. (1999). ChildrenÕs 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. |
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: 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 formulaÕs 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. |
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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: 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.
-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. |
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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: 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.
-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. |
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(d) exploration, development, analysis, and comparison of algorithms designed
to accomplish a task or solve a problem; |
K A |
TEACH: 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. |
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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: 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. |
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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: 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. |
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(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. |
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(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. |
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(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. |
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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
PolyaÕs problem solving steps
1.
Understand
the problem
Lesson Sequencing
Intuitions
Þ Concrete ķ Semi-Concrete ķ Abstract
Dr. Glen RichgelsÕ Teaching/Learning
Principles
1.
Teach the
way students learn
2.
Use group
work, heterogeneous, 3-4, change monthly
3.
Communication
student ķ student
4.
Communication
student ķ 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 Grades
A: 100
– 90% B: 89 – 80% C: 79 – 70% D: 69 – 60%
Course Policies
Attendance:
Daily attendance is expected
Participation: Class participation and group work is expected
Tentative Course Calendar
Day 1 |
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 |
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. |
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; CraigÕs 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 |
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 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
PascalÕs 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 |
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