# Exam 1 Review by L91Lstwl

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```									Final Exam Review
Chapter 1: Viewing Mathematics
   Mathematics as Problem Solving
o The Role of Problem Solving
 The Meaning of a Problem p. 38
 A problem is a situation for which the following conditions exist
 It involves a question that represents a challenge for the individual
 The question cannot be answered immediately by some routine procedures known
to the individual
 The individual accepts the challenge
 The Meaning of Problem Solving p. 39
 Problem solving is a process by which an individual uses previously learned
concepts, facts, and relationships, along with various reasoning skills and strategies,
 Algorithms are known steps used for solving different types of equations – the
problem solving process CANNOT be made into an algorithm
 Solution – process used to find the answer
o Importance of Problem Solving
 Mathematics is primarily used to solve problems in mathematics and in the real world
 Learning to solve problems is the principal reason for studying mathematics
 Mathematics is MUCH more than algorithms
 Problem solving applies to all aspects of our lives, NOT just mathematics

Chapter 2: Sets and Whole-Number Operations and Properties
   Sets and Whole Numbers
o Connecting Sets to Whole Number Ideas
 Sets and their elements
 set: any collection of objects or ideas that can be listed or described
 element: each individual object in a set, they are separated by commas
 empty set or null set: a set with no elements in it can be designated by either of the
following symbols, but NEVER both at the same time – { } or  but NEVER {}
 finite set: a set with a limited or countable number of elements
 infinite set: a set with an unlimited number of elements
 One-to-one Correspondence
 Definition of One-to-one Correspondence: Sets A and B have one-to-one
correspondence if and only if each element of A can be paired with the exactly one
element of B and each element of B can be paired with exactly one element of A.
 Equal and Equivalent Sets
 Definition of Equal Sets: Sets A and B are equal sets, symbolized by A = B (read “A
is equal to B”), if and only if each element of A is also an element of B and each
element of B is also an element of A.
 Definition of Equivalent Sets: Sets A and B are equivalent sets, symbolized by A ~
B (read “A is equivalent to B”), if and only if there is one-to-one correspondence
between A and B.
o Using Sets to Define Whole Numbers
 Definition of a Whole Number: A whole number is the unique characteristic embodied in
each finite set and all sets equivalent to it. The number of elements in set A (the cardinality
of set A) is expressed as n(A).
 The set of whole numbers is an infinite set designated: W = {0, 1, 2, 3, …}
 Counting: the process that enables people systematically to associate a whole number
with a set of objects
o Using Sets to Compare and Order Whole Numbers
 Procedure for using one-to-one correspondence to compare whole numbers
 Look at the sets for each of the numbers
 One-to-one correspondence cannot be made between the elements of two sets
ө set with left over elements has more elements
ө whole number for set with more is greater than for other set
 Using Subsets to Describe Whole-Number Comparisons
 Definition of a subset of a set: For all sets A and B, A is a subset of B, symbolized
as A  B, if and only if each element of A is also an element of B.
 Venn diagram: representation of sets using circles, where the elements of a set are
contained within a circle
 Definition of a proper subset of a set: For all sets A and B, A is a proper subset of B,
symbolized A  B, if and only if A is a subset of B and there is at least one element of
B that is not an element of A.
 A is NOT a subset of B is symbolized: A  B
 Definition of less than and greater than: For whole umbers a and b and sets A and
B, where n(A) = a and n(B) = b, a is less than b, symbolized by a < b, if and only if A
is equivalent to a proper subset of B. Note that a is greater than b, written a > b,
whenever b < a.
 Ordering Whole Numbers
 Increasing: the next whole number is 1 greater than the number it follows
 n+1
 Decreasing: the next number is 1 less than the number that follows
 n–1
o Three types of Numbers
 Nominal: one, two, three, …
 Ordinal: first, second, third, …
 Cardinal: 1, 2, 3, …

   Addition and Subtraction of Whole Numbers
o Using Models and Sets to Define Addition
 Union of Two Sets
 Definition of the union of two sets: The union of two sets A and B is the set
containing every element belonging to set A or set B and is written A  B (read “the
union of A and B”)
 Intersection of Two Sets
 Definition of the intersection of two sets: The intersection of two sets A and B is
the set containing every element belonging to both set A and set B and is written A
 B (read “the intersection of A and B”)
 Two sets are said to be disjoint or mutually exclusive if their intersection is the
empty set
 Definition of the addition of whole numbers: In the addition of whole numbers, if A and B
are two disjoint sets, and n(A) = a and n(B) = b, then a + b = n(A  B). In the equation a +
b = c, a and b are addends, and c is the sum
 Definition of Greater Than (>) and Less Than (<) for Whole Numbers
o   Modeling Subtraction
 Taking away
 Separating
 Comparing
o   Using Addition to Define Subtraction
 Subtraction as the Inverse of Addition
 Subtraction is the inverse operation of addition
 Definition of subtraction of whole numbers: In the subtraction of whole numbers, a
and b, a – b = c if and only if c is a unique whole number such that c + b = a. In the
equation, a – b = c, a is the minuend, b is the subtrahend, and c is the difference
 Using this model: if you know addition facts, then you know subtraction facts
o   Addition and Subtraction Facts and Fact Families
 Fact is a digit plus a digit
 100 subtraction facts
 Fact Families
 Another way of organizing and learning addition facts
 Fact families deal with sums and the facts that make up those sums

   Multiplication and Division of Whole Numbers
o Using Models and Sets to Define Multiplication
o Sets arranged in equal rows and columns are called rectangular arrays
 To name an array we say the number of rows first, then the number of columns – RC cola –
rows then columns
o Area Model
 The two numbers being multiplied represent the dimensions of a rectangle
 The area of the rectangle is the result of the multiplication
o Using repeated addition to define multiplication
o Using the language of sets to define multiplication
 An ordered pair is a pair of elements where order is important: (a, b) is NOT the same as
(b, a) as long as a  b
 a x b (read “a cross b”) is called a Cartesian product
 Definition of Cartesian product: The Cartesian product of two sets A and B, A x B (read “A
cross B”) is the set of all ordered pairs (x,y) such that x is an element of A and y is an
element of B
 Definition of multiplication of whole numbers: In the multiplication of whole numbers, if A
and B are finite sets with a = n(A) and b = n(B), then a x b = n(A x B). In the equation a x b
= n(A x B), a and b are called factors and n(A x B) is called the product
o Properties of Multiplication
 Closure property: For whole numbers a and b, a x b is a unique whole number
 Identity property: There exists a unique whole number, 1, such that 1 x a = a x 1 = a for
every whole number a. Thus 1 is the multiplicative identity element
 Commutative property: For whole numbers a and b, a x b = b x a
 Associative property: For whole numbers a, b, and c, (a x b) x c = a x (b x c)
 Zero property: For each whole number a, a x 0 = 0 x a = 0
 Distributive property of multiplication over addition: For whole numbers a, b, and c, a x
(b + c) = (a x b) + (a x c)
o Modeling Division
 Finding how many subsets model
 How many subsets of a known quantity can be removed from a set?
 Repeated subtraction idea
 This method also called the measurement interpretation of division
 Finding how many in each subset model
 How many elements are in a known amount of subsets?
 A sharing interpretation of division
o Using Multiplication to Define Division
 Division as the inverse of multiplication
 Multiplication model: factor x factor = product
 Division model: product  factor = missing factor
 Revised Division model: factor x missing factor = product
 Definition of division: In the division of whole numbers a and b, b  0, a  b = c if
and only if c is a unique whole number such that c x b = a. In the equation, a  b =
c, a is the dividend, b is the divisor, and c is the quotient. The operation a  b
c
a
may also be written as  c , b a , or a:b = c
b
 Division as finding the missing factor
 takes advantage of the inverse relationship between multiplication and division
 If I know my multiplication facts, I know my division facts
 Comparing Division to Multiplication
 The Division Algorithm for Whole Numbers
 The division algorithm: For any two numbers a and b, b  0, a division process for a  b
can be used to find unique whole numbers q (quotient) and r (remainder) such that a = bq +
r and 0  r < b.
o Multiplication and Division Facts and Fact Families
 Fact is a digit times a digit
 100 multiplication facts
 Division Facts
 90 division facts
 Fact Families
 Another way of organizing and learning multiplication facts
 Fact families deal with products and the facts that make up those products

   Numeration
o Numeration Systems
 2 is a symbol, it is NOT a number
 Symbol representing a number is a numeral
 Definition of a numeration system: A numeration system is an accepted collection of
properties and symbols that enables people to systematically write numerals to represent
numbers.
o The Hindu-Arabic Place-Value Numeration System
 Most familiar example of a numeration system
 Grouping by tens and place value are the cornerstones of this system
 The group size used determines the base of the numeration system
 Base 2 results in groups of 2
 Base 5 results in groups of 5
 Base 10 sometimes called the base-ten place-value numeration system
 Expressing numerals with different bases
 102 base 5 is written 1025
 Models of base-ten place value
 Proportional models for place value actually exhibit the proportional differences in
the values of the digits in the numerals, i.e. base-ten blocks
 Non-proportional models of place value actual quantities are not visible, i.e. place
value is represented by some object or set of objects – colored chips
 Using expanded notation
 A numeral written to show the sum of its digits times the value of each place
 1025 = 1 x 52 + 0 x 51 + 2 x 50 = 1 x 25 + 0 x 5 + 2 x 1 = 25 + 0 + 2 = 2710
 Using place value to compare numbers
 Use the comparative method of subtraction
o Other Early Numeration Systems
 Roman numeration system
 developed between 500 B.C. and 100 A.D.
 still used today – super bowl games; year a movie was made; some clocks; etc.
 seven basic symbols
Symbol Whole Number
I               1
V               5
X              10
L              50
C              100
D              500
M             1000
 combined and repeated as necessary to form a number
 NO more than 3 of any symbol are used in a numeral
 A bar over a letter represents multiples of 1000
 A symbol representing a smaller number placed in front of number representing a
larger number reduces the number by that amount, i.e. IV = 4; IX = 9; XL = 40; etc.
o Comparing Numeration Systems
 Use of zero
 Babylonian and Hindu-Arabic systems only ones that use zero

Chapter 3: Estimation and Computation
 Integer rod fact families
 The standard algorithm
ө RL
ө Regrouping at the top
ө No regard for place value
 Partial sums
 RL
 Emphasis on place value
 No regrouping
 Denominate numbers
 RL
 Regroup at the top
 Emphasis on place value
 Expanded notation
 RL
 Re-expand as necessary
 Regroup expansion
 Emphasis on place value
 LR
 Emphasis on place value
 No regrouping
 Similar to partial sums
 Scratch method
 LR
 No regard for place value
 Regrouping at the bottom
 Any column first
 No order required
 Emphasis on place value
 No regrouping
 Similar to partial sums
 Only need to know addition facts
o Subtraction
 Denominate numbers
 RL
 Regroup at the top
 Emphasis on place value
 Expanded notation
 RL
 Regroup at the top
 Emphasis on place value
 Standard algorithm
 RL
 Regroup at the top
 No regard for place value
 Left to right subtraction
 LR
 Emphasis on place value
 Regroup at the bottom
 Similar to expanded notation
 Scratch method
 LR
 No regard for place value
 Regrouping at the bottom
 Any column first subtraction
 No order required
 No regard for place value
 Regrouping at the bottom
 Similar to scratch method
 Integer subtraction
 RL
 Emphasis on place value
 Uses integer values (positives and negatives to determine missing addend)
   Multiplication and Division
o Multiplication
 Standard algorithm
 RL
 Regrouping at the top
 No regard for place value
 Partial product method
 RL
 Emphasis on place value
 No regrouping
 Foil
 Horizontal method
 Emphasis on place value
 No regrouping
 Makes use of expanded notation
 Lattice multiplication
 Only need to know multiplication facts and how to add
 Regrouping for addition done outside edge of large box (left and bottom)
 Left to right multiplication
 LR
 Emphasis on place value
 No regrouping
 Similar to partial product
 Russian peasant multiplication
 need to know how to
ө halving
ө doubling
o Division
 Standard division algorithm
 estimation skills required
 need to know how to
ө multiply
ө subtract
 No regard for place value
 looks weird
 Repeated subtraction division
 limited estimation skills required
 emphasis on place value

 Chapter 4: Number Theory
   Factors and Divisibility
o Connecting Factors and Multiples
 Definition of factor and multiple: If a and b are whole numbers and ab = c, then a is a
factor of c, b is a factor of c, and c is a multiple of both a and b.
 Factors of 12: 1, 2, 3, 4, 6, 12 (finite set)
 Multiples of 12: 12, 24, 36, 48, … (infinite set)
 Prime Factorization of 12: 22  3
  Finding factors and multiples
 Theorem: Factor Test – To find all of the factors of a number n, test only those
natural numbers that are no grater than the square root of the number, n .
o Defining Divisibility
 Definition of Divisibility: For whole numbers a and b, a  0, a divides b, written a|b, if and
only if there is a whole number x so that ax = b. Also, a is a divisor of b or b is divisible by
a. Further, a | b means that a does not divide b.
o Techniques for Determining Divisibility
 Theorem: Divisibility of Sums – For natural numbers a, b, and c, if a|b and a|c, then a|(b
+ c)
 Theorem: Divisibility of 2, 5, and 10 –
 A natural number n is divisible by 2 if and only if its units digit is 0, 2, 4, 6, or 8.
 A natural number n is divisible by 5 if and only if its units digit is 0 or 5.
 A natural number n is divisible by 10 if and only if its units digit is 0.
 Need to be able to do demonstration of why divisible by 2 AND 5
 Definition of Even and Odd Numbers:
 A whole number is even if and only if it is divisible by 2.
 A whole number is odd if and only if it is not divisible by 2.
 Theorem: Divisibility Tests for 3 and 9 –
 A natural number n is divisible by 3 if and only if the sum of its digits is divisible by 3.
 A natural number n is divisible by 9 if and only if the sum of its digits is divisible by 9.
 Need to be able to do demonstration of why divisible by 3 OR 9
 Theorem: Divisibility of Products – For natural numbers a, b, and c, if a|c and b|c, and a
and b have no common factors except 1, then ab|c.
 Theorem: A divisibility test for 6 – A natural number n is divisible by 6 if and only if it is
divisible by both 2 and 3.
 Theorem: Divisibility tests for 4 and 8 –
 A natural number n is divisible by 4 if and only if the number represented by its last two
digits is divisible by 4.
 A natural number n is divisible by 8 if and only if the number represented by its last
three digits is divisible by 8.
 Need to be able to do demonstration of why divisible by 4 OR 8
 Theorem: Divisibility tests fro 7 and 11 –
 A natural number n is divisible by 7 if and only if the number formed by subtracting
twice the last digit from the number formed by all digits but the last is divisible by 7.
 A natural number n is divisible by 11 if and only if the sum of the digits in the even-
powered places minus the sum of the digits in the odd-powered places is divisible by
11.
 Need to be able to apply these only
 Theorems: Divisibility – for natural numbers a, b, and c,
 If a|b and a|c, then a|(b-c)
 If a|b and c is any natural number, then a|bc
 If a|(b + c), and a|b, then a|c
 If a|(b – c), and a|b, then a|c
   Prime and Composite Numbers
o Defining Prime and Composite Numbers
 Definition of Prime and Composite Numbers: A natural number that has exactly two
distinct factors is called a prime number. A natural number that has more than two distinct
factors is called a composite number
 The number ONE is NOT prime or composite because it has only one distinct factor
o Techniques for finding prime numbers
 Sieve of Erastosthenes – Greek mathematician (200 B.C.)
o The role of prime numbers in mathematics
 Prime numbers as building blocks
 Prime numbers are building blocks for composite numbers
 Fundamental Theorem of Arithmetic (Unique Factorization Theorem): Each
composite number can be expressed as the product of prime numbers in exactly one
way, disregarding the order of the factors
 Finding the prime factorization of a number
 When a number is expressed as a product of only prime numbers it is called the
prime factorization of that number
o Greatest common factor (GCF) and least common multiple (LCM)
 Greatest common factor
 GCF
 Definition of Greatest Common Factor: The greatest common factor of two natural
numbers is the greatest natural number that is a factor of both numbers
 Least Common Multiple
 LCM
 Definition of the Least Common Multiple: The least common multiple of two
natural numbers is the smallest natural number that is a multiple of both the natural
numbers
o Prime and composite numbers and relationships
 Relationships involving GCF and LCM
 Theorem: The GCF-LCM Product – The product of the GCF and the LCM of two
numbers is the product of the two numbers

Chapter 5: Understanding Integer Operations and Properties
   Addition, Subtraction, and Order Properties of Integers
o Integer Uses and Basic Ideas
o 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.
o The opposite of an integer is the mirror image of the integer around zero on the number line
o 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
 Using Counters Model
 A black counter and a red counter cancel each other
 Concrete way of representing the addition of integers
 Using a Charge Field Model
 Another model for adding integers
 + cancels out –
 Using the Number Line
 Allows students an opportunity to “act out” the mathematics
 Great for kinesthetic/visual learners (most kids)
 Using a Calculator :Great tool for exploring patterns and ideas associated with integers
 Formulating procedures for Adding Integers
ө Adding two positive integers: Add the digits and keep the sign
ө Adding two negative integers: Add the digits and keep the sign
ө Adding a positive and a negative integer: Subtract the smaller from the larger
digit and keep the sign of the larger digit
 Additive Inverse Property: For each integer a, there is a unique integer, -a, such that a +
(-a) = 0
 Closure Property: For integers a and b, a + b is a unique integer
 Additive Identity Property: Zero is the unique integer such that for each integer a, a + 0 =
0+a=a
 Commutative Property: For all integers a and b, a + b = b + a
 Associative Property: For all integers a, b, and c, (a + b) + c = a + (b + c)
 Using the Basic Ideas of Integer Addition in a Proof :See page 240 Example 5.5
o Modeling Integer Subtraction
 Using Counters Model
 A black counter and a red counter cancel each other
 Concrete way of representing the addition of integers
 Using a Charge Field Model
 Another model for adding integers
 + cancels out –
 Using the Number Line
 Allows students an opportunity to “act out” the mathematics
 Great for kinesthetic/visual learners (most kids)
 Using Mathematical Relationships and Patterns
 Apply “Addend + Missing Addend = Sum” model to integer subtraction (See example
5.9)
 Definition of Integer Subtraction: For all integers a, b, and c, a – b = c if and only if
c+b=a
 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.
 Procedures for Subtracting Integers p. 274
 Take Away: To find 5 – (-2), take 2 red counters from a counter model for 5
 Missing Addend: To find 5 – (-2), think, “What number adds to -2 to give 5?”
 Add the Opposite: To find 5 – (-2), find 5 + 2
o Comparing and Ordering Integers
 Using the Number Line to Order Integers
 Numbers on the right of a given point on the number line are larger than numbers to
the left of that point
 Using Addition to Order Integers
 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
   Multiplication, Division, and Other Properties of Integers
o Modeling Integer Multiplication
 Using the Number Line Model :Good way of showing and explaining integer multiplication
 Using Mathematical Relationships, Patterns, and Reasoning
 Repeated addition model can be used to explain multiplication of integers
 Patterns can be established to show why a negative times a negative is a positive –
see example p. 259 (STILL a good test question)
 Procedures for Multiplying Integers
ө Multiplying two positive integers: Multiply digits, keep the sign (+)
ө Multiplying two negative numbers: Multiply digits, change the sign to (+)
ө Multiplying a positive and a negative: Multiply digits, change the sign to (-)
o Properties of Integer Multiplication
 Basic Properties of Integer Multiplication
 Closure Property – For all integers a and b, ab is a unique integer
 Multiplicative Identity Property – 1 is the unique integer such that for each integer
a, a x 1 = 1 x a = a
 Commutative Property – For all integers a and b, ab = ba
 Associative Property – For all integers a, b, and c, (ab)c = a(bc)
 Distributive Property – For all integers a, b, and c, a(b + c) = ab + ac and (b +c)a =
ba + ca
 Zero Property of Multiplication – For all integers a, a(0) = 0(a) = 0
 Basic Properties of Integer Multiplication in a Proof – p. 263
o Explaining Integer Division
 Use “Factor x Missing Factor = Product” model from before
 Definition of Integer Division – For all integers a, b, and c, b  0, a  b = c if and only if c
xb=a
 Procedure for Dividing Integers
 Dividing two positive integers: Divide digits, keep the sign (+)
 Dividing two negative integers: Divide digits, change the sign to (+)
 Division with one positive and one negative integer: Divide digits, change the sign to (-)
o More Properties of Integer Multiplication and Division
 Some Integer Division Properties
 when a  0, then a  a = 1
 a1=a
 0a=0
 a  0 is undefined for all a
o 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
 Some Properties of Opposites for Integers
 For all integers a and b,
ө –(-a) = a
ө –a(-b) = ab
ө (-a)b = a(-b) = -(ab)
ө a(-1) =(-1)a = -a
 Some Distributive Properties for Integers
 Distributive Property for Multiplication over Subtraction – For all integers a, b,
and c, a(b – c) = ab – ac
 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
   Rational Number Ideas and Symbols
o Modeling Rational Numbers
 Used to describe a quantity between 0 and 1
 identify the whole representing the numeral 1
 separate the whole into equal parts
 use an ordered pair of numbers to describe the portion of the whole under consideration
 Identifying the whole and separating it into equal parts
 Egg carton fractions
 Integer Rods
 Using two integers to describe part of a whole
 Need more language to describe part-whole relationship
 number of pieces of interest vs. number of pieces found in the original whole
o Defining Rational Numbers
 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
o Using Fractions to Represent Rational Numbers
 Fractions and Equivalent Fractions
a
 Definition of a fraction: A fraction is a symbol, , where a and b are numbers and
b
b  0. Here, a is the numerator of the fraction and b is the denominator of the
fraction
 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
 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
 Definition of equivalent fractions : Two fractions,        and , are equivalent
b       d
fractions if and only if ad = bc
 Using fractions to represent rational numbers
 every rational number can be represented by an integer in the numerator and the
denominator
0.25 1
 sometimes rational numbers are represented by non-integers             
0.50 2
o Properties of Fractions
a                       a ac
 The Fundamental Law of Fractions: Given a fraction            and a number c  0, 
b                       b bc
 Fractions in simplest form
 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.
ө Finding equivalent fractions on the number line
ө Folding paper
ө Using a calculator
ө Using Integer rods
o Using Decimals to Represent Rational Numbers
 Decimals
 Description of a decimal: A decimal is a symbol that uses a base-ten place-value
system with tenths and multiples of tenths to represent rational numbers
 decimal point divides the decimal portion of the number from the whole number
portion of the number
 Expanded notation
 1  1 
 23 .85  210   31  8   5       
 10   100 
 Writing a decimal for a fraction
3
       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 4 x 25  100
 terminating decimals – rational numbers that have a finite number of decimal
4
places when written as decimals:  0.8
5
 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
 Generalization about decimals for rational numbers: Every rational number can
be expressed as a terminating or a repeating decimal
 Scientific notation
 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
o Connecting Rational Numbers to Whole Numbers, Integers, and Other Numbers
a
 a=a1=
1
 The set of rational numbers is denoted by Q
 The set of real numbers is denoted by R
 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.
 The real numbers are composed of the rational numbers and the irrational numbers ( 2 ,
1.23223222322223…, , etc.)
 The number line is dense – there are no holes in the number line
 Between any pair of rational numbers is an irrational number
 Between any pair of irrational numbers there is a rational number
   Adding and Subtracting Rational Numbers
o Modeling Addition and Subtraction of Rational Numbers
 Modeling Adding and Subtracting: Fractions with like denominators
a       b a b a  b            a b a  b
rational numbers      and ,                  and  
c       c c c          c         c c      c
 Using integer rods to add and subtract fractions with like denominators
 Modeling Adding and Subtracting: Fractions with unlike denominators
 Using integer rods to add and subtract fractions with unlike denominators
 Using paper folding method
o Adding and Subtracting Rational Numbers in Fraction Form
 Procedure for adding and subtracting rational numbers represented by fractions: For
rational numbers      and ,                             and            
b       d b d bd bd               bd         b d bd bd          bd
o Properties of Rational Number Addition and Subtraction
a
 Definition of rational number subtraction in terms of addition: For rational numbers
b
c a c e                      e                                         e c a
and ,   if and only if             is the unique rational number such that  
d b d f                      f                                         f d b
 Properties of Addition of rational numbers
a       c a c
 Closure property – For rational numbers           and ,  is a unique rational
b       d b d
number
a a         a
 Identity property – A unique rational number, 0, exists such that 0    0 
b b         b
a
for every rational number ; 0 is the additive identity element
b
a       c a c c a
 Commutative property – For rational numbers        and ,   
b       d b d d b
a c          e
 Associative property – For rational numbers , , and ,
b d          f
a c e a c e
      
b d f b  d f 
a
 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
o Adding and Subtracting Rational Numbers in Decimal Form
 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

Chapter 6: Rational Number Operations and Properties
   Multiplying Rational Numbers
o Modeling rational number multiplication
 Repeated addition can be used when we have a whole number times a rational number:
4 4 4 4 12               2
3                 or 2
5 5 5 5 5                5
 Area model can also be used for multiplying a mixed number times a rational number:
1 1
2   ? see figure 6.15, p. 308
2 2
 Additionally the area model can be used to show multiplication of a rational number
times a rational number.
o Multiplying rational numbers in fraction form
 Fraction with a numerator of one is called a unit fraction
 Generalization about multiplying rational numbers represented by unit fractions: For
rational numbers a and b , a  b  ab
1      1  1 1    1

   Procedure for multiplying rational numbers in fraction form: For rational numbers        a
b
and c , b  c  ac
d
a
d   bd
 Using paper folding to show multiplication of rational numbers
 Integer rod steps (ALWAYS use least number of rods possible)
 Represent the factors of the original problem
 Run a race to a tie
 Represent the 2nd factor only using the new base and find the part of the numerator
indicated by the original 1st factor
 The top and bottom rods now form the answer
o Properties of rational number multiplication
 Basic properties of rational numbers
 Basic properties for multiplication of rational numbers
 Closure property: For rational numbers b and c , b  c is a unique rational number
a
d
a
d
 Identity property: A unique rational number, 1, exists such that 1 b  b  1  b ; 1 is
a   a       a

the multiplicative identity element
 Zero property: For each rational number                 a
b
, 0 b  b 0  0
a   a

 Commutative property: For rational numbers                         cb
a c
a
b
b d
andd
a
c
d
,
   Associative property: For rational numbers b , c , and e , b  c  e  b  c  e
a
d       f
a
d    f
a
d f
                        
   Distributive property: For rational numbers b da , c , and e , a
f   b
 d f  bd  be
c   e    a c   a
f
                     
                                                                   a
Multiplicative inverse: For every nonzero rational number b , a unique rational
number, b , exists such that b  b  b  b  1
a
a
a   a
a

  Property for multiplying an integer by a unit fraction: For any integer a and any unit
fraction b , a  b  b
1      1   a

o Multiplying rational numbers in decimal form
 Procedure for multiplying rational numbers in decimal form
 Multiply the factors as if they are whole numbers, then place the decimal point into
the appropriate place in the product
 The product of a factor with m places to the right of the decimal point and a factor
with n places to the right of the decimal point will have m + n decimal places to the
right of the decimal point in the product (including zeros that are part of the product).
   Dividing, Comparing, and Ordering Rational Numbers
o Modeling rational number division
 no remainders with rational number division
 Integer rods
o Defining rational number division
 Definition of Rational Number Division in terms of multiplication: For rational numbers
a
b
and c , c  0, b  c  e if and only if e is a unique rational number such that e  c  b
d
a
d  f                f                                        f  d
a

 Closure property of division for nonzero rational numbers: For nonzero rational
numbers b and c , b  c is a unique nonzero rational number
a
d
a
d
o Dividing rational numbers in fraction form
 Procedure for dividing rational numbers – Multiply by the reciprocal method: For
rational numbers b and c , where c, b, d  0, b  c  b  c
a
d
a
d
a d

 The common denominator method
 choose a common denominator for each fraction
 find the quotient of the numerators
 this is the solution to the original problem
 Procedure for dividing rational numbers – common denominator method: For
rational numbers b and c , where c  0, b  c  bd  bc  bc
a
d
a
d
bd

 Integer rod division of rational numbers
 The complex fraction method
 rewrite the original division problem as a fraction
 find an equivalent fraction with a denominator of one
 Complex fraction: fraction numerator and fraction denominator
 Procedure for dividing rational numbers – complex fractions method: For
a       a d       a d
rational numbers   a
b
and   c
d
, where c  0,   a
b
c 
d
b
c
   b c
c d
   b c
1
bc
d       d c
   Dividing rational numbers in decimal form
 Based on the Fundamental Law of Fractions
 Place value
 Idea is convert the divisor (the denominator) into an integer using equivalent
fractions
 Procedure for dividing rational numbers in decimal form: To find the quotient of
two decimals, in which the divisor has n places to the right of the decimal point,
multiply both the divisor and the dividend by 10n to make the divisor an integer and
to locate the decimal point in the quotient. Then use the whole number division
algorithm to find the digits in the quotient
o Comparing rational numbers
 models
 common denominators
 place value
 Using models to compare rational numbers
 basis for explaining how rational numbers compare
 fraction wall composed of bar fractions – see fig. 6.23, p. 331
 fraction wall can be condensed into a number line
 helps to compare and order rational numbers
 Using common denominators to compare rational numbers
 Denominators can be used as the basis for comparing fractions
 Can build models to show that if denominator the same, largest numerator is largest
rational number
 Generalization about comparing rational numbers that have like denominators: For
rational numbers b and c , where b > 0, b  b if and only if a > c
a
d
a   c

 Generalization about comparing rational numbers that have unlike denominators: For
rational numbers b and c , where b > 0 and d > 0, b  c ( or bd  bc ) if and only if ad > bc
a
d
a
d
bd
 In other words – look at the results of cross-multiplying
ө ad > bc, then b  c
a
d
ө ad < bc, then    a
b
   c
d
ө ad = bc, then     a
b
c
d
o Other characteristics of rational numbers
 multiplicative inverses
 closure for division of nonzero rational numbers
 Denseness of rational numbers
 rational numbers are dense
 Denseness property for rational numbers: For any two rational numbers,              a
b
 c , at
d
least one rational number         e
f
ec
exists such that
f   d
a
b
 this is sometimes referred to as the betweenness property of rational numbers
   Repeating decimals and fractions
 all rational numbers are either repeating or terminating decimals
 converting repeating decimals into fractions – see example p. 335

 Chapter 7: Proportional Reasoning
   The Concept of Ratio
o Meaning of ratio
 A ratio as a comparison
 compares two like or unlike quantities
 Definition of ratio: A ratio is an ordered pair of numbers used to show a
comparison between like or unlike quantities, written x to y, x , x  y, or x : y (y  0)
y
 x and y are called the terms of the ratio
 ALL ratios can be written as fractions
 Equivalent ratios
 Definition of equivalent ratios: Two ratios are equivalent ratios if their respective
fractions are equivalent or if the quotients of the respective terms are the same
(Fundamental Theorem of Fractions)
o Uses of ratio
 Using ratios to compare like quantities of measure
 Three (3) types of comparison possible
ө part to part – nurses to doctors – 6:2
ө part to whole – nurses to staff – 6:8
ө whole to part – staff to nurses – 8:6
 Using ratios to show rates involving time
 rate of speed – distance to time
 Using ratios to show rates involving unit price
 Using ratios to show comparisons of other unlike quantities
 comparisons with different units
o Ratios in decimal form
 sometimes ratios are reported as a single number in decimal form
 fraction is implied by the decimal
   Proportional Variation and Solving Proportions
o Recognizing proportional variation
 ratio remains constant – Fundamental Theorem of Fractions idea
 Definition of quantities varying proportionally: Two quantities vary proportionally if and
only if, as their corresponding values increase or decrease, the ratios of the two quantities
are always equivalent
 Tables are an effective means of showing proportional variation
o Characteristics of quantities that vary proportionally
 A multiplicative relationship
 another use of the Fundamental Theorem of Fractions - b  bc  a   ac

 Tables helpful in seeing these relationships
 Multiplicative property of quantities that vary proportionally: When quantities a
and b vary proportionally, a nonzero number k exists such that b  k , or a  b  k for
a

all corresponding values a and b
 This type of variation is called direct proportionality
 A constant change relationship
 numerator increases by causes constant increase in denominator
 Constant change property of quantities that vary proportionally: When
quantities a and b vary proportionally and the ratio of a to b is 1 to n, a unit change in
a value of a always evokes a constant change of n in the corresponding value of b
 The idea of a proportion
 setting two ratios equal to each other produces an equation known as a proportion
 Definition of a proportion: A proportion is an equation stating that two ratios are
equivalent
o Proportional variation in geometry
 Similar figures and proportional variation
 Geometric figures with the same shape are called similar figures
 Lengths of corresponding sides of similar figures are proportional to each other
 These relationships are used to solve many different types of problems
o Properties of proportions
 First property of proportions: Cross-Product property of proportions: For integers a, b, c,
and d (b  d  0), b  c if and only if ad = bc
a
d
  Second property of proportions: Reciprocal property of proportions: For nonzero integers a,
b, c, and d, b  c if and only if b  c
a
d                a
d

o Solving proportional problems
 can be solved in a variety of ways
 methods are related but require different procedures
 Which method works best depends on what is needed and of course your comfort level in
using that method
 Using properties of equations to solve proportions
 multiply both sides of the proportion by the same number
 Using cross products to solve proportions
 cross multiply to simplify the proportion into a one or two step equation
 Equating numerators or denominators to solve proportions
 Similar to the common denominator method for division
 Converting one ratio to a decimal to solve proportions
 convert one ratio to a decimal (usually the one consisting of just numbers)
   Solving Percent Problems
o Definition of percent
 Definition of percent: a percent is a ratio with a denominator of 100
 represented by the symbol %
 100% means 100 or ALL of the amount
100
 One circle or one grid represents 100%
 to show less than 100% shade in the fractional amount of the circle or grid
equivalent to the desired percent
 to show a percent larger than 100% shade in one whole circle or grid for each
multiple of 100% and then the fractional amount of another circle or grid equivalent
to the desired percent
o Connecting percents, ratios, and decimals
 Converting a percent to a decimal – divide the percent by 100 and drop the percent sign
 Converting a percent to a fraction – place the percent over 100, drop the percent sign,
then simplify the fraction
 Converting a fraction to a decimal or percent – write an equivalent fraction with a
denominator of 100
o Types of percent problems
 Three (3) basic types
 A% x B = C
 Finding a percent of a number – know A% and B
 Finding a number when a percent of it is known – know A% and C
 Finding the percent that one number is of another – know B and C
o Solving percent problems
 Finding a percent of a number
 Finding a number when a percent of it is known – know A% and C
 Finding the percent that one number is of another – know B and C
o Percent increase or decrease
 Increase or decrease relative to the original situation
 Procedure for finding the percent increase or decrease: Step 1 – Determine the
amount of increase or decrease; Step 2 – Divide this amount by the original amount; and
Step 3 – Convert this fraction or decimal to a percent
o Simple and compound interest
 rate of return or rate of payment = interest rate
 amount of money invested or borrowed = principal
 amount of money received or amount of money paid = interest
   Simple interest
 simple interest is computed by multiplying the interest rate (a percent) by the
principal
 amount of interest does not change from year to year
 Assume simple interest is paid annually
 Formula used to calculate simple interest: i = prt (formula will NOT be given)
ө i = amount of interest
ө p = principal
ө r = rate of interest
ө t = time in terms of years
   Compound interest
 interest is paid on the principal and any amount of interest previously earned
 NOT always compounded annually – sometimes quarterly (4 times a year) or even
daily (365 days a year)
 Difference between simple and compound interest rates can be substantial
 Formula used to calculate compound interest: a  p1  n  (formula will be
i nt

given)
ө a = total amount of interest plus principal
ө p = principal
ө i = annual rate of interest
ө n = number of times the interest is to compounded per year
ө t = number of years
Recommend review appropriate questions from chapter reviews.

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