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Principles of Calculus

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Week 1-14

Fractions

Definition:

When a whole is divided into equal pieces, if fewer equal pieces are needed to make up the whole, then each piece must be larger. When two positive fractions have the same numerator, they represent the same number of parts, but in the fraction with the smaller denominator, the parts are larger.

Rule:

K, C, F or K, C, I = Keep, Change, Flip (Inverse)

*Follow same produce as Multiplication of Fractions with unlike denominators

Multiply

Simplify first = top to bottom, and across

Multiply straight across then change to a Mixed number

Subtraction

You have to change the denominators to be the same number by finding out what is missing in each fraction (denominator).

Simplify = to to bottom

Then you subtract straight across.

Change to a mixed number

Addition

Example:

2 + 5 = 2 ^{x 6} + 5 ^{x 2} = 12 + 10 = 22 ^{(divide by 2)} = 11

3 + 9 = 3 ^{x 6} + 9 ^{x 2} = 18 + 18 = 18 ^{(divide by 2)} = 9

Answer is:

11 or simplify 11 = 1 2/9

9 or simplify 9

Unlike Denominators

You have to change the denominators to be the same number by finding out what is missing in each fraction (denominator).

Simplify = to to bottom

Then you add straight across.

Change to a mixed number

Simplify

Example:

12 ^{(divided by 4)} = 3

20 ^{(divided by 4)} = 5

OR

12 = 2 x 2 x 3 = 3

20 = 2 x 2 x 5 = 5

OR

y ² = y x y x 1 = 1

y ³ = y x y x y = y

Equivilency

Example (Set Model):

2 = 4

3 = 6

OR

2 = 6

3 = 9

OR

2 = * * *

3

OR

2 = * * *

3

OR

2 = + + +

3

OR

2 = + + +

3

Greater Than or Less Than

Notes:

• The aligator/pacman eats the the bigger number.
• The lower number is inside pacman's face or the aligator's body.

• If the denominators are different numbers, then the fraction with the bigger denominator is greater (bigger size of pieces.)
• If the denominators are the same numbers, then the fraction with the bigger numerator is greater (bigger amount of pieces.

Example:

4 is greater than 4

7 is greater than 11

5 is greater than 3

12 is greater than 12

4 ? 10

5 ? 11

4 ^x11 is less than 10 ^x11 = 44 is less than 55

5 ^x11 is less than 11 ^x11 = 55 is less than 55

OR

4 ^{= 44} is less than 10 ^{= 50}

5 ^{= 50} is less than 11 ^{= 44}

Linear Model

Example:

5 is greater then 12

9 is greater then 25

l---------l----------l----------l----------l

0----- ^12/25------ ^1/2------- ^5/9-------- 1

Area Model

Example:

4 is less then 10

5 is less then 11

l l l l l l is less then l l l l l l l l l l l l

l l l l l l is less then l l l l l l l l l l l l

l l l l l l is less then l l l l l l l l l l l l





Scientific Notation

• position of a decimal, bigger or equal to 1, multiply by the power of 10
• if the exponent is negative the answer will have a decimal, a small number
• if the exponent is positive the answer will not have a decimal, a big number

Example:

134000 = 1.34 x 10⁵

0.00000761 = 7.61 x 10^-6

Properties

Distributive for Multiplication

A = l x w

Examples:

25(15) = 375

10 + 15

20 l 200 l 100 l

+

5 l 50 l 25 l

300 + 75 = 375

4(-3 + 5) + 8

-3 + 5

4 l -12 l 20 l

4(x + 5) = 4x + 20

x + 5

4 l 4x l 20 l

3(x + 7) = 3x + 21

x + 7

3 l 3x l 21 l

10(2x² - 4x) = 20x² - 40x

2x² - 4x

10 l 20x² l -40x l

4(x² + 3x -1) = 4x² + 12x - 4

x² + 3x - 1

4 l 4x² l 12x l -4 l

(x + 2)(x + 3) = x² + 5x + 6

x + 3

x l x² l 3x l

+

2 l 2x l 6 l

(x + 2)(x + 5) = x² + 7x + 10

x + 5

x l x² l 5x l

+

2 l 2x l 10 l

Difference of Squares

Example:

(x + 2)(x - 2) = x² - 4

x - 2

x l x² l - 2x l

+

2 l 2x l - 4 l

x² + ox - 4 = x² - 4

(3x - 7)(3x + 7) = 9x² - 49

3x - 7

3x l 9x² l - 21x l

+

7 l 21x l - 49 l

9x² + ox - 49 = 9x² - 49

Factoring

Example:

6x + 15 = 3(2x + 5)

3 l 6x l +15 l

2x + 5

x² - 3x = x(x - 3)

x l x² l - 3x l

x - 3

3x²y - 9xy + 6y = 3y(x² - 3x + 2)

3y l 3x² y l -9xy l + 6y l

x² - 3x + 2

Communitive For Addition

• order changes

Example:

5 + 7 + 2 or 5 + 2 + 7

Associative for Addition

• identifies what numbers are being associated with - order does not change or the answer, but the numbers grouped does change.

Example:

(4 + 2 + 5) or (4 + 2) + 5 or 4 + (2 + 5)

-5 + (5 + 3) or (-5 + 5) + 3

• same sign = positive number
• different sign = negative number

JUST SOLVE

Rules

• same sign = positive number
• different sign = negative number

• add a zero bank if the coefficent is negative

• tiles (show) if numbers are less than 10
• just solve if numbers are greater than 10

Negative Coefficent

Example:

-3(2) = -6

++ ++ ++ ++++

------ ----

-3(-2) = 6

++++++ ++++

-- -- -- ----

-1(-3) = 3

+++ +++++++

--- -------

-2(4) = -8

++++ ++++ ++++

-------- ----

Positive Coefficent

Example:

6(2) = 12

l l l l l l l l l l l l = 12

6(1) = 6

l l l l l l = 6

3(-2) = -6

(- -)(- -)(- -) = -6

5(-4) = -20

(- - - -)(- - - -)(- - - -)(- - - -)(- - - -) = -20

Numberlines

Number Lines: a line with numbers placed in their correct positions

• Useful fro addition and subtraction
• Useful for showing relations of numbers

Absolute Value

Absolute Value: how far away a number is from zero

Example:

6 is 6 units from zero

Absolute value of 6 = 6

-6 is 6 units from zero

Absolute value of -6 = 6 or l-6l

Diagram

Draw Diagram: use when you have any numbers bigger than ten.

Examples:

-15 + 436 = +421

- sub. ++

-15 + 436

- 15

+421

Examples:

-47 + (26) = -73

-- add. -

-47 + (-26) 26

+ 47

-73

Tiles

Draw Tiles: use when you have any numbers less than ten.

Examples: Addition:

2 add 4 = -2

+ +

----

-5 add -2 = -7

----- --

Examples: Subtraction:

4 take away 3 = 1

+ + + +

-5 take away -2 = -3

+ + + + +

-5 take away 1 = -4

+ + + + +

Division

6 divided by 6 = 1

6 = Total Number

6 = Number of Groups

1 = Number of Units Inside 1 Group

*If divisor gets smaller = answer gets bigger (Inverse Relationship)

Divisibility Rules

• Divisible by 2 = look at the 1's
• Divisible by 4 = look at 10's and 1's (last 2 numbers)
• Divisible by 5 = look if it ends in 5 or 0
• Divisible by 8 = look at the last 3 numbers
• Divisible by 3 = add numbers and if the sum is divisible by 3
• Divisible by 10 = ends in 0
• Divisible by 6 = divisible by 2 and 3
• Divisible by 9 = add numbers and if the sum is divisible by 9

Upwards

382 divided by 3 = 127 1/3

-3 -6 -21 = 1

3 8 2 = 127 1/3

3

Repeated Subtraction

382 divided by 3 = 127 1/3

127 1/3

3 l 382

-30

352

-30

322

-300

22

-15

7

-6

1

Traditional

382 divided by 3 = 127 r1

127 r1

3 l 382

-3

08

-6

22

-21

10

-9

10

-9

Multiplication

6(6) = 36

6 = Number of Groups

(6) = Number of Units Inside 1 Group

36 = Total Number of Whole Groups

*ORDER MATTERS*

• Read left to right

Lattice

Expanded

Area Model for Multiplication

1.) A Rectangle with a length of 10 + 1 and a width of 4 - A = 44

Distributive Property:
(4)(10 + 1)
40 + 4 =44

2.) A Rectangle with a length of 10 + 3 and a width of 10 + 2 - A = 156

10 + 3 = 13
10 + 2 = 12
A = 12 x 13 = 156

Alternative Algorithims: Subtraction

Traditonal

47 - 12 = 35

47
- 12
35

Equal Add Ons

47 - 12 = 35

47 + 8 = 55
- 12 + 8 = - (20)
35

Alternative Algorithims:Additon Estimating

31 + 24 + 15 + 42 + 39

80
40
+ 30
150

Front End

31 + 24 + 15 + 42 + 39

30
20
10
40
+ 30
130

Alternative Algorithims: Addition

Compatible Numbers

31 + 24 + 15 + 42 + 39 = 151

2
31 1 + 9 = 10
24 4 + 5 + 2 = 11
15
42 3 + 2 + 1 + 4 = 10
+ 39 3 + 2 = 5
151

Scratch Method

31 + 24 + 15 + 42 + 39 = 151

1 2
31
24
15
2 42
+ 39 1
151

5.) Tradtional

46 + 28 = 74

1
46
+ 28
74

4.) Lattice

3.) Left to Right

46 + 28 = 74

46
+ 28
74

2.) Partial Sum

46 + 28 = 74

46 + 20 = 66
66 + 8 = 74

1.) Expanded

46 + 28 = 74

40 + 6
+ 20 + 8
60 + 14
60
+ 10 + 4
70 + 4 =74

Other Bases

Never have any number bigger than the base number

Other bases to base ten

1.) Convert 23 four to ten = 11
23 four
2 long 3 unit
2(4) 3
8 + 3 = 11

2.) Convert 42 eight to ten = 34
42 eight
4 long 2 unit
4(8) 2
32 + 2 = 34

3.) Covert 123 five to ten = 38
123 five
1 flat 2 long 3 unit
25 + 10 + 3 = 38

Base ten to other bases

Examples:
1.) 15 to base five = 3 long, 0 units = 30 five
2.) 15 to base three = 1 flat, 2 longs, 0 units = 120 three
3.) 17 to base six = 2 long, 5 units = 25 six
4.) 11 to base four = 2 long, 3 units = 23 four
5.) 14 to base three = 1 flat, 1 long, 2 units = 112 three
6.)356 to base four = 11210 four
4 l356
4 l89 r 0
4 l22 r 1
4 l5 r 2
1
7.) 14 to base five = 24 five
5 l14
2 r 4
8.) 14 ro base three = 112 three
- 3 l14
3 l4 r 2
1 r 1

Vocab

Order of Operations

G = groups (identified by an additon or subtraction symbol)

E = exponents

D M = (left to right) divide/multiply

S A = (left to right) subtract/addition

DO NOT use or teach PEMDAS = confusing!!!!!!!

Denominaror

Denominator: tells us the size of each whole or piece.

Example:

4

5 = the denominator

Numerator

Numerator: tells us how many pieces we have of a whole.

Example:

4 = the numerator

5

Standard Form

• the standard way to write a number (the normal way)

Example:

2 , 345 , 1,112 , 4 , 300 , 32 , 250

Irrational Numbers

• a "number" that cannot be written as a ratio (fraction), never stops and never repeats

Rational Numbers

• can always be written as a ration (fraction), a number that stops or repeats

Example:

4.12 , x = + 5 , x = + 7 , x² = 49 , x² = 25

Trinomial

• three "numbers" that cannot be simplified

Binomial

• two "numbers" that cannot be simplified

Volumw

Volume = l x w x d

• any three "numbers" together

Area

A = l x w (any two "numbers" together)

- generic rectangle

- base 10 recrangle

Zero Bank

Zero Bank: adding any equal number of pairs of positive and negative numbers.

Zero Pairs

Zero Pairs: add a positive and a negative (cancels each other out).

Intergers

Intergers: whole counting numbers

Venn Diagram

Venn Diagram:  sets are represented by shapes; usually circles or ovals. The elements of a set are labelled within the circle. They are especially useful for showing relationships between sets.

Double Bubble

GCF:

(2 x 2 x 2) x 3 x (5 x 5) and (2 x 2) x 5 x (7 x 7) = 20

Example:

#1 2 2 #2

2 2

2 5

3 7

5 7

5

2 x 2 x 5 = 20

LCM:

(2 x 2 x 2) x 3 x (5 x 5) and (2 x 2) x 5 x (7 x 7) =

(2 x 2 x 2) x 3 x (5 x 5) x (7 x 7)

Example:

#1 2 2 #2

2 2

2 5

3 7

5 7

5

(2 x 2 x 2) x 3 x (5 x 5) x (7 x 7)

Upside Down Division

Upside Down Division: is one of the techniques used in Prime Factorization method to factor numbers.



LCM

LCM: Least Common Multiple (bigger numbers)

LCM:

28 and 60 = 420

Example:

28 60

(4) (7) (5) (12)

(2) (2) (6) (2)

(3) (2)

(2)(2)(7)(5)(3)(2)(2)

2 x 2 x 7 x 5 x 3 = 420

GCF or GCD

GCF = Greatest Common Factor (small numbers)

OR

GCD = Greatest Common Divisor (small numbers)

Factor Tree

GCF or GCD:

18 and 30 = 6

Example:

18 30

(9) (2) (3) (10)

(3) (3) (2) (5)

2 x 3 x 3 2 x 3 x 5

2 x 3 = 6 2 x 3 = 6

List

GCF or GCD:

18 and 30 = 6

Example:

Factors of 18 = 1, 2, 3, 6, 9, 18

Factors of 30 = 1, 2, 3, 5, 6, 10, 15, 20

Composite Numbers

Composite Numbers: Divisible by more than 2 factors

Example:

385 = Composite

385 divided by 5 = YES

285 divided by 385 = YES

384 divided by 1 = YES

Prime Numbers

Prime Numbers: Divisible by 1 and itself

• Prime Numbers = 2, 3, 5, 7, 11, 13, 19 ...

Example:

257 = Prime

257 divided by 7 = NO

257 divided by 11 = NO

257 divided by 13 = NO

Base Ten

Unit = 1
Long = 10
Flat = 100
Cube = 1000
Repeats on and on...

UnDevCarLo

1.) Understand the problem
2.) Develop a plan
- your way to solve with pictures, guess & check, equations
3.) Carry out plan
- do the work & solve
4.) Look back- check your work

CCR