MATH 156

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MATH 156 by Mind Map: MATH 156

1. Test One

1.1. Chapter 1-1

1.1.1. George Polya's Four Step Problem Solving Process 1.Understanding the problem 2. Devise a plan 3. Carry out the plan 4. Look back

1.1.2. Gauss's Approach n(n+1)/2

1.2. Chapter 1-2

1.2.1. Arithmetic Sequence an=a1+(n-1)d

1.2.2. Geometric Sequence an=a1xrn-1

1.2.3. Fibonacci Sequence 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 Hindu-Arabic 0,1,2,3,4,5,6,7,8,9 Talley IIII Roman Numeral I,V,X,L,C,D,M Base Five 1030five = 140

1.4. Chapter 2-2 & 2-3

1.4.1. Language of Set a collection of objects, elements, or members

1.4.2. One to One Corresspondence each element of the set is represented once

1.4.3. Equivalent Sets same amount of elements in each set

1.4.4. Cardinal Numbers the number of elements represented in a set

1.4.5. Sets a group or collection of objects

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

1.4.7. Set Intersection the common element

1.4.8. Set Union all the elements

1.4.9. Set Difference the different elements

1.4.10. Properties of Set Operations Associative Property distributive Property

1.4.11. Cartesian Products the amount of different combinations

2. Test Two

2.1. Chapter 3-1

2.1.1. Inverse Operations (-,+) (x,/)

2.1.2. Mixing Addend Model 3+5=8 8-3=5

2.1.3. Number Line Model ALWAYS START AT ZERO

2.1.4. Ordering Whole Numbers > greater than < less than

2.1.5. Closure Property 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 a+b=b+a

2.1.7. Associative Property (a+b)+c=a+(b+c) Same order

2.1.8. Identity Property Any number added to zero is that number

2.1.9. How to Master Basic Addition Facts Count On Doubles Making Ten Counting Back Fact Famlies

2.2. Chapter 3-2

2.2.1. Algorithms A systematic procedure used to accomplish an operation Step One: Use Manipulatives Step two: paper/pencil Lattice Algorithm for addition Regroup or Trade aka "carrying" Expanded algorithm Expanded w/ Regrouping Left to Right Algorithm

2.3. Chapter 3-3

2.3.1. Multiplication of Whole Numbers Repeated Addition Model 3x4=12 3+3+3+3=12 Cartesian Product Model use of tree diagrams Modeled as: axb ab a(b)

2.3.2. Properties of Whole Number Multiplication Closure Property of Multiplication of Whole Numbers if you mult. two whole numbers you get a new or "unique" whole number Commutative Property of Multiplication of Whole Numbers axb=bxa Associative Property of Multiplication of Whole Numbers aka-groupong a(bc)=(ab)c Identity Property of Multiplication of Whole Numbers any number times one is that number Zero Multiplication Property of Multiplication of Whole Numbers any number times zero is zero Distributive Property of Multiplication over addition and subtration a(b+c)=ab+ac

2.3.3. Division of Whole Numbers Set (Partition) Model Like Tree Model Missing Factor Model by using multiplication we can see the relationship to division Repeated Subtraction Model 12/4= 12-4-4-4=3

2.3.4. Division Algorithm Divide Does Multiply McDonald's Subtract Sell Check Cheeseburgers Circle Correctly

2.3.5. Order Operations Perenthesies Exponents Multplication Division Addition Subtraction

2.4. Chapter 3-4

2.4.1. Multiplication Algorithms single times double ones times ones tens times number add double times double times ones times tens add left to right time tenes times ones add lattice know your L's

2.4.2. Division Algorithms Divide Does Multiply McDonald's Subtract Sell Check Cheeseburgers Circle Correctly

2.5. Chapter 4-3

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

2.5.2. Relations a correspondence between two elements

3. Test Three

3.1. Chapter 5-1

3.1.1. Adding Intergers Chip Model represented by chips Charged Field Model positive and negative charges Number Line Model ALWAYS START AT ZERO! Pattern Model

3.1.2. Properties of Adding Intergers Closure Property a+b= unique number Communitive Property a+b=b+a Associative Property (a+b)+c=a+(b+c) Identity Elements a+0=a

3.1.3. Subtracting Intergers Chip Model Charged Field Model Number Line Model Pattern Model Missing Addends Adding Opposite Approach "keep, change, change"

3.1.4. Properties of Subtracting Intigers Cannot do Communtive nor Associative Properties

3.2. Chapter 5-2

3.2.1. Multiplying Intigers Pattern Model Chip Model grouping using chips Number Line Model

3.2.2. Properties of Interger Multiplication Closure Property axb=unique number Commutative Property ab=ba Associative Property (ab)c=a(bc) Identity Property 1xa+a Distributive Property a(b+c)=ab+ac Zero Multiplication Property ax0=0

3.2.3. Dividing Intergers Same signs = positive Different Signs = negative

3.3. Chapter 5-3

3.3.1. Divisibility 2 Even Number 3 Sum of ALL digits divisible by 3 4 Last TWO digits divisible by 4 5 End in 5 or 0 6 If 2 AND 2 are divible 8 Last THREE digits are divisible by 8 9 Sum of ALL digits divisible by 9 10 Ends in a 0 11 add even powers, add odd powers, then subtract

3.4. Chapter 5-4

3.4.1. Prime Factorization Prime Factorization (Tree Model) Ladder Model Number of Divisors

3.4.2. Prime 1 times itself 2 3 5 7 11 13 17 19 23

3.4.3. Composite more than two factors (1 and itself)

3.5. Chapter 5-5

3.5.1. GCD Intersection of Sets Model list out divisors Prime Factorization Model Keep lowest exponent and all common primes Ladder Method Birthday Cake

3.5.2. LCM Number Line Method Intersection of Sets Method list out multiples Prime Factorization Method Keep HIGHEST exponent All Common primes and ALL the Left overs Ladder Method

4. Test Four

4.1. Chapter 6-1

4.1.1. Rational Numbers Are Fractions Portions of a whole

4.1.2. Ratio

4.1.3. Probability

4.1.4. Proper Fraction numerator is less than denominator

4.1.5. Improper Fraction numerator is greater than denominator

4.1.6. Equivalent / Equal Fractions 25/100 = 1/4

4.1.7. Simplest form 2/6 = 1/2

4.2. Chapter 6-2

4.2.1. Adding and Subtracting Rational Numbers Find LCM multiply the numerator then add or subtract left to right

4.3. Chapter 6-3

4.3.1. Multiplying Ration Numbers Multiply numerators Multiply denominator simplify

4.3.2. Dividing Rational Numbers Keep Dot Flip Keep the 1st fraction the same Multiply by the reciprocal

4.4. Chapter 7-1

4.4.1. Decimals Tenths Hundereths Thousandths Ten-Thousandths Hundred-Thousandths

4.5. Chapter 7-2

4.5.1. Adding and Subtracting Decimals line up decimals add or sub like normal

4.5.2. Multiplying Decimals line up digits multiply like normal count up all numbers to the right of a decimal place decimal the number of places to the left

4.5.3. Dividing Decimals set up division problem move divisor decimal to the far right Move the dividends decimal the same number of places carry decimal up divide as normal

4.5.4. Scientific Notation short version 0.00056 = 5.6x10 (exponent) -4

4.5.5. Expanded Notation long version 5.6x10 (exponent) -4 = 0.00056

4.6. Chapter 7-3

4.6.1. Nonterminating decimals "repeating " how to write as fration 0.323232...= 0.32 1n=0.323232... multiply each side by 100 then subract 32/99