# MATH 156

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MATH 156

## 1. Test One

### 1.1. Chapter 1-1

1.1.1. George Polya's Four Step Problem Solving Process

1.1.1.1. 1.Understanding the problem

1.1.1.2. 2. Devise a plan

1.1.1.3. 3. Carry out the plan

1.1.1.4. 4. Look back

1.1.2. Gauss's Approach

1.1.2.1. n(n+1)/2

### 1.2. Chapter 1-2

1.2.1. Arithmetic Sequence

1.2.1.1. an=a1+(n-1)d

1.2.2. Geometric Sequence

1.2.2.1. an=a1xrn-1

1.2.3. Fibonacci Sequence

1.2.3.1. The sum of the first two numbers become the third EX:1,1,3,3,5,8,13,21

### 1.3. Chapter 2-1

1.3.1. Numeration Systems

1.3.1.1. Hindu-Arabic

1.3.1.1.1. 0,1,2,3,4,5,6,7,8,9

1.3.1.2. Talley

1.3.1.2.1. IIII

1.3.1.3. Roman Numeral

1.3.1.3.1. I,V,X,L,C,D,M

1.3.1.4. Base Five

1.3.1.4.1. 1030five = 140

### 1.4. Chapter 2-2 & 2-3

1.4.1. Language of Set

1.4.1.1. a collection of objects, elements, or members

1.4.2. One to One Corresspondence

1.4.2.1. each element of the set is represented once

1.4.3. Equivalent Sets

1.4.3.1. same amount of elements in each set

1.4.4. Cardinal Numbers

1.4.4.1. the number of elements represented in a set

1.4.5. Sets

1.4.5.1. a group or collection of objects

1.4.6. Subsets

1.4.6.1. has something in common with the main set but at least one element is different

1.4.7. Set Intersection

1.4.7.1. the common element

1.4.8. Set Union

1.4.8.1. all the elements

1.4.9. Set Difference

1.4.9.1. the different elements

1.4.10. Properties of Set Operations

1.4.10.1. Associative Property

1.4.10.2. distributive Property

1.4.11. Cartesian Products

1.4.11.1. the amount of different combinations

## 2. Test Two

### 2.1. Chapter 3-1

2.1.1. Inverse Operations

2.1.1.1. (-,+)

2.1.1.2. (x,/)

2.1.2. Mixing Addend Model

2.1.2.1. 3+5=8

2.1.2.2. 8-3=5

2.1.3. Number Line Model

2.1.3.1. ALWAYS START AT ZERO

2.1.4. Ordering Whole Numbers

2.1.4.1. > greater than

2.1.4.2. < less than

2.1.5. Closure Property

2.1.5.1. If a and b are whole numbers the a+b is a whole number also so long the answer is in the set

2.1.6. Commutative Property

2.1.6.1. a+b=b+a

2.1.7. Associative Property

2.1.7.1. (a+b)+c=a+(b+c) Same order

2.1.8. Identity Property

2.1.8.1. Any number added to zero is that number

2.1.9. How to Master Basic Addition Facts

2.1.9.1. Count On

2.1.9.2. Doubles

2.1.9.3. Making Ten

2.1.9.4. Counting Back

2.1.9.5. Fact Famlies

### 2.2. Chapter 3-2

2.2.1. Algorithms

2.2.1.1. A systematic procedure used to accomplish an operation

2.2.1.2. Step One: Use Manipulatives

2.2.1.3. Step two: paper/pencil

2.2.1.3.1. Lattice Algorithm for addition

2.2.1.3.2. Regroup or Trade aka "carrying"

2.2.1.3.3. Expanded algorithm

2.2.1.3.4. Expanded w/ Regrouping

2.2.1.3.5. Left to Right Algorithm

### 2.3. Chapter 3-3

2.3.1. Multiplication of Whole Numbers

2.3.1.1. Repeated Addition Model

2.3.1.1.1. 3x4=12

2.3.1.1.2. 3+3+3+3=12

2.3.1.2. Cartesian Product Model

2.3.1.2.1. use of tree diagrams

2.3.1.3. Modeled as:

2.3.1.3.1. axb

2.3.1.3.2. ab

2.3.1.3.3. a(b)

2.3.2. Properties of Whole Number Multiplication

2.3.2.1. Closure Property of Multiplication of Whole Numbers

2.3.2.1.1. if you mult. two whole numbers you get a new or "unique" whole number

2.3.2.2. Commutative Property of Multiplication of Whole Numbers

2.3.2.2.1. axb=bxa

2.3.2.3. Associative Property of Multiplication of Whole Numbers

2.3.2.3.1. aka-groupong

2.3.2.3.2. a(bc)=(ab)c

2.3.2.4. Identity Property of Multiplication of Whole Numbers

2.3.2.4.1. any number times one is that number

2.3.2.5. Zero Multiplication Property of Multiplication of Whole Numbers

2.3.2.5.1. any number times zero is zero

2.3.2.6. Distributive Property of Multiplication over addition and subtration

2.3.2.6.1. a(b+c)=ab+ac

2.3.3. Division of Whole Numbers

2.3.3.1. Set (Partition) Model

2.3.3.1.1. Like Tree Model

2.3.3.2. Missing Factor Model

2.3.3.2.1. by using multiplication we can see the relationship to division

2.3.3.3. Repeated Subtraction Model

2.3.3.3.1. 12/4= 12-4-4-4=3

2.3.4. Division Algorithm

2.3.4.1. Divide

2.3.4.1.1. Does

2.3.4.2. Multiply

2.3.4.2.1. McDonald's

2.3.4.3. Subtract

2.3.4.3.1. Sell

2.3.4.4. Check

2.3.4.4.1. Cheeseburgers

2.3.4.5. Circle

2.3.4.5.1. Correctly

2.3.5. Order Operations

2.3.5.1. Perenthesies

2.3.5.2. Exponents

2.3.5.3. Multplication

2.3.5.4. Division

2.3.5.5. Addition

2.3.5.6. Subtraction

### 2.4. Chapter 3-4

2.4.1. Multiplication Algorithms

2.4.1.1. single times double

2.4.1.1.1. ones times ones

2.4.1.1.2. tens times number

2.4.1.1.3. add

2.4.1.2. double times double

2.4.1.2.1. times ones

2.4.1.2.2. times tens

2.4.1.2.3. add

2.4.1.3. left to right

2.4.1.3.1. time tenes

2.4.1.3.2. times ones

2.4.1.3.3. add

2.4.1.4. lattice

2.4.1.4.1. know your L's

2.4.2. Division Algorithms

2.4.2.1. Divide

2.4.2.1.1. Does

2.4.2.2. Multiply

2.4.2.2.1. McDonald's

2.4.2.3. Subtract

2.4.2.3.1. Sell

2.4.2.4. Check

2.4.2.4.1. Cheeseburgers

2.4.2.5. Circle

2.4.2.5.1. Correctly

### 2.5. Chapter 4-3

2.5.1. Function

2.5.1.1. a relationship between that assigns exactly one output for each input

2.5.2. Relations

2.5.2.1. a correspondence between two elements

## 3. Test Three

### 3.1. Chapter 5-1

3.1.1. Adding Intergers

3.1.1.1. Chip Model

3.1.1.1.1. represented by chips

3.1.1.2. Charged Field Model

3.1.1.2.1. positive and negative charges

3.1.1.3. Number Line Model

3.1.1.3.1. ALWAYS START AT ZERO!

3.1.1.4. Pattern Model

3.1.2. Properties of Adding Intergers

3.1.2.1. Closure Property

3.1.2.1.1. a+b= unique number

3.1.2.2. Communitive Property

3.1.2.2.1. a+b=b+a

3.1.2.3. Associative Property

3.1.2.3.1. (a+b)+c=a+(b+c)

3.1.2.4. Identity Elements

3.1.2.4.1. a+0=a

3.1.3. Subtracting Intergers

3.1.3.1. Chip Model

3.1.3.2. Charged Field Model

3.1.3.3. Number Line Model

3.1.3.4. Pattern Model

3.1.3.5. Missing Addends

3.1.3.6. Adding Opposite Approach

3.1.3.6.1. "keep, change, change"

3.1.4. Properties of Subtracting Intigers

3.1.4.1. Cannot do Communtive nor Associative Properties

### 3.2. Chapter 5-2

3.2.1. Multiplying Intigers

3.2.1.1. Pattern Model

3.2.1.2. Chip Model

3.2.1.2.1. grouping using chips

3.2.1.3. Number Line Model

3.2.2. Properties of Interger Multiplication

3.2.2.1. Closure Property

3.2.2.1.1. axb=unique number

3.2.2.2. Commutative Property

3.2.2.2.1. ab=ba

3.2.2.3. Associative Property

3.2.2.3.1. (ab)c=a(bc)

3.2.2.4. Identity Property

3.2.2.4.1. 1xa+a

3.2.2.5. Distributive Property

3.2.2.5.1. a(b+c)=ab+ac

3.2.2.6. Zero Multiplication Property

3.2.2.6.1. ax0=0

3.2.3. Dividing Intergers

3.2.3.1. Same signs = positive

3.2.3.2. Different Signs = negative

### 3.3. Chapter 5-3

3.3.1. Divisibility

3.3.1.1. 2

3.3.1.1.1. Even Number

3.3.1.2. 3

3.3.1.2.1. Sum of ALL digits divisible by 3

3.3.1.3. 4

3.3.1.3.1. Last TWO digits divisible by 4

3.3.1.4. 5

3.3.1.4.1. End in 5 or 0

3.3.1.5. 6

3.3.1.5.1. If 2 AND 2 are divible

3.3.1.6. 8

3.3.1.6.1. Last THREE digits are divisible by 8

3.3.1.7. 9

3.3.1.7.1. Sum of ALL digits divisible by 9

3.3.1.8. 10

3.3.1.8.1. Ends in a 0

3.3.1.9. 11

3.3.1.9.1. add even powers, add odd powers, then subtract

### 3.4. Chapter 5-4

3.4.1. Prime Factorization

3.4.1.1. Prime Factorization (Tree Model)

3.4.1.2. Ladder Model

3.4.1.3. Number of Divisors

3.4.2. Prime

3.4.2.1. 1 times itself

3.4.2.1.1. 2

3.4.2.1.2. 3

3.4.2.1.3. 5

3.4.2.1.4. 7

3.4.2.1.5. 11

3.4.2.1.6. 13

3.4.2.1.7. 17

3.4.2.1.8. 19

3.4.2.1.9. 23

3.4.3. Composite

3.4.3.1. more than two factors (1 and itself)

### 3.5. Chapter 5-5

3.5.1. GCD

3.5.1.1. Intersection of Sets Model

3.5.1.1.1. list out divisors

3.5.1.2. Prime Factorization Model

3.5.1.2.1. Keep lowest exponent and all common primes

3.5.1.3. Ladder Method

3.5.1.3.1. Birthday Cake

3.5.2. LCM

3.5.2.1. Number Line Method

3.5.2.2. Intersection of Sets Method

3.5.2.2.1. list out multiples

3.5.2.3. Prime Factorization Method

3.5.2.3.1. Keep HIGHEST exponent All Common primes and ALL the Left overs

3.5.2.4. Ladder Method

## 4. Test Four

### 4.1. Chapter 6-1

4.1.1. Rational Numbers

4.1.1.1. Are Fractions

4.1.1.2. Portions of a whole

4.1.2. Ratio

4.1.3. Probability

4.1.4. Proper Fraction

4.1.4.1. numerator is less than denominator

4.1.5. Improper Fraction

4.1.5.1. numerator is greater than denominator

4.1.6. Equivalent / Equal Fractions

4.1.6.1. 25/100 = 1/4

4.1.7. Simplest form

4.1.7.1. 2/6 = 1/2

### 4.2. Chapter 6-2

4.2.1. Adding and Subtracting Rational Numbers

4.2.1.1. Find LCM

4.2.1.2. multiply the numerator

4.2.1.3. then add or subtract left to right

### 4.3. Chapter 6-3

4.3.1. Multiplying Ration Numbers

4.3.1.1. Multiply numerators

4.3.1.2. Multiply denominator

4.3.1.3. simplify

4.3.2. Dividing Rational Numbers

4.3.2.1. Keep Dot Flip

4.3.2.1.1. Keep the 1st fraction the same

4.3.2.1.2. Multiply

4.3.2.1.3. by the reciprocal

### 4.4. Chapter 7-1

4.4.1. Decimals

4.4.1.1. Tenths

4.4.1.2. Hundereths

4.4.1.3. Thousandths

4.4.1.4. Ten-Thousandths

4.4.1.5. Hundred-Thousandths

### 4.5. Chapter 7-2

4.5.1. Adding and Subtracting Decimals

4.5.1.1. line up decimals

4.5.1.2. add or sub like normal

4.5.2. Multiplying Decimals

4.5.2.1. line up digits

4.5.2.2. multiply like normal

4.5.2.3. count up all numbers to the right of a decimal

4.5.2.4. place decimal the number of places to the left

4.5.3. Dividing Decimals

4.5.3.1. set up division problem

4.5.3.2. move divisor decimal to the far right

4.5.3.3. Move the dividends decimal the same number of places

4.5.3.4. carry decimal up

4.5.3.5. divide as normal

4.5.4. Scientific Notation

4.5.4.1. short version

4.5.4.1.1. 0.00056 = 5.6x10 (exponent) -4

4.5.5. Expanded Notation

4.5.5.1. long version

4.5.5.1.1. 5.6x10 (exponent) -4 = 0.00056

### 4.6. Chapter 7-3

4.6.1. Nonterminating decimals

4.6.1.1. "repeating "

4.6.1.2. how to write as fration

4.6.1.2.1. 0.323232...= 0.32

4.6.1.2.2. 1n=0.323232...

4.6.1.2.3. multiply each side by 100

4.6.1.2.4. then subract

4.6.1.2.5. 32/99