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% symbolizes a ratio. amount per 100
30% is 30 / 100
Using the logic that it is out of 100, we can draw 10 frame boxes as a way to show percent. Each box is equal to a tenth of the total (100).
40% would have four boxes colored in
70% would have seven boxes colored in
35% would have 3 and one half boxes
20% of 40
10% of 40 is 4 (move the decimal to the left one spot)
A 10 frame that represents 40 would have 4 in each box
20% would be two of the boxes
4 + 4 = 8
20% of 40 = 8
45% of 80
10% of 80 is 8
40% of 80 would be 8x4 = 32
5% of 80 would be 4 (half of 10%)
32 + 4 = 36
45% of 40 = 36
42% of 120
10% of 120 is 12
1% of 120 is 1.2 (move the decimal one more spot to the left of the 10% total)
40% is 12x4 = 48
1.2 + 1.2 = 2.4
48 + 2.4 = 50.4
42% of 120 = 50.4
Groups
Exponents
D/M divide/multiple, left to right
S/A subtract/add, left to right
groups are made with an add or subtract sign
-12 - 4(2) + 5
bring down the - and +
-12-8+5
add the inverse
-12+ (-8)
-20 + 5
-15
-24 / 2 x 4
-12 x 4
-48
Remember to go LEFT TO RIGHT
20 - 4^2 x 3 / 2
20 - 16 x 3 / 2
20 - 48 / 2
20 - 24
-4
-4^2 + 8 - 3^2 - 4
-16 + 8 - 9 - 4
-8 - 9 - 4
-17 -4
-21
(-7)^2 = 49
-(7^2) = -49
(-7^2)= -49
-7^2 = -49
-(7)^2 = -49
Building Integer Multiplication
3 (-5)
3 groups of 5 negatives
OOOOO OOOOO OOOOO
3 (-5) = -15
-4 (3)
0 TAKE AWAY 4 groups of 3 postives
0 - 4 (3)
ZERO BANK
OOO OOO OOO OOO
OOO OOO OOO OOO
take away 4 groups of 3 positives
12 negative is left
-4 (3) = -12
-3 (-2)
zero take away 3 groups of 2 negatives
OO OO OO
OO OO OO
take away 3 groups of 2 negatives
6 positives are left
-3 (-2) = 6
Solving Multiplication and Division
If the signs are the same, the answer is positive
if the signs are different, the answer is negative
-25 (-30)
same sign, answer will be positive
-25 (-30)=750
-100/20
signs are different, answer will be negative
-100/20 = -5
Showing Subtraction - use + and - instead of color counters
5 - (-2)
+++++++
[--]
5 - (-2) = 7
Solving Subtraction
Add the inverse - this is the same as making a zero bank
-18 - (-15)
-18 + (+15)
Now I can use Hector's Method
-18 + (+15)
- - +
subtract the two numbers
18-15=3
a negative is still left outside the circle
-18 + (+15) = -3
-53 - 20
-53 + (-20)
- - -
add
53 + 20 = 73
negative sign left outside the circle
-53 + (-20) = -73
5 - 3
5 take away 3
OO [OOO]
OO
5-3=2
-6 - (-2)
6 negatives take away 2 negatives
OOOO [OO]
-6 - (-2) = -4
4 - (-3)
OOOO
Wait, we have no negatives, how can I do this??
ZERO BANK!
OOOOOOO
[OOO]
4 - (-3) = 7
3 - 6
OOO
I don't have enough positives to take away 6. I need a zero bank
[OOOOOO] O
OOO O
3 - 6 = -3
Instead of color counters, use + or -
-5 + 7
+++++++
- - - - -
zero bank leaves you with 2
-5 + 7 = 2
Hector's Method - only works for addition
-30 + 27
- - +
The bigger pile gets two of whatever it's sign is
Circle together one sign from each side
If the signs are different, we subtract
If the signs are the same, we add
The sign outside of the circle will be the sign for the answer
30-27=3
-30 + 27 = -3
-320 + (-430)
- - -
circle, same sign means add
320+430=750
a - is on the outside of the circle so the answer will be -
-320 + (-430)= -750
Build
Use two color counters
RED is negative
1 positive with 1 negative is a ZERO PAIR
Build -5 using 9 tiles
OO
OOOOOOO
The two zero pairs make a ZERO BANK
4 + (-3)
OOOO
OOO
Zero bank cancel each other out
The answer is 4 + (-3) = 1
2 + (-5)
OO
OOOOO
Zero bank cancels out
2 + (-5) = -3
Solving Fractions - use logic!
20 + (7/8)
Just add them together
20 + 7/8 = 20 7/8
13 - 1/5
Remember that a number over the same number is 1
12 5/5 - (5/5) is the same as the original problem
= 12 4/5
14 - 6 3/8
take away your wholes first
14 - 6 = 8
8 - 3/8
7 8/8 - 3/8 = 7 5/8
Unlike denominators, use fraction trees
7/10 - 5/12
10 - 2 * 5
12 - 2 * 6
2 in common to both
Multiple the fraction by 1, using what it is missing from the other fraction (REMEMBER THAT ANY NUMBER OVER THE SAME NUMBER IS 1)
7/10 (6/6)
5/12 (5/5)
42/60 - 25/60 = 17/60
FUNKY ONES, simplify first, then multiply - I love this!!
14/36 * 7/21
factor each number - try to see if you can find numbers in common
2*7/9*4 x 9*3/3*7
Remember that a number OVER the same number is 1
I can make a 1 with the two 7s (numbers can't both be on the top or both on the bottom numerator/denominator)
1 with the two 9s
1 with the two 3s
I now have (2*1*1*1)/(1*4*1*1) = 2/4
14/36 * 7/21 = 2/4 = 1/2
5 * 8/15
You'll want to make the whole number into a fraction and then do Funky Ones
5/1 * 8/15
3 3/5 * 2 2/4
same as Funky Ones but make it into an improper fraction first.
denominator times the whole number, add that to the numerator, that total goes over the original denominator
18/5 * 10/4
continue with Funky Ones
(8/15) / (4/5)
Keep Change Flip - Fact Families is why we can do this
8/15 * 5/4
Funky Ones
Use color counters
Building Multiplication
(2/3) * (1/4)
2/3 groups of 1/4 are red
multiply the denominators
12 pieces total
oooo
oooo
oooo
Only need 2/3 of the group
oooo
oooo
oooo
of the 2/3 we only need 1/4 to be red
oooo
oooo
oooo
(2/3) * (1/4) = 2/12
Showing Multiplication
Area Model
use rectangles
Draw one less line than your denominator - if you need 4ths, then you will draw 3 lines.
I like to make my bigger number vertically since that seems to be easier to draw than the horizontal lines
With multiplication you will only have one rectangle
The answer is the part that will have both colors shaded in
TAKE AWAY
Build subtraction
3/4 - 1/3
build the first fraction on the board and the second on off the board
3/4 take away 1/3
check if the take away number fits into the first number
Convert both fractions using the same size pieces
We would use 1/12 tiles
3/4 would convert to 9/12
1/3 would convert to 4/12
We now have 9/12 - 4/12 and we can do this because they are now the same size pieces
We take away 4/12 and are left with 5/12
3/4 - 1/3 = 5/12
Show Subtraction
Area Model
use rectangles
Draw one less line than your denominator - if you need 4ths, then you will draw 3 lines.
I like to make my bigger number vertically since that seems to be easier to draw than the horizontal lines
With subtraction you will have two boxes
2/3 - 1/2
Build Addition
2/5 + 1/2
Show Addition
Area Model
use rectangles
Draw one less line than your denominator - if you need 4ths, then you will draw 3 lines.
I like to make my bigger number vertically since that seems to be easier to draw than the horizontal lines
With addition, you will have three rectangles
3/4 + 1/5
Area Model
Linear Model
Set Model
Which is bigger?
[3/8] or 3/11 - same number of pieces but the smaller denominator means the pieces are bigger
[9/17] or 7/15 - 1/2 anchor fraction, 9 is more than half of 17 and 7 is NOT more than half of 15
13/23 or [20/23] - more pieces of 23 - same denominator
15/17 or [27/29] - both missing 2 pieces to become whole, 29 is a smaller piece so more of this one is filled in
Numerator: number of pieces you have
Denominator: tells me the size of my piece
Inverse relationship
17 is bigger than 5 BUT 1/17 is smaller than 1/5 - the pieces are smaller in 1/17 than in 1/5
LCD for adding or subtraction fractions but you don't need to for multiplication, why?
Multiplication is _____ groups of ______
so 2/3 x 3/4 is, 2/3 groups of 3/4, and does not need a LCD
Divisibility Rules
2: even
3: sum of the digits can divide by 3
4: last two digits divide by 4
5: ends in 5 or 0
6: #2 and #3 works
7: NONE
8: last 3 digits can divide by 8
9: sum of the digits can divide by 9
10: number ends in 0
Prime Factorization
Prime number - only factors are 1 and itself (i.e. 2, 3, 5, 7)
Factor Tree
48 < 4 x 12
4< 2 x 2
12< 3 x 4
4< 2 x 2
2x2x2x2x3
2^4 x 3
Upside down division - use an upside down "house", use only prime numbers
40 < 5 x 8
8< 2 x 4
4< 2x2
2x2x2x52^3 x 5
Use prime factorization to find LCD
use whichever is bigger
2^2 x 3 x 5^4 x 11 and 3^3 x 5^2 x 7^2
LCD = 2^2 x 3^3 x 5^4 x 7^2 x 11
Build - 12/3
Show - 129/25
Repeated Subtraction
Subtraction Area Model
Upwards Division
3 (4) - 3 groups of 4 units
Build
Show - drawing, no blocks, flats, longs, units - 3 x 4
Teach multiplication facts in this order:
1s, 2s, 5s, 10s, 3s, 9s, doubles, 4s, 6s, 7s, 8s
What makes a good algorithm?
Is it expandable, efficient, and based on solid math principles?
Addition
Expanded Form
Left-to-Right
Friendly Numbers = numbers that end in 0
Trading Off - like Friendly numbers but only moving part
Subtraction - take away
Expanded Form
Left-to-right
Equal Addends - this works because subtraction is the measurement, amount, distance between two numbers.
squares = flats
lines = longs
dots = units
Addition
if you have enough longs to make a flat, close the 10 longs with a Z
Subtraction
Addition
i.e. 3 + 2 would have 3 units + 2 units = 5 units (you did not fill a long)
In non-base 10
Subtraction
i.e. 17-9
In non-base 10
12 base 8 take away 6 base 8 would need to exchange a long for units. A long equals 8 units (NOT 10)
UnDevCarLo
Method for solving problems
i.e. Base 6 is 6 unites make a long and 6 longs make a flat
i.e. 356 base 4 is impossible but 356 base 7 is ok
