av Roxanne Cavalera 7 år siden
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Mer som dette
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
Simplify first = top to bottom, and across
Multiply straight across then 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 bottom
Then you subtract straight across.
Change to a mixed number
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
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
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
Example (Set Model):
2 = 4
3 = 6
OR
2 = 6
3 = 9
OR
2 = * * *
3
OR
2 = * * *
3
OR
2 = + + +
3
OR
2 = + + +
3
Notes:
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
Example:
5 is greater then 12
9 is greater then 25
l---------l----------l----------l----------l
0----- 12/25------ 1/2------- 5/9-------- 1
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
Example:
134000 = 1.34 x 105
0.00000761 = 7.61 x 10-6
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
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
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
Example:
5 + 7 + 2 or 5 + 2 + 7
Example:
(4 + 2 + 5) or (4 + 2) + 5 or 4 + (2 + 5)
-5 + (5 + 3) or (-5 + 5) + 3
JUST SOLVE
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
Number Lines: a line with numbers placed in their correct positions
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
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
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
+ + + + +
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)
382 divided by 3 = 127 1/3
-3 -6 -21 = 1
3 8 2 = 127 1/3
3
382 divided by 3 = 127 1/3
127 1/3
3 l 382
-30
352
-30
322
-300
22
-15
7
-6
1
382 divided by 3 = 127 r1
127 r1
3 l 382
-3
08
-6
22
-21
10
-9
10
-9
6(6) = 36
6 = Number of Groups
(6) = Number of Units Inside 1 Group
36 = Total Number of Whole Groups
*ORDER MATTERS*
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
47 - 12 = 35
47
- 12
35
47 - 12 = 35
47 + 8 = 55
- 12 + 8 = - (20)
35
31 + 24 + 15 + 42 + 39
80
40
+ 30
150
31 + 24 + 15 + 42 + 39
30
20
10
40
+ 30
130
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 = 151
1 2
31
24
15
2 42
+ 39 1
151
46 + 28 = 74
1
46
+ 28
74
46 + 28 = 74
46
+ 28
74
46 + 28 = 74
46 + 20 = 66
66 + 8 = 74
46 + 28 = 74
40 + 6
+ 20 + 8
60 + 14
60
+ 10 + 4
70 + 4 =74
Never have any number bigger than the base number
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
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
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!!!!!!!
Denominator: tells us the size of each whole or piece.
Example:
4
5 = the denominator
Numerator: tells us how many pieces we have of a whole.
Example:
4 = the numerator
5
Example:
2 , 345 , 1,112 , 4 , 300 , 32 , 250
Example:
4.12 , x = + 5 , x = + 7 , x2 = 49 , x2 = 25
Volume = l x w x d
A = l x w (any two "numbers" together)
- generic rectangle
- base 10 recrangle
Zero Bank: adding any equal number of pairs of positive and negative numbers.
Zero Pairs: add a positive and a negative (cancels each other out).
Intergers: whole counting numbers
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.
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: is one of the techniques used in Prime Factorization method to factor numbers.
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 = Greatest Common Factor (small numbers)
OR
GCD = Greatest Common Divisor (small numbers)
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
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: 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: Divisible by 1 and itself
Example:
257 = Prime
257 divided by 7 = NO
257 divided by 11 = NO
257 divided by 13 = NO
Unit = 1
Long = 10
Flat = 100
Cube = 1000
Repeats on and on...
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