Review for Exam 2
Numeration Systems and Whole Number Computations
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Egyptian, Babylonian, Mayan, and Roman Systems
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Number Bases 2, 4, 5, 12
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Example: Convert TE4twelve to base 5.
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Addition Algorithms—Concrete models; expanded, standard, left-to-right, lattice,
and scratch algorithms
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Subtraction Algorithms—Concrete models; equal addends
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Adding/Subtracting in Other Bases
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Multiplication Algorithms—Concrete models; Lattice; Using Distributive Property
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Division Algorithms—Base 10 Blocks; Repeated Subtraction; Inverse of
Multiplication; Short Division
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Multiplying/Dividing in Other Bases
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Mental Addition Strategies—Adding from the left; Breaking up and bridging,
Trading off; Using compatible numbers; Making compatible numbers
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Mental Subtraction Strategies—Breaking up and bridging, Trading off, Drop the
zeroes
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Mental Multiplication Strategies—Front-End, Using compatible numbers,
Thinking money
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Mental Division Strategies—Breaking up the dividend, Using compatible numbers
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Estimation—Front-End, Grouping to nice numbers, Clustering, Rounding
Integers and Number Theory

Natural Numbers, Whole Numbers, and Integers (negative integers are opposites
of positive integers or natural numbers)
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Integer Addition Models—Chip, Charged field, Patterns, Number Line
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Absolute Value
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Integer Addition Properties—Closure, Commutative, Associative, Identity
element, Additive inverse
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Integer Subtraction Models—Chip, Charged field, Patterns, Number Line
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Integer Subtraction—Definition; Closure
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Order of Operations
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Integer Multiplication Models—Chip, Charged field, Patterns, Number Line

Integer Multiplication Properties—Closure, Commutative, Associative, Identity
element, Zero multiplication
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Distributive Property of Multiplication over Addition
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Difference of Squares
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Division of Integers
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Definition of Less Than

Extending the Coordinate System (x, y)
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Divisibility
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Theorem 4-1 If d|a, n  I, then d|(a  n)

Theorem 4-2
If d|a, and d|b, then d|(a+b).
If d|a, and d does not |b, then d does not |(a+b).
If d|a, and d|b, then d|(a-b).
If d|a, and d does not |b, then d does not |(a-b).
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Divisibility Rules for 2, 3, 4, 5, 6, 8, 9, 10, and 11
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Prime Numbers and Composite Numbers
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Prime Factorization
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Fundamental Theorem of Arithmetic: Each composite number can be written as
a product of primes in one, and only one, way (except for order)
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Number of Divisors
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Fundamental Counting Principle
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Theorem 4-4: If d is a divisor or n, then n/d is also a divisor of n.
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Theorem 4-5: If n is composite, then n has a prime factor p such that p2  n.
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Theorem 4-6: If n > 1 and not divisible by any prime such that p2  n, then n is
prime.

Creating a Sieve of Prime Numbers

Example: The local record store sold x copies of their newest CD, Spring Break
Memories, for a total of $4539. The next day they sold y copies of the same CD,
and collect $8245. How many were sold each day?
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Greatest Common Factor/Divisor (GCF or GCD)—Colored Rods, Intersection of
Sets, Prime Factorization
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Relatively Prime
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GCF(0, a) = a
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Euclidean Algorithm—Theorem 4-7: If a  b, a, b > 0,
then GCF (a, b) = GCF (r, b) where r is the remainder when a is divided by b.

Least Common Multiple (LCM)—Colored Rods, Intersection of Sets, Prime
Factorization
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Theorem 4-8: If a, b  N, then GCF(a, b)  LCM(a, b) = a  b.

Division by Primes Method