Graphic Organizers Used at Landmark EMS © 2016 C. Woodin and Landmark School
OODIN ATH
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RedLaser
SCAN
OODIN
Multidigit Multiplicationl
ATH
2016
× 2D x 2D with J-Highlight QR code
e
w h ol
#
e
1
4
w h ol
.2
5
1 .0
0
All the parts
8 2 0 2 0 0
Percentage
to m a
4
100% 1
5 0%
25 __% 1 0. 0 % 0%
w h ol
e
to m a
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#
Decimal ---->
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4
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1
to m a
w h ol
Decimal
ke
1
m ak
Fraction
4
Fraction -----> All the parts
to
e
1
w h ol
Decimal 4 1.0 0
e.g., “1 hoof”
#
4
Parts
#
1
m ak
e
Fraction
1
to
https://www.youtube.com/watch?v=ccmUcgRl1G8 e
1
Parts
#
Dynamic Conversion Model
2
_______ All the parts
1 .40 _______ 0
1 92 6 1 0 18 2 1 12 00 8 1 2 0
Continue by adding: • Factors (1+2=3) • Products (12+24=36) This will produce 3x12=36 3) Continue the process of adding pairs of factors and products. Move the index card down so you only see the 2 numbers you are adding.
Mask the top of the Chart with an index card. Move it down one space after each 1 2addition step. 1 2 2 4 3 6
Index
6 0 9 6 card.
# 1
1 1 2 3 5 8 1 9 √ 10
w h ol
e
6 30 6 9 66 0
to m a
ke
1 12 2 2 14 2 3 26 4
1) 1 x 12 = 12 Write this “12” twice in the chart. Add the factors (1+1) to get “2.” and their products (12+12) This will produce 2 x 12 = 24 2) Mask (cover) the top of the chart (the top 1 x 12 = 12) with an index card,
Ladder Charts
1 2
Make a Ladder Chart to find multiples of a divisor. e.g., for 12:
2 2 4 6 10 2 5 7
1 2 1 01 82 1 21 02 2 4 3 6 6 0 9 6 1 2
You should get 60 here. It is 5 x 12.
1 0 8 1 2 0
4) After reaching 8x12, write in (1 x 12) = “12” Add this to the 96 This will produce 9x12 = (108)
https://www.youtube.com/watch?v=GxfRWDJjKYE
1 2 1 2 Index card. 2 4 3 6
6 0 9 6 1 2 1 0 8 1 2 0
4) Check multiples of 2, 3, 5, 9, 10 with divisibility rules as you go. The chart ends at 10 x 12 which should be recognized as the correct 10x product (120).
OODIN ATH
2016
1
All the stuff you have
ATH 2016
w h ol
e
Subtract
OODIN
to m a
ke
}
Compare divisor to dividend Divide Multiply
#
Long Division.
to m a
5
1
ke
e
5
e
w h ol
to m a
to m a
5
w h ol
to m a
5
ke
w h ol
e
1
w h ol
e
1
#
A B C
5
ke
#
1
ke
#
1
6 3 1
to m a
ke
#
w h ol
3 1
e
#
Check Subtraction by adding up Bring down the next digit
Divide 31 ÷ 5 = 6 Write the 6 in the quotient area. Your pencil tip should already be in the correct location. Leave your pencil tip on the 6!
6 3 1 3 0
6 3 1 3 0 1
6 3 1 3 0 1
Compare divisor to dividend. Track with your pencil point as you work through the problem: “Can you get an entire 5 out of 3?” “No.” • Move pencil tip to the 1 of 31. “Can you get an entire 5 out of 31?” “Yes.” Move pencil tip straight up to the quotient area. Leave pencil tip there!
1 5
Multiply Your pencil tip should be on the 6. Track your pencil tip through this “C” - like path: Multiply: 6 x 5 (tap the 5) = 30 (write 30). Leave pencil tip on the 30! A) Subtract Subtract the 30 from the 31. The difference is 1. Write “1.” Leave your pencil tip on the 1. 3 1 B) √- Check subtraction by adding up: 3 0 + 1 + 0 = 1, 1 “0” + 3 = 3 C) Bring down the next digit from the dividend. In this case there is no additional digit to bring down. Compare the divisor (5) to the dividend (1). • Is the 1 smaller than the divisor? If no: the quotient was too small. Go back and select a larger quotient. If yes, box the “remainder.” Express the remainder as a fraction using the graphic organizer:
Three Types of Fraction Addition/ Subtraction Problems
Numerator
3 8
Denominator
+ 1 8 4 8
Different Denominators
×2 4= 2
1 4
+ 1 2
= × 2 =
=
× 2 =
1 4 2 4 3 4
• Circle the bigger denominator (e.g., 4). “Can you use multiplication to make the smaller denominator match the bigger denominator?”
“Yes.”
The circled denominator will be the common denominator. • Rename the fraction with the smaller denominator to match the circled common denominator.
“No.”
The larger denominator is not a multiple of the smaller.
https://www.youtube.com/watch?v=v5IsH3OHLok
3×2=6 ×
OODIN ATH
2016
1 + 3
× 2 =
1 2
× 3 =
=
× 2 =
=
× 3 =
2 6 3 6 5 6
Mixed Numbers: A Whole Number with a Fraction
OODIN
e.g., 3 pairs of shoes and 1 loose shoe. Number of whole things.
ATH
Ta k
Number of parts to make 1 whole.
ol e wh
f r o m th e w h o l e
nu
ber
3
1
Number of parts you have.
m
e
3
1 2
2016
1 2
A dd
1
wh
2
= o le
( 2/ 2) to t h e fr a c t
3 1
1 2 2 2
io n
e.g., 2 pairs of shoes and 3 loose shoes.
3 2
https://www.youtube.com/watch?v=M2f89t9dBI8
= =
2 1 1
3 2 2 2 1 2
Fraction Multiplication
OODIN ATH
Numerator
2015
Denominator
1) Both 18 and 27 pass the divisibility test of 9: their digits add to 9 (1 + 8 =9, and 2 + 7 = 9) 2) Divide the fraction by 9/9 ... a fraction worth 1. Simplify
18 27
= 2 ÷9 = 3 ÷9
1) Both 18 and 27 pass the divisibility test of 9: their digits add to 9 (1 + 8 =9, and 2+7=9). Circle both of these numbers. 2) Divide both by 9: 3) As you divide, “Cancel.” Put a slash Ask, “How many 9’s? through each number. Slash both with a similar (matching) slash.
2
18 25
×
5 27 3
5) Both 25 and 5 pass the divisibility test of 5: their digits have a 0 or 5 in the ones place. Circle both of these numbers. 6) Divide both by 5: Ask, “How many 5’s?”
2
18 25 5
1
×
5 27 3
=
2 15
7) As you divide, put a distinctive, matching slash through each number. 8) Write the answers (quotients) above the numerator and below the denominator.
9) Multiply numerators: 2×1= 2 Multiply denominators: 5 × 3 = 15
Divisibility RuleS
OODIN ATH
2016
• A number is divisible by 2 if it has 0, 2, 4, 6, or 8 in the ones place. * A number is divisible by 2 if it is an even number. Examples: 220 112 34 106 2008
• A number is divisible by 3 if it’s digits add up to 3, 6, or 9. Examples: 12 1+2=3 24 2 + 4 = 6 45 4 + 5 = 9 291 2 + 9 + 1 = 12
1+2=3
A number is divisible by 4 if . . .
• The last two digits are divisible by 4. [Dividing them by 2 results in an even quotient (answer).] Example: Is 128 divisible by 4? Step 1: Step 2: Underline, or write down Divide the 28 by 2. the last two digits: 1 2 8. (Cut 28 in half.) “28”
Step 3: Is the quotient an even number? 1 4 Yes.
2 2 8 This quotient (14) is even. 128 is divisible by 4!
• A number is divisible by 5 if it has 5 or 0 in the ones place. Examples: 25
60
75
300
1115
• A number is divisible by 6 if it divisible by BOTH 2 and 3 Examples: 18 even? √ 1 + 8 = 9 √ Yes- 18 is divisible by 2 and 3, so it’s divisible by 6. 42 even? √ 4 + 2 = 6 √ 120 even? √ 1 + 2 + 0 = 3 √ 2442 even? √ 2 + 4 + 4 + 2 = 12 1+2=3
• A number is divisible by 9 if it’s digits add up to 9.
Examples: 18 1+8=9 27 2 + 7 = 9 459 4 + 5 + 9 =18 1+8=9 8991 8 + 9+ 9 + 1 = 27 2+7=9
• A number is divisible by 10 if it has 0 in the ones place.
Examples: 20
60
170
300
4510
http://www.landmarkoutreach.org/publications/teaching-guides-resources/multiplication-and-division-facts
Multiplication and Division of Mixed Numbers
OODIN
Example: Two cars and 1 extra tire = 9 tires.
ATH
2016
Whole number. The number of whole things: e.g., 2 whole cars NumeratorNumber of parts. e.g., 1 extra tire.
1 4
2
Denominator Number of parts needed https://www.youtube.com/watch?v=I186x9-1PJw to make 1 whole thing. (e.g., It takes 4 tires to make 1 whole car.)
+
1
×
4
2
(4 × 2 + 1 = 9)
=
9 4
9 Loose tires. Keep the same denominator (4). (i.e., It still takes 4 tires to make 1 whole car.)
× 2
1 4
×
1 3
9 4
×
1 3
3
9 4
1
3
9 4
1 3
×
×
1 = 3 3 4 1
1 4
2
÷ ÷
1 2
9 4
÷
1 2
9 4
×
2 1 1
9 4 2
× 9 4 2
2 1
1
×
2 = 1
9 2
Prime Factoring: Use A Venn Diagram to Calculate GCF, or LCM. OODIN Prime Factors: ATH
2016
* Prime numbers have exactly 2 factors: 1 and themselves: e.g., 3 = 3 x 1 * 0 is not prime. It has an infinite number of factors: e.g., 0 = 0 x 3, 0 = 0 x 10000. * 1 is not a prime number. It has only one factor (1). • The first 9 prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23.
• Factor each number using prime numbers: 2, 3, 5, 7... Prime #s
12
20
2
6
2
10
2
3
2
5
3
1
5
1
• Complete the Venn Diagram that relates these two numbers in terms of their prime factors.
12
L.C.M. =
12
d# 2n
by uncommon 1st # fact y l p ors i ult To find L.C.M. of M
3
2 2
20 5 L.C.M. =
× 5 = 60
Multiply these common factors to find the Greatest Common Factor: G.C.F. =
2×2=4
20
× 3 = 60
Integers Water Level
0 -1 -2 -3
+3 +2 +1 0
Integers are like the steps of a ladder. They are whole numbers. e.g., 2, -4, +5 are integers. 1/2, π, -3.2 are not integers Positive integers are greater than 0. Negative numbers are less than 0.
+1
-1
Think of positive (+) integers as being “hot” and negative (-) integers as being “cold.” Integers with the same sign work together. Opposing signs work against each other. Generally think of single “+” and “-” signs as adjectives (hot or cold) instead of verbs. Compound signs necessitate using a verb:- (-2): “Subtract 2 cold cubes”; + (-2): “Add 2 cold.”
+2 +3 = +5
-2 -3 = -5
+2
-(-3) = +3
+1 -1
+2 -(+1) = +1
+3 -3 = 0 Taking away 3 cold makes the mixture hotter by 3.
0
-2
- 3-(-1) = -2 Adding hot to hot makes it really hot! Adding cold to cold makes it really cold!
1 hot neutralizes 1 cold.
Taking away 1 hot will make the mixture colder by 1.
Taking away 1 cold will make the mixture hotter by 1.
OODIN ATH
2016
https://www.youtube.com/watch?v=zLzRF-ey3LE
Combining Integers Adding or Subtracting (not X or ÷)
1) Express each term with one operational sign. e.g., - ( -3) = +3 “Taking away 3 cold makes it hotter by 3.” ( -2) “Two cold things.” ( +2) = “Two hot things” 2) Commute to group positive terms (+ hot) and negative (- cold) terms. • Combine all positive terms. Combine all negative terms. • Find the difference between the positive and negative terms. The result will take the sign of the largest + or - term.
- ( - 3)
-2
+2
+3
-2
+2
+3
+2
-2
+5
+3
-2
x
NTEGERS
Multiplying or Dividing Integers
OODIN
Multiplying or dividing by a negative is like flipping a switch. A switch is either on (+) or off (-). Every time you multiply or divide by a negative, change the sign.
ATH
2016
Use your thumb like a switch to demonstrate the effect of multiplying by a negative. • Start with your thumb in the up (+) position. • Flip if you multiply by a negative: • Don’t flip (keep the same sign) when you multiply by a positive number.
Example 1: Start +
2
No Change
•
Now you try- Use your thumb! Example 2:
-1 = -2
Start +
Flip Negative Product!
-1 Flip
• -1 = +1 Flip again!
Positive Product!
Order of Operations. [(
eX
)]
× ÷ + -
First, perform all operations within parentheses. Work from the (inside) out. Next, evaluate all terms involving exponents. After that, work left to right, performing all multiplication or division. Finally, work left to right performing all addition or subtraction.
List each operation performed:
Document Each Step
Solving Single Step Equations
https://www.youtube.com/watch?v=zCNAq5wOxag
OODIN ATH
2015
1) Draw a vertical line over the = sign to separate the problem in two. 2) State what is happening to the variable.
3) State the opposite, then do this to both sides.
“Subtract 2 from A.” “Subtract 2 from the other A + 2 = 45 side too.” -2 =-2
“A is having 2 added to it.”
“The A is having 2 subtracted from it.” So, do the opposite operation to both sides: “Add 2 to both sides.”
A - 2 = 45 +2 =+2 A = 47
A
= 43 “The A is being multiplied by 2.” So, do the opposite operation to both sides: “Divide both sides by 2.”
“The A is being divided by 2.” So, do the opposite operation to both sides: “Multiply both sides by 2.”
A = 46 2
2 • A = 46 • 2 1 2
2A = 48 2A = 48 2 2 A = 24
A = 92
Related Equations and the Linear Diagram
Write three equations for each diagram. Solve each equation for the variable. The solution should be A=18 for all 3 equations.
8 A = 8 + 10
A
10 = A - 8
10 8 = A - 10
Translate Word Problems into Equations
1) Locate the word or phrase that indicates the “=” sign within the equation. Write the “=” sign over that word or phrase. Start formulating the equation by writing an “=” sign. 2) Locate and define the variable. In a key sentence: Write the variable, what it stands for, followed by an “ =” sign, followed by a blank where you will write the value of that variable. 3) Usually one side of the = sign is relatively easier to translate- oftentimes it does not contain the variable. Select this side and translate it. 4) Translate the side with variable by first writing the variable, then perform the prescribed operations described in the problem. Write these operations using the correct order of operations. 5) solve the equation, then write the solution in the Key Phrase that defined the variable ( Step 2) to answer the problem.
e.g.,
Five times a number is 5 less than 50
=
(1) Five times a number is =
(3) Translate the
=
“easy side.”
Translate the “harder side.” First write the variable:
5 less than 50
N
(2) N =
Define the Variable
9 A number = ________
50 - 5
=
50 - 5
(4a)
Translate the “harder side.” First write the variable, then perform the prescribed operation(s) to the variable.
(4b) Solve the equation and answer the problem by completing the Key Phrase above.
5N
=
50 - 5
5N = 45 5 5 N = 9
OODIN ATH
201
Proportions
https://www.youtube.com/watch?v=H9ZznBI3iaI
1) Label headings on left. One of these labels should describe the variable for which you are solving. 2) Fill-in the data table following the headings. 3A) Direct Variation: Going left to right, or top to
bottom, both data pairs get proportionally larger or smaller. Cross multiply to create an equation to
Circle one: Direct Proportion
solve.
3B) Inverse Variation: As one data pair gets
proportionally larger from left to right, the other gets smaller. Multiply vertically (straight down) to
×
Inverse Proportion
×
Known pair Headings
create an equation to solve.
4) After solving, put the answer into the table to answer the problem with units. Check to make sure the data follow the expected pattern. Direct Variation
3 birds have 6 wings. How many wings do 5 birds have?
×
Birds
3
5
Wings
6
W = 10
Equation
OODIN ATH
2016
=
Equation:
5•6 =3W 30 = 3 W 10 = W
It takes 5 men 60 minutes to paint a fence. How long will it take 10 men to paint the same fence at the same rate? Inverse Variation
×
Men Minutes Equation:
5
10
60
5 • 60 300
30
T = 30
=
10 T
= 10 T =T
Proportions Involving Percentages What is 20% of 50? P = part 100% OODIN
50
P
20%
10% x 2 = 10
0%
5.0 _______ 0
10 is 20% of what number? W = whole 100% W=
1 0. 0 %
10
Whole
W = 50
0
50
Percent
Headings Part
10
P = 20%
Whole
50
100
50 P
= 10 • 100
50 P 50
=
Equation:
20%
1 0. 0 % 0%
25
_______
( 2 x 5 =10)
5.0
_______ 0
100
1000 = 20 W 20 20 50 = W
_______
10 is what percent of 50? P = percentage
5 0%
20
100 •10 = 20 W
Equation:
= 10 ( given) 5.0 ( 1/2 of 10)
0%
Percent
Part
_______
20%
100%
Headings
25
5 0%
100
100 P = 1 0 0 0 100 100 P = 10
2015
1 0. 0 %
20
= 10
100 P = 20 • 50
Equation:
25 _______
Percent
50
Whole
5 0%
ATH
Headings Part
1000 50 P = 20%
https://www.youtube.com/watch?v=AhNFl8pg7VM
Find the slope of the line. “ ” means change.
Slope:
Y X
=
Y X
Write the coordinates of two points that are on the line. https://www.youtube.com/watch?v=aiQeLvn91Os Subtract to find the change in Y and X values.
( (
X
X
Y
) )
Make a Slope fraction:
Slope: M =
Simplify this slope fraction. Y X
=
Y
X
Y
3
1
(3 (0
https://www.youtube.com/watch?v=hvG1-x-T418
X Make a fraction:
Slope =
Y = X
Simplify this slope fraction.
Y
1 3
(Rise) (Run)
1 3
m=
b =+1
) 1) 2
Equation of a Line: Y = mx + b Run = +3
Rise = +1
M = Slope = 1/3
b = Y intercept = +1
Substitute the slope (m) and Y- intercept values into “y=mx+b to express this line as an equation.
y = m x+ b y= (
1 3
)x+ 1
OODIN ATH
2015
D=R•T
Rate
R=D÷T
Time Distance
Distance
=
Rate
•
Time
Headings
Time
T=D÷R
Distance
=
Rate
Rate
Distance =
Rate
=
Distance Time
• Time
Distance Total Equation1_____________________________
Equation2 ______________________________
Combine _______________________________________
%
100%
Part
100
Whole
5 0%
1 0. 0 % 0%
=
_______
_______ 0
√
c
=
√
a2 + b2 = c2
√
( , )
( , ) ( (
) )
X
Y X
II
I
Y
=
X
III
© C.Woodin and Landmark School EMS 2016 Approved MCAS Template
Y
IV
Pythagorean Theorem: a2 + b2 = c2
Write benchmarks: • Higher perfect square on top • Lower perfect square on bottom. • Lower square root in circles • Draw arrow to approximate tenths place.
c2 a2
c
c
Estimate √ 80
a2 + b2 = c2
b
2
Estimate Square Roots
9
√ 81 a= 3
C=5
8 . _9
C 1) Mark the right angle with a square. 2) Draw an arrow from the square to the opposite side. Label that side as “C.” 3) Label the other sides a and b. 4) Substitute values and Solve • Substitute values. • Square values
• Solve for the variable:
√
32 + b2 = 52
9 + b2 = 25
• Square root both sides: • Label diagram with solution. C must be larger than a or b. C must be smaller than a+ b.
._
64
.9 .8 .7 .6
.5
.4 .3 .2 .1
√
b2 =
16
√
b
4
Estimate √ 80
=
.4 .3 .2 .1
8
√
a2 + b2 = c2
9 + b2 = 25 -9 -9
.5
√
b
9 81
8.9 OODIN ATH
2016
.9 .8 .7 .6
80
.9 .5
8 64
