Math156

George Poylas 4 steps

Step 1. Understand the problem

Step 2. Devising a plan

Step 3. Carrying out the plan

Step 4. Looking back

Sum of sequential numbers

n(n+1)/2

Arithmetic Sequence

Addition of subtraction of a fixed number

Fibonacci Sequence

Sequence when the sum of the first two numbers becomes the third number, and so on. ex 1,1,2,3,5,8,13,21...

Geometric Sequence

Each successive term of a geometic sequence is obtained from its predecessor by multiplying by a fixed number, the ratio. (Ratio can never =0) ex2,4,8,16,32...

Hindu-Arabic

All numbers come from the 10 digits (0,1,2,3,4,5,6,7,8,9)

Tally Numberation System

This method uses single strokes or tally marks to represent each item being counted. ( l,ll,lll)

Roman Numeration System

l=1, V=5, X=10, L=50, C=100, D=500, M=1000

Base Five

The following website gives a detailed description of base five

One to One Correspondence

If the elements of sets P and S can be paired so that each element of P there is exactly one element of S, then the two sets P and S are considered one to one correspondence

Equivalent Sets

Two sets A and B are equivalent, if and only if there exists a one-to-one correspondence between the sets.

Cardinal Numbers

The cardinal number of a set denotes the number of elements in a set. ex. (a,b) (r,s),@,#) are equivalent to each other because they each have two elements in their set.

Universal Set

Subset

B is a subset of A, if every element of B is an element of A.

Set Intersection

The intersection of two sets, A and B, is the set of all elements common to both A and B.

Set Union

The union of two sets A and B, is the set of all elements in A or B.

Set Difference

The colplement of A relative to B, is the set of all elements in B that are not in A.

Properties of Set Opperations

1. Associative Property 2. Distributive Property

Cartesian Pruducts

For any sets A and B, the Cartesian produce of A and B, is the set of all ordered pairs such that the first componet of each pair is an element of A and the second componet of each is an element of B.

Vocabulary

Addends: Numbers added together

Sum: Total number after adding two or more numbers

Dividend: A number divided by another

Divisor: A number by which another number is to be divided

Quotient: The answer after dividing one number into another

Difference: The result of subtracting one number from another.

Product: The result of multiplying two or more numbers

Multiplication: An operation that gives the total number when you join two or more numbers

Subtraction: The operation of taking away/subtracting two or more numbers.

Addition: The opperation of adding two or more numbers.

Division: The operation of dividing multiple numbers into each other.

Function: A relation in which every input value is paired with exactly one output value.

Relations: A set of input and output values. Usually represented in ordered pairs.

Functions

Ordered pairs

The rule

Make table

Make a graph

Arrow diagram

Teaching Addition and Subtraction Facts

Numbers Blocks

Counting On

Number Line

Fact Family

Counting Backwards

Counting by 10s

Countin by Doubles

Order of Operation

Parenthesis

Exponents

Division

Multiplication

Addition

Subtraction

Order of Long Division

Divide

Multiply

Subtract

Check

Bring Down

Correct/Circle

Properties

Distributive: a(b+c) =ab+ac

Associative: a+(b+c)=(a+b)+c

Commutaive: a+b=b+a

Closure: The set of whole numbers is closed under multiplication.

Identity: Any number multiplied by 1 is the same number 2x1=2

Zero multiplication: Any number multiplied by 0 is 0 2x0=0

Integer Addition

1. Chip model

2. Number line

3. Absolute value

Closure property

Communitive property

Associative property

Identity property

Integer Subtraction

1. Chip model

2. Number line

3. Pattern model for subtraction

Adding opposites: "Keep change change"

Missing addends

Integer Multiplication

Multiply a positive and a positive the answer is positive

Multiply a positive and a negative the answer is negative

Integer Division

Divide a positive and a positive the answer is positive

Divide and positive and a negative the answer is negative

Divisibility

A number is divisible by 2 if the number ends with an even number. ex. 202

A number is divisible by 3 if the sum of the digits is divisible by 3. ex. 123

A number is divisible by 4 if the last two digits are divisible by 4. ex 424

A number is divisible by 5 if the number ends with 0 or 5. ex. 205

A number is divisible by 6 if the number is divisible by both 2 and 3. ex. 126

A number is divisible by 8 if the last 3 digits are divisible by 8. ex. 1,888

A number is divisible by 9 if the sum of digits is divisible by 9. ex. 126

A number is divisible by 10 if the number ends with 0. ex. 200

A number is divisible by 11 if the sum of the digits in the places that are even powers of 10 minus the sum of the digits in the places that odd powers of 10 is divisible by 11. ex. 57,729,364,583

Prime numbers

Prime numbers are numbers that only have 2 factos or positive divisors. ex. 2,3,5,7,11,13,17,19

Composite numbers

Composite numbers are numbers that have more than 2 factors or positive divisors. ex. 4,6,8,9,10,12,14,15,16,18,20

Greatest Common Divisor

The greatest common divisor of two natural numbers a and b is the greatest natural number that divides both a and b.

1. The intersection of Sets Method

2.Prime factorization method

3. Ladder method

Least Common Multiple

The least common multiple of a and b is the least natural number that is simultaneously a multiple of a and multiple of b.

1. Number line method

2. Intersection of Sets Method

3. Prime factorization method

4. Ladder method

Scientific Notation: Used when you have either a very large or small number. EX large: 93,000,000=9.3x10power of 7 EX small: 0.000078=7.8x10-power of 5.