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Exam 3 Review
Chapter 5: Understanding Integer Operations and Properties
5.1 Addition, Subtraction, and Order Properties of Integers
5.1.1. Integer Uses and Basic Ideas
5.1.1.1. Definition of Integers: The set of integers, I (more often seen as Z), consists of
the positive integers (the Natural numbers), the negative integers (the opposites of
the Natural numbers), and zero.
5.1.1.2. The opposite of an integer is the mirror image of the integer around zero on the
number line
5.1.1.3. Definition of Absolute Value: The absolute value of an integer is the number of
units the integer is from 0 on the number line. The absolute value of an integer n is
written |n|, and is positive for all n  0
5.1.1.3.1. |x| = x, when x > 0
5.1.1.3.2. |-x| = x, when x > 0
5.1.1.3.3. -|x| = -x, when x > 0
5.1.1.3.4. |x| = -x, when x < 0
5.1.1.3.5. |-x| = -x, when x < 0
5.1.1.3.6. -|x| = x, when x < 0
5.1.1.3.7. |x| = |-x| = -|x| = x, when x = 0
5.1.2. Modeling Integer Addition
5.1.2.1. Using Counters Model
5.1.2.1.1. A black counter and a red counter cancel each other
5.1.2.1.2. Concrete way of representing the addition of integers
5.1.2.2. Using a Charge Field Model
5.1.2.2.1. Another model for adding integers
5.1.2.2.2. + cancels out –
5.1.2.3. Using the Number Line
5.1.2.3.1. Allows students an opportunity to “act out” the mathematics
5.1.2.3.2. Great for kinesthetic/visual learners (most kids)
5.1.2.4. Using a Calculator :Great tool for exploring patterns and ideas associated with
integers
5.1.2.5. Formulating procedures for Adding Integers
5.1.2.5.1. Procedures for Adding Integers
5.1.2.5.1.1.
Adding two positive integers: Add the digits and keep the sign
5.1.2.5.1.2.
Adding two negative integers: Add the digits and keep the sign
5.1.2.5.1.3.
Adding a positive and a negative integer: Subtract the smaller from
the larger digit and keep the sign of the larger digit
5.1.3. Properties of Integer Addition
5.1.3.1. Additive Inverse Property: For each integer a, there is a unique integer, -a,
such that a + (-a) = 0
5.1.3.2. Closure Property: For integers a and b, a + b is a unique integer
5.1.3.3. Additive Identity Property: Zero is the unique integer such that for each integer
a, a + 0 = 0 + a = a
5.1.3.4. Commutative Property: For all integers a and b, a + b = b + a
5.1.3.5. Associative Property: For all integers a, b, and c, (a + b) + c = a + (b + c)
5.1.3.6. Using the Basic Ideas of Integer Addition in a Proof :See page 240 Example 5.5
5.1.4. Modeling Integer Subtraction
5.1.4.1. Using Counters Model
5.1.4.1.1. A black counter and a red counter cancel each other
5.1.4.1.2. Concrete way of representing the addition of integers
5.1.4.2. Using a Charge Field Model
5.1.4.2.1. Another model for adding integers
5.1.4.2.2. + cancels out –
5.1.4.3. Using the Number Line
5.1.4.3.1. Allows students an opportunity to “act out” the mathematics
5.1.4.3.2. Great for kinesthetic/visual learners (most kids)
5.1.4.4. Using Mathematical Relationships and Patterns
5.1.4.4.1. Apply “Addend + Missing Addend = Sum” model to integer subtraction (See
example 5.9)
5.1.4.4.2. Definition of Integer Subtraction: For all integers a, b, and c, a – b = c if
and only if c + b = a
5.1.4.4.3. Theorem: Subtracting an Integer by adding the Opposite – For all
integers a and b, a – b = a + (-b). That is, to subtract an integer, add its
opposite.
5.1.4.5. Procedures for Subtracting Integers p. 274
5.1.4.5.1. Take Away: To find 5 – (-2), take 2 red counters from a counter model for 5
5.1.4.5.2. Missing Addend: To find 5 – (-2), think, “What number adds to -2 to give 5?”
5.1.4.5.3. Add the Opposite: To find 5 – (-2), find 5 + 2
5.1.5. Comparing and Ordering Integers
5.1.5.1. Using the Number Line to Order Integers
5.1.5.1.1. Numbers on the right of a given point on the number line are larger than
numbers to the left of that point
5.1.5.1.2. Graphing guys help us to mark the number line appropriately
5.1.5.2. Using Addition to Order Integers
5.1.5.2.1. Definition of Greater Than (>) and Less Than (<) for Integers: b > a if and
only if there is a positive integer p such that a + p = b. Also, a < b whenever b > a
5.1.5.2.2. See example 5.13
5.2 Multiplication, Division, and Other Properties of Integers
5.2.1. Modeling Integer Multiplication
5.2.1.1. Using a Counters Model: See example p. 255
5.2.1.2. Using a Charged Field Model :See example p. 256
5.2.1.3. Using the Number Line Model :Good way of showing and explaining integer
multiplication; See example p. 257
5.2.1.4. Using Mathematical Relationships, Patterns, and Reasoning
5.2.1.4.1. Repeated addition model can be used to explain multiplication of integers
5.2.1.4.2. Patterns can be established to show why a negative times a negative is a
positive – see example p. 259 (good test question)
5.2.1.4.3. Procedures for Multiplying Integers
5.2.1.4.3.1.
Multiplying two positive integers: Multiply digits, keep the sign (+)
5.2.1.4.3.2.
Multiplying two negative numbers: Multiply digits, change the sign
to (+)
5.2.1.4.3.3.
Multiplying a positive and a negative: Multiply digits, change the
sign to (-)
5.2.2. Properties of Integer Multiplication
5.2.2.1. Basic Properties of Integer Multiplication
5.2.2.1.1. Closure Property – For all integers a and b, ab is a unique integer
5.2.2.1.2. Multiplicative Identity Property – 1 is the unique integer such that for each
integer a, a x 1 = 1 x a = a
5.2.2.1.3. Commutative Property – For all integers a and b, ab = ba
5.2.2.1.4. Associative Property – For all integers a, b, and c, (ab)c = a(bc)
5.2.2.2. Distributive Property : Ties integer addition and multiplication together
5.2.2.2.1. For all integers a, b, and c, a(b + c) = ab + ac and (b +c)a = ba + ca
5.2.2.3. Zero Property of Multiplication – For all integers a, a(0) = 0(a) = 0
5.2.2.4. Basic Properties of Integer Multiplication in a Proof – p. 263
5.2.3. Explaining Integer Division
5.2.3.1. Use “Factor x Missing Factor = Product” model from before
5.2.3.2. Definition of Integer Division – For all integers a, b, and c, b  0, a  b = c if
and only if c x b = a
5.2.3.3. Procedure for Dividing Integers
5.2.3.3.1. Dividing two positive integers: Divide digits, keep the sign (+)
5.2.3.3.2. Dividing two negative integers: Divide digits, change the sign to (+)
5.2.3.3.3. Division with one positive and one negative integer: Divide digits, change the
sign to (-)
5.2.4. More Properties of Integer Multiplication and Division
5.2.4.1. Some Integer Division Properties
5.2.4.1.1. when a  0, then a  a = 1
5.2.4.1.2. a  1 = a
5.2.4.1.3. 0  a = 0
5.2.4.1.4. a  0 is undefined for all a
5.2.4.1.5. Properties of Integer Division – For all integers a and b, a  0, a  a = 1, a 
1 = a, 0  a = 0, and ab  a = b. You cannot divide an integer by zero because
no unique quotient exists
5.2.4.2. Some Properties of Opposites for Integers
5.2.4.2.1. For all integers a and b,
5.2.4.2.1.1.
–(-a) = a
5.2.4.2.1.2.
–a(-b) = ab
5.2.4.2.1.3.
(-a)b = a(-b) = -(ab)
5.2.4.2.1.4.
a(-1) =(-1)a = -a
5.2.4.3. Some Distributive Properties for Integers
5.2.4.3.1. Distributive Property for Multiplication over Subtraction – For all integers
a, b, and c, a(b – c) = ab – ac
5.2.4.3.2. Distributive Property for Opposites over Addition – For all integers a, b,
and c, -(a + b) = -1(a + b) = -a + (-b) = -a – b
Chapter 6: Rational Number Operations and Properties
6.1 Rational Number Ideas and Symbols
6.1.1. Modeling Rational Numbers
6.1.1.1. Used to describe a quantity between 0 and 1
6.1.1.1.1. identify the whole representing the numeral 1
6.1.1.1.2. separate the whole into equal parts
6.1.1.1.3. use an ordered pair of numbers to describe the portion of the whole under
consideration
6.1.1.2. Identifying the whole and separating it into equal parts
6.1.1.2.1. Egg carton fractions
6.1.1.2.1.1.
http://pegasus.cc.ucf.edu/~mathed/egg.html
6.1.1.2.2. Integer Rods
6.1.1.2.2.1.
http://pegasus.cc.ucf.edu/~mathed/crods.html
6.1.1.2.2.2.
http://arcytech.org/java/
6.1.1.2.3. Make your own fraction kit
6.1.1.2.3.1.
http://pegasus.cc.ucf.edu/~mathed/fk.html
6.1.1.3. Using two integers to describe part of a whole
6.1.1.3.1. Need more language to describe part-whole relationship
6.1.1.3.2. number of pieces of interest vs. number of pieces found in the original whole
6.1.2. Defining Rational Numbers
6.1.2.1. Description of a rational number: A rational number is the relationship
represented by an infinite set of ordered pairs, each of which describes the same
quantity – (1,2); (2,4); (3,6); (6,12); etc.
6.1.3. Using Fractions to Represent Rational Numbers
6.1.3.1. Fractions and Equivalent Fractions
a
6.1.3.1.1. Definition of a fraction: A fraction is a symbol, , where a and b are
b
numbers and b  0. Here, a is the numerator of the fraction and b is the
denominator of the fraction
6.1.3.1.2. Proper fraction: when the numerator of the fraction is less than the
denominator of the fraction and both the numerator and the denominator are
integers
6.1.3.1.3. Improper fraction: when the numerator of the fraction is greater than the
denominator of the fraction (fractions with non-integers in the numerator or
denominators are also improper)
a
c
6.1.3.1.4. Definition of equivalent fractions : Two fractions,
and , are equivalent
b
d
fractions if and only if ad = bc
6.1.3.2. Using fractions to represent rational numbers
6.1.3.2.1. every rational number can be represented by an integer in the numerator and
the denominator
0.25 1

6.1.3.2.2. sometimes rational numbers are represented by non-integers
0.50 2
6.1.4. Properties of Fractions
a
6.1.4.1. The Fundamental Law of Fractions: Given a fraction
and a number c  0,
b
a ac

b bc
6.1.4.2. Fractions in simplest form
6.1.4.2.1. Description of the simplest form of a fraction: a fraction representing a
rational number is in simplest form when the numerator and the denominator are
both integers that are relatively prime and the denominator is greater than zero.
6.1.4.2.1.1.
Finding equivalent fractions on the number line
6.1.4.2.1.2.
Folding paper
6.1.4.2.1.3.
Using a calculator
6.1.4.2.1.4.
Using Integer rods
6.1.4.2.1.5.
Using NCIS Mountain
6.1.5. Using Decimals to Represent Rational Numbers
6.1.5.1. Decimals
6.1.5.1.1. Description of a decimal: A decimal is a symbol that uses a base-ten placevalue system with tenths and multiples of tenths to represent rational numbers
6.1.5.1.2. decimal point divides the decimal portion of the number from the whole
number portion of the number
6.1.5.1.3. Using base ten blocks to explore decimals – see p. 287-8
6.1.5.2. Expanded notation
 1  1 
6.1.5.2.1. 23.85  210   31  8   5

 10   100 
6.1.5.3. Writing a decimal for a fraction
3
6.1.5.3.1.  0.75 - divide 3 by 4 to get the decimal equivalent
4
3 3 x25 75


 0.75 - use the Fundamental Law of Fractions
4 4x25 100
6.1.5.3.3. terminating decimals – rational numbers that have a finite number of
4
decimal places when written as decimals:  0.8
5
6.1.5.3.4. repeating decimals – rational numbers that have an infinite number of
decimal places filled by the same number or a fixed number of digits repeated
over an infinite number of decimal places:
1
 0.33333   0. 3
6.1.5.3.4.1.
3
5
 0.454545  0.45
6.1.5.3.4.2.
11
6.1.5.3.4.3.
Generalization about decimals for rational numbers: Every
rational number can be expressed as a terminating or a repeating decimal
6.1.5.4. Scientific notation
6.1.5.4.1. Description of scientific notation: A rational number is expressed in
scientific notation when it is written as a product where one factor is a decimal
grater than or equal to 1 and less than 10 and the other factor is a power of 10
6.1.5.4.1.1.
0.1234 = 1.234 x 10-1
6.1.5.4.1.2.
0.0000001234 = 1.234 x 10-7
6.1.6. Connecting Rational Numbers to Whole Numbers, Integers, and Other Numbers
a
6.1.6.1. a = a  1 =
1
6.1.6.2. The set of rational numbers is denoted by Q
6.1.6.3. The set of real numbers is denoted by R
6.1.6.4. R(Q(Z(W(N)))) or N  W  Z  Q  R – All of the natural numbers are contained
within the whole numbers which are contained within the integers which are
contained within the rational numbers which are contained within the real numbers.
6.1.6.5. The real numbers are composed of the rational numbers and the irrational
numbers ( 2 , 1.23223222322223…, , etc.)
6.1.6.6. The number line is dense – there are no holes in the number line
6.1.6.6.1. Between any pair of rational numbers is an irrational number
6.1.6.6.2. Between any pair of irrational numbers there is a rational number
6.2 Adding and Subtracting Rational Numbers
6.2.1. Modeling Addition and Subtraction of Rational Numbers
6.2.1.1. Modeling Adding and Subtracting: Fractions with like denominators
6.2.1.1.1. Generalization about rational number addition and subtraction: In general, for
a
b a b a  b 
a b a  b 
rational numbers
and ,  
and  
c
c c c
c
c c
c
6.2.1.1.2. Using integer rods to add and subtract fractions with like denominators
6.2.1.2. Modeling Adding and Subtracting: Fractions with unlike denominators
6.2.1.2.1. Using integer rods to add and subtract fractions with unlike denominators
6.2.1.2.2. Using paper folding method
6.2.2. Adding and Subtracting Rational Numbers in Fraction Form
6.2.2.1. Procedure for adding and subtracting rational numbers represented by fractions:
a
c a c ad cb ad  cb 


For rational numbers
and ,  
and
b
d b d bd bd
bd
a c ad cb ad  cb 
 


b d bd bd
bd
6.1.5.3.2.
6.2.3. Properties of Rational Number Addition and Subtraction
6.2.3.1. Definition of rational number subtraction in terms of addition: For rational
a
c a c e
e
numbers
and ,   if and only if
is the unique rational number such that
f
b
d b d f
e c a
 
f d b
6.2.3.2. Properties of Addition of rational numbers
a
c a c
6.2.3.2.1. Closure property – For rational numbers
and ,  is a unique
b
d b d
rational number
6.2.3.2.2. Identity property – A unique rational number, 0, exists such that
a a
a
a
0    0  for every rational number ; 0 is the additive identity element
b b
b
b
a
c a c c a
6.2.3.2.3. Commutative property – For rational numbers
and ,   
b
d b d d b
a c
e
6.2.3.2.4. Associative property – For rational numbers , , and ,
f
b d
a
c
e
a
c
e




      
b d f b  d f 
a
6.2.3.2.5. Additive Inverse Property – For every rational number , a unique rational
b
a
a  a
a a
number  exists such that         0
b
b  b
b b
6.2.4. Adding and Subtracting Rational Numbers in Decimal Form
6.2.4.1. Procedure for adding and subtracting rational numbers represented by
decimals: When adding and subtracting rational numbers in decimal form, align the
place-value positions by aligning the decimal points and apply the algorithms for
addition and subtraction of whole numbers