Math 1321-Exam One, Chapters 1, 2, & 3

3.2-Repeated Addition Approach Multiplication

Let a and b be any whole numbers where a≠0. Then ab= b + b + b +..... +b. If a=1, then ab= 1*b=b; also 0*b=0 for all b.

3.2-Cartesian Product Approach

Let a and b be any whole numbers. If a=n(A) and b=n(B), then ab=n(A X B).

3.2-Communitive Property Multiplication

Let a and b be any whole numbers. The ab=ba.

3.1-Communitive Property for Whole-Number Addition

Let a and b be any whole numbers. Then (a+b)+c = a+ (b+c)

3.2-Associative Property Multiplication

Let a, b, and c be any whole numbers. Then a(bc)=(ab)c

3.2-Distributive Property Multiplication

Let a,b, and c be any whole numbers. Then a(b+c)=ab+ac.

3.3-Theorem Two

Let a,b, and m be any whole numbers where m is nonzero. Then a^m * b^m = (ab)^m. Ex.) 2^3 * 3^3 = (2*2*2)(3*3*3) =(2*3)^3

3.2-Distributive Property of Multiplication over Subtraction

Let a,d, and c be any whole numbers where b ≥ c. Then a(b-c)=ab-ac.

3.3-Theorem Four

Let a,m, and n be any whole numbers were m > n and a, m, and n are nonzero. Then a^m ÷ a^n = a^m-n. EX.) 12^8 ÷ 12^4 = 12^8-4 = 12^4

3.3-Theorem Three

Let a,m, and n be any whole numbers where m and n are nonzero. Then (a^m)^n = a^mn. EX.) (10^3)^11 = 10^33

3.3-Theorem One

Let a,m, and n be any whole numbers where m and n are nonzero. Then a^m * a^n + a^m+n. EX.) 4^3 * 4^5= 4^8

2.2-Ordering Whole Numbers

Let a=n(A) and b=n(B). Then a < b or b > a if A is equivalent to a proper subset of B.

1.2-Three Additional Solving Strategies

Look for a Pattern, Make a List, Solve a Simpler Problem, and Combine Strategies to Solve Problems.

2.1- Are all equivalent sets equal sets?

No.

2.1-Are all subsets proper subsets?

No.

3.3-Proper Order of Operations

PEMDAS Parenthesis Exponents Multiplication Division Addition Subtraction GEMS Groupings (Parenthesis) Exponents Multiplication OR Division Subtraction OR Addition OR=Solve which ever comes first from left to right.

3.1-Take-away Approach

Presented by using Set Model or Measurement Model.

3.1-Missing-addend Approach

Presented by using Set Model, Measurement Model, or a Comparison. a and b=any whole numbers. The a-b=c is and only if a=b+c for some whole number c. C would be called the missing-addend. Shows how to relate subtraction to addition via the four-fact families. (Look up four-fact families)

3.1-Identity Property for Whole-Number Addition

There is a unique whole number, namely 0, such that for all whole numbers a, a+0 = a = 0+a.

3.1-Approaches to Subtraction

Two distinct. 1.) Take-Away Approach 2.) Missing-Addend Approach

2.3-Base Four

0,1,2,3 are the digits, skip 4. 1,2,3,10,11,12,13,20,21,22,23,30... 213four-Two One Three Base Four

2.3-Base Five

0,1,2,3,4 are the digits, skip 5. 1,2,3,4,10,11,12,13,14,20,21,22,23,24,30... 43five-Four Three Base Five Cant be a 35five, becomes 40 five.

2.3-Digits

0,1,2,3,4,5,6,7,8,9

2.1-Three Ways of Defining Sets

1.) A Verbal Description. 2.) A Listing of Elements Separated by Commas and Braces. 3.) Set-builder Notation.

3.1-Thinking Strategies for Learning the Addition Facts

1.) Community 2.) Adding Zero 3.) Counting on by 1 and 2 4.) Combinations to Ten 5.) Doubles 6.) Adding Ten 7.) Associativity 8.) Doubles Plus or Minus 1 and 2 Packet for examples on how to use them.

2.1-Two Inherent Rules Regarding Sets

1.) The same element is not allowed to be listed more than once within a set 2.) The order of the elements in a set is immaterial.

2.1-Examples of Sets

1.) The set of all seasons in Chicago. 2.) {Spring, Summer, Winter, Fall} 3.) {x|X is a season in Chicago}

2.3-Base Ten Place Values

10^5, 10^4, 10^3, 10^2, 10^1, 10^0 Hundred Thousands, Ten Thousands, Thousands, Hundreds, Tens, Ones

2.3-Base Four Place Values

4^3, 4^2, 4^1, 4^0 64, 16,4,1 Packet.

2.3-Base Five Place Values

5^4, 5^3, 5^2, 5^1, 5^0 625, 125, 25, 5, 1 Look at how to convert in packet.

2.1-Cartesian Product

A X(Cross) B, is the set of all ordered pairs (a,b), where a ∈ A and b ∈ B. EX.) {3,4,5} X {a,b} -Solution: (3,a), (3,b), (4,a), (4,b), (5,a), (5,b)

2.1-Set

A collection of objects.

1.1-Cryptarithm

A collection of words where the letters represent numbers. EX.)ABCD X 4 = DCBA

2.3- Place-Value

A device used to represent numbers written in place value is a chip abacus-a piece of paper, chips are written to represent unit values.

3.1-Binary Operations

Addition and Multiplication for numbers. Intersection, Union, and Set Difference for sets. When two numbers are combined to a produce a unique number (Only ONE number).

3.1-Zero

Additive identity or the identity for addition.

2.2-Tally Numeration System

Before grouping of 5. Single strokes for each object being counted. |||||||||| (10)

3.2-Number 1 in Multiplication

Called the multiplicative identity or the identity for multiplication.

3.2-Why is Dividing by Zero Undefined?

Case One: Missing Factor Idea Case Two: It is not Unique Look at Packet.

3.2-Thinking Strategies for Learning and Multiplication Facts

Community Multiplication by 0 Multiplication by 1 Multiplication by 2 Multiplication by 5 Multiplication by 9 Associativity and Distributivity

3.1-Measurement Model

Draw line segments over the top to represent the numbers given then circle the number that you end up at for the answer.

2.1-Equal Sets and Not Equal Sets

Equal Sets: Two sets A and B are equal (A=B), if and only if they have precisely the same elements. EX.) {1,2,3} = {2,3,1} = {1,2,2,1,3,3} = {3,1,2} Not Equal Sets: Two sets A and B are equal of every element in A is in B and vice versa. If set A does not equal set B, write A≠B.

3.3-Transitive Property of "Less Than" for whole numbers

For all whole numbers a,b, and c, if a < b and b < c, then a < c.

3.2-Multiplication Property of Zero

Fore every whole number, a, a*0=0*a=0.

2.2-Roman Numerals

Grouping and additivity. A subtractive system since it permits simplifications using combinations of basic Roman Numerals. A positional system since the position of a numeral can affect the value of the number being represented. I=1 V=5 X=10 L=50 C=100 D=500 M=1,000

2.1- 1-1 Correspondence

Happens between two sets A and B and is the pairing of the elements of A with the elements of B so that each element of A corresponds to exactly one element of B, and vice versa. Written as A~B, and say that A and B are equivalent or matching sets. Ex.) {1,2,3}~{Red,Blue,Yellow}~{A,B,C} -Look at the number of items in each set, there are 3 that's why they are equivalent, the items inside the set do not matter.

2.1-Proper Subset

If A is the subset of B, and B has an element that is NOT in A, write A ⊂ B, and say that A is a proper subset of B. EX.) {1,2,3} ⊂ {1,2,3,4,5,6}

3.3-Less Than and Multiplication Property

If a < b and c ≠ 0, then ac < bc.

3.3-Less Than and Addition Property

If a < b, then a + c < b + c.

3.2-Missing-Factor Approach Division

If a and b are any whole numbers with b ≠ 0, then a÷b=c if and only if a=bc for some whole number c.

3.2-The Division Algorithm

If a and b are any whole numbers with b≠ 0, then there exists unique whole numbers q and r such that a=bq+r, where 0≤r<b. Here b is called the divisor, q is called the quotient, and r is called the remainder.

3.2-Division Property of Zero

If a ≠ 0, the 0÷a=0. Division by 0 is UNDEFINED!!

2.1-Infinite Set

If it goes on without end. EX.) {1,2,3,4,5...}

2.1-Finite Set

If it is empty or can have its elements listed where they eventually will end. EX.) {1,3,5,7,9,11}

2.1-What does the Symbol "∈" Indicate?

Indicates that a object IS an element of a set.

2.1-What does the Symbol "∉" Indicate?

Indicates that an object is NOT an element of a set.

2.1-What does the Symbol "∅" Indicate?

Indicates that it is a set without elements. Also known as an empty set or a null set. It can also be denoted by {}. *Note: The empty set is NOT {∅}.

2.3-6 Place Values

Quadrillion, Trillion, Billion, Million, Thousands, Hundreds

3.2-Define Multiplication

Repeated addition.

3.2-Different Approaches to Multiplication

Repeated-Addition Rectangular Array Measurement Model Cartesian Product Tree Diagram

2.1-Subest

Set A is said to be a subset of set B (A⊆B), if and only if every element in A is also an element in B. Ex.) {1,2} ⊆ {1,2,3}

2.1-Set Difference (Relative Compliment)

Set difference of set B from set A (A-B), is the set of all elements in A that are NOT in B. Only write what's different in set A as the solution. EX.) {3,4,7,8} - {1,3,4,9} -Solution: {7,8}

3.2-Drawing for Division

Sharing and Measurement Repeated-Subtraction and Missing-Factor

2.2-Identificaion Numbers

Social Security, Drivers License number.

2.2-Whole Number Line

Starts at zero and increases to the right.

1.1-Polya's Four-Step Process

Step 1: Understand the Problem Step 2: Devise a Plan Step 3: Carry Out the Plan Step 4: Look Back

2.1-Compliment of Sets

The compliment of set A (A'), is the set of all elements in the universe (U), that are NOT in A. The opposite. EX.) U={0,1,2,3,4,5,6,7}, A={1,3,5,7} -Solution: A'={0,2,4,6}

2.1-Intersection of Sets

The intersection of two sets A and B (A∩B), is the set of all elements that are common to sets A and B. EX.) {0,1,2,3,4,5} ∩ {2,3,4,5,6} -Solution: {2,3,4,5}

2.2-Cardinal Number

The most common use of a whole number. Used to describe how many elements are in a finite set.

3.2-Identity Property Multiplication

The number 1 is the unique whole number such that every whole number a, a*1=a=1*a.

2.3-Grouping by Tens

The number of objects grouped together is called the base of the system (Base ten system). Models-Bundles of sticks, and Dienes Blocks (Individual cubes that are are called units, longs, flats, and blocks)

2.1-Elements

The objects in the set. Can also be called members of the sets.

2.2-Ordinal Number

The order of things.

3.2-Closure Property for Multiplication

The product of two whole numbers is a whole number.

3.2-What do you notice about the divisor?

The remainder MUST always be LESS THAN the divisor.

3.1-Closure Property for Whole-Number Addition

The sum of any two whole numbers is a whole number.

2.1-Union of Sets

The union of two sets A and B (A∪B), is the set that consists of all elements belonging to either A or to B or to both. EX.) {0,1,2,3,4,5} ∪ {2,3,4,5,6,7} -Solution: {0,1,2,3,4,5,6,7}

2.3-Additive and Multiplicative

The value of a numeral in this system is found by multiplying each place value by its corresponding digit and then adding all the resulting problems.

2.1-Disjoint Sets

Two sets that have no elements in common.

1.2-Inductive Reasoning

Used to draw conclusions or make predictions about a large collection of objects or numbers, based on a small representative sub-collection.

2.3-Hindu-Arabic System

What is used today. Attributes of the system: Digits-10 symbols that can be used in combination to represent all possible numbers, Grouping by tens-Decimal system, Place-vale (positional)- each of the various places in a numeral has its own value, Additive and multiplicative-expressing numbers in expanded form or expanded notation.

2.1-Are all proper subsets subsets?

Yes.

2.1_Are all equal sets equivalent sets?

Yes.

3.2-a÷b

a divided by b. a is called the dividened b is called the divisor c is called the quotient of missing factor

3.2-ab Multiplication

a times b. Called the product of a and b. Numbers a and b are called factors of ab. Product can also be written as a*b and a X b.

3.1-Set Model

a+b=n(AUB) a and b are called the addends or summands.

3.1- a-b

a-b is called the difference. a is called the minuend. b is called the subtrahend.

3.3-Zero as an Exponent

a^0=1 for all whole numbers a ≠ 0 (except zero).