Week 1-14

1.) Understand the problem2.) Develop a plan- your way to solve with pictures, guess & check, equations3.) Carry out plan- do the work & solve4.) Look back- check your work

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

Prime Numbers: Divisible by 1 and itselfPrime Numbers = 2, 3, 5, 7, 11, 13, 19 ...Example:257 = Prime257 divided by 7 = NO257 divided by 11 = NO257 divided by 13 = NO

Composite Numbers: Divisible by more than 2 factorsExample:385 = Composite385 divided by 5 = YES285 divided by 385 = YES384 divided by 1 = YES

GCF = Greatest Common Factor (small numbers)ORGCD = Greatest Common Divisor (small numbers)

GCF or GCD:18 and 30 = 6Example:Factors of 18 = 1, 2, 3, 6, 9, 18Factors of 30 = 1, 2, 3, 5, 6, 10, 15, 20

GCF or GCD:18 and 30 = 6Example: 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

LCM: Least Common Multiple (bigger numbers)

LCM:28 and 60 = 420Example: 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

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

GCF:(2 x 2 x 2) x 3 x (5 x 5) and (2 x 2) x 5 x (7 x 7) = 20Example:#1 2 2 #2 2 2 2 5 3 7 5 7 52 x 2 x 5 = 20LCM:(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)

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.

Intergers: whole counting numbers

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

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

A = l x w (any two "numbers" together) - generic rectangle - base 10 recrangle

Volume = l x w x dany three "numbers" together

two "numbers" that cannot be simplified

three "numbers" that cannot be simplified

can always be written as a ration (fraction), a number that stops or repeatsExample:4.12 , x = + 5 , x = + 7 , x2 = 49 , x2 = 25

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

the standard way to write a number (the normal way)Example:2 , 345 , 1,112 , 4 , 300 , 32 , 250

Numerator: tells us how many pieces we have of a whole.Example:4 = the numerator5

Denominator: tells us the size of each whole or piece.Example:45 = the denominator

G = groups (identified by an additon or subtraction symbol)E = exponentsD M = (left to right) divide/multiplyS A = (left to right) subtract/additionDO NOT use or teach PEMDAS = confusing!!!!!!!

Never have any number bigger than the base number

Examples:1.) 15 to base five = 3 long, 0 units = 30 five2.) 15 to base three = 1 flat, 2 longs, 0 units = 120 three3.) 17 to base six = 2 long, 5 units = 25 six4.) 11 to base four = 2 long, 3 units = 23 four5.) 14 to base three = 1 flat, 1 long, 2 units = 112 three6.)356 to base four = 11210 four 4 l356 4 l89 r 0 4 l22 r 1 4 l5 r 2 17.) 14 to base five = 24 five 5 l14 2 r 48.) 14 ro base three = 112 three- 3 l14 3 l4 r 2 1 r 1

1.) Convert 23 four to ten = 11 23 four2 long 3 unit2(4) 38 + 3 = 112.) Convert 42 eight to ten = 34 42 eight4 long 2 unit4(8) 232 + 2 = 343.) Covert 123 five to ten = 38 123 five1 flat 2 long 3 unit25 + 10 + 3 = 38

46 + 28 = 74 40 + 6+ 20 + 8 60 + 14 60+ 10 + 4 70 + 4 =74

46 + 28 = 7446 + 20 = 6666 + 8 = 74

46 + 28 = 74 46+ 28 74

46 + 28 = 74 1 46+ 28 74

31 + 24 + 15 + 42 + 39 = 1511 2 31 24 15 2 42+ 39 1 151

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

31 + 24 + 15 + 42 + 39 30 20 10 40+ 30 130

31 + 24 + 15 + 42 + 39 80 40+ 30 150

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

47 - 12 = 35 47 - 12 35

6(6) = 366 = Number of Groups(6) = Number of Units Inside 1 Group36 = Total Number of Whole Groups*ORDER MATTERS*Read left to right

1.) A Rectangle with a length of 10 + 1 and a width of 4 - A = 44Distributive Property: (4)(10 + 1) 40 + 4 =442.) A Rectangle with a length of 10 + 3 and a width of 10 + 2 - A = 15610 + 3 = 1310 + 2 = 12A = 12 x 13 = 156

6 divided by 6 = 16 = Total Number6 = Number of Groups1 = Number of Units Inside 1 Group*If divisor gets smaller = answer gets bigger (Inverse Relationship)

382 divided by 3 = 127 r1 127 r13 l 382 -3 08 -6 22 -21 10 -9 10 -9

382 divided by 3 = 127 1/3 127 1/33 l 382 -30 352 -30 322 -300 22 -15 7 -6 1

382 divided by 3 = 127 1/3 -3 -6 -21 = 1 3 8 2 = 127 1/3 3

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 0Divisible by 8 = look at the last 3 numbersDivisible by 3 = add numbers and if the sum is divisible by 3Divisible by 10 = ends in 0Divisible by 6 = divisible by 2 and 3Divisible by 9 = add numbers and if the sum is divisible by 9

Intergers: whole counting numbers

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+ + + + +

Draw Diagram: use when you have any numbers bigger than ten.Examples: -15 + 436 = +421 - sub. ++-15 + 436 - 15 +421Examples: -47 + (26) = -73 -- add. - -47 + (-26) 26 + 47 -73

Absolute Value: how far away a number is from zeroExample:6 is 6 units from zeroAbsolute value of 6 = 6-6 is 6 units from zeroAbsolute value of -6 = 6 or l-6l

Number Lines: a line with numbers placed in their correct positionsUseful fro addition and subtractionUseful for showing relations of numbers

Example:6(2) = 12l l l l l l l l l l l l = 126(1) = 6l l l l l l = 63(-2) = -6(- -)(- -)(- -) = -65(-4) = -20(- - - -)(- - - -)(- - - -)(- - - -)(- - - -) = -20

Example:-3(2) = -6++ ++ ++ ++++------ -----3(-2) = 6++++++ ++++-- -- -- -----1(-3) = 3+++ +++++++--- --------2(4) = -8++++ ++++ ++++-------- ----

same sign = positive numberdifferent sign = negative numberadd a zero bank if the coefficent is negativetiles (show) if numbers are less than 10just solve if numbers are greater than 10

same sign = positive numberdifferent sign = negative numberJUST SOLVE

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

order changesExample:5 + 7 + 2 or 5 + 2 + 7

A = l x wExamples:25(15) = 375 10 + 15 20 l 200 l 100 l+5 l 50 l 25 l300 + 75 = 3754(-3 + 5) + 8 -3 + 5 4 l -12 l 20 l4(x + 5) = 4x + 20 x + 54 l 4x l 20 l3(x + 7) = 3x + 21 x + 73 l 3x l 21 l10(2x2 - 4x) = 20x2 - 40x 2x2 - 4x 10 l 20x2 l -40x l4(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

Example:6x + 15 = 3(2x + 5)3 l 6x l +15 l2x + 5x2 - 3x = x(x - 3)x l x2 l - 3x lx - 33x2y - 9xy + 6y = 3y(x2 - 3x + 2)3y l 3x2 y l -9xy l + 6y lx2 - 3x + 2

Example:(x + 2)(x - 2) = x2 - 4 x - 2 x l x2 l - 2x l+ 2 l 2x l - 4 lx2 + ox - 4 = x2 - 4(3x - 7)(3x + 7) = 9x2 - 49 3x - 7 3x l 9x2 l - 21x l+ 7 l 21x l - 49 l9x2 + ox - 49 = 9x2 - 49

position of a decimal, bigger or equal to 1, multiply by the power of 10if the exponent is negative the answer will have a decimal, a small numberif the exponent is positive the answer will not have a decimal, a big numberExample:134000 = 1.34 x 1050.00000761 = 7.61 x 10-6

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.

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 115 is greater than 3 12 is greater than 124 ? 10 5 ? 114 x11 is less than 10 x11 = 44 is less than 55 5 x11 is less than 11 x11 = 55 is less than 55OR4 = 44 is less than 10 = 505 = 50 is less than 11 = 44

Example:4 is less then 10 5 is less then 11l l l l l l is less then l l l l l l l l l l l ll l l l l l is less then l l l l l l l l l l l ll l l l l l is less then l l l l l l l l l l l l

Example:5 is greater then 129 is greater then 25l---------l----------l----------l----------l0----- 12/25------ 1/2------- 5/9-------- 1

Example (Set Model):2 = 43 = 6OR2 = 63 = 9OR2 = * * *3OR2 = * * *3OR2 = + + +3OR2 = + + +3

Example:12 (divided by 4) = 320 (divided by 4) = 5OR12 = 2 x 2 x 3 = 320 = 2 x 2 x 5 = 5ORy 2 = y x y x 1 = 1y 3 = y x y x y = y

Example:2 + 5 = 2 x 6 + 5 x 2 = 12 + 10 = 22 (divide by 2) = 113 + 9 = 3 x 6 + 9 x 2 = 18 + 18 = 18 (divide by 2) = 9Answer is:11 or simplify 11 = 1 2/99 or simplify 9

You have to change the denominators to be the same number by finding out what is missing in each fraction (denominator).Simplify = to to bottomThen you add straight across.Change to a mixed number

You have to change the denominators to be the same number by finding out what is missing in each fraction (denominator).Simplify = to to bottomThen you subtract straight across.Change to a mixed number

Simplify first = top to bottom, and acrossMultiply straight across then change to a Mixed number

Rule:K, C, F or K, C, I = Keep, Change, Flip (Inverse)*Follow same produce as Multiplication of Fractions with unlike denominators