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A Work in Progress....
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Outside of foldable Under Flaps of foldable
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Pages TitlePg. 13 Coordinate Plane Pg. 1415 Terminating and Repeating DecimalsPg. 16 Adding FractionsPg. 17 Adding DecimalsPg. 18Pg. 19
Subtracting FractionsSubtracting Decimals
Pg. 20Pg. 21
Adding IntegersSubtracting Integers
Pg. 22 Multiplying and Dividing Rules
Pg. 24 Multiplying FractionsPg. 23
Pg. 25 Dividing FractionsPg. 26Pg. 27
Multiplying DecimalsDividing Decimals
Pg. 28 Properties of Addition and Multiplication
Order of Operations PEMDAS
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Pages TitlePg. 29 Expressions with Algebra TilesPg. 30 Simplifying Expressions (Addition and Subtraction)Pg. 31 Pg. 32 32a
Solving OneStep EquationsSolving TwoStep and MultiStep Equations
Pg. 33 Inequalities
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Pages Title
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Pg. 34 Unit Rates Pg. 35 Proportional Relationships Pg. 36 Constant of Proportionality and Direct VariationPg. 36a Pg. 37 Pg. 38
Positive/Negative Correlation & Discrete/Continuous Data
Scale Drawings w/ Scale FactorCongruent and Similar Figures
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Pages Title
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P. 39 Lines, Angles, and TrianglesP. 40 Types of Angle PairsP. 41 Exterior Angle Theorem
Line segments and TrianglesP. 42P. 43P. 44
Perimeter of a TriangleArea of a Triangle
P. 45P. 46
Volume of a Triangular PrismSurface Area of a Triangular Prism
P. 47 Circumference and Area of a Circle
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Pages Title
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P. 48 Collecting DataP. 49 Measure of Central Tendancy
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Pages Title
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13Coordinate Plane
GGraph
The origin is the point where the axis intersect.
IVIV
I
III
A CB D A
B
CD
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14 Fractions to Decimals to Percents
move the decimal two places right and add the percent
3 8
tip it on it's side and divide
8 3.0000.375
-24 60 -56 40
Yes, you CAN have a percent
that still has a decimal in it.
= 3.75%
Percent to decimalmove the decimal two places left and take away the percent
Some fractions are easy to write as decimals:
Some you have to convert:
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15 Terminating and Repeating Decimals
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16 Adding Fractions
Must have a common denominator!The LCM of 2 and 3 is 6
*Stack them to add
4
*Stack them to add
Stack them to add Must have a common denominator!
The LCM of 3 and 9 is 9
BIG
TopBott
om
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17 Adding Decimals
Stack them to add and line up the decimal points.
Put zeros in missing place values.
.00
Stack them to add and line up the decimal points.
Put zeros in missing place values.
0
535.49
9.71
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18 Subtracting Fractions
Must have a common denominator!The LCM of 2 and 3 is 6
*Stack them to subtract
4
*Stack them to subtract
Stack them to subtract Must have a common denominator!
The LCM of 3 and 9 is 9
-
--
- -
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19 Subtracting Decimals
Stack them to subtract and line up the decimal points.
Put zeros in missing place values.
.00
-
-
Stack them to subtract and line up the decimal points.
Put zeros in missing place values.
-
-0
520.51
1.29
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20 Adding IntegersOnly adding Positives = +
2 + 6 + 8 = 16Only adding Negatives = -
(-2) + (-6) + (-8) = (-16)Adding Positives and Negatives
Put together positives Put together negatives2 + 3 + (-7) + 4 + (-8)
2 + 3 + 4 + (-7) + (-8)
9 + (-15) = Decide if you have more positives or more negatives.
There are 6 more negatives so 9 + (-15) = -6
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21 Subtracting Integers
Every Subtraction can be written as an addition.
The Rule is to : Add the oppositeStep 1) Change minus to plus
Step 2) Change sign of # after the minus to it's opposite.
2 - (-7) =
2 + 7 = 9
Change minus to plus
Change negative 7 to positive 7
Solve with addition
- 6 - 5 =
-6 + (-5) = -11
Change minus to plus
Change positive 5 to negative 5
Solve with addition
-8 - (-5)
4 - 7 4 - 7--8 - (-5)=
=When you write on your paper just make sure you change two signs.
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22 Multiplication and Division Rules
Same signs Positive Answer 5 x 4 = 20
6 x 8 = 48
Different signs Negative Answer
5 x 5 = 25
3 x 8 = 24
Multiplication Man
Answer
Answer
2 x 4 = +8 3 x 5 = 15
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23Order of operations
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24 Multiplying Fractions
Make all whole numbers fractions.
1 2 x ¾ becomes 2 x 3 1 4
Turn all mixed numbers into improper fractions.
2 1½ x 2¼ becomes 3 x 9 2 4
Simplify before you multiply.3
5 x 6 12 10becomes 1 x 1 2 2
Multiply the numerators then the denominators. 4
Multiply straight across the top then straight across the bottom.
1 x 3 2 5
3 10
=
Simplify your answer.5
Find the largest number that can go into both the numerator and the denominator.
6 27
3 is a factor of both so divide both by 3
6 27÷
33
2 9
=
around the world
1 ½x
2 + 3
=
=
Multiply the denominator with the whole number and add to the numerator.
cross simplify 5 x 6 10 12
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25 Dividing Fractions
2 32
2 31 2
=x = 6
3= 2 = 7 6
1842 8
21 4
= = 5 1 4
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26 Multiplying Decimals
5.00 x 0.105.00
x 0.10.5000
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27 Dividing Decimals
0.4 2 0.3 )
Step 2: If you move the decimal outside the box, move the decimal inside the box the same number of places.
0.42 ÷ 0.3
Step 1: If there is a decimal outside the box move the decimal point to the right until you create a WHOLE number.
Step 3: Bring the decimal point of the number in the box above the box.
Step 4: Divide as you would divide whole numbers.
4. 2 3 )1.4
-3 12-12 0
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28 Properties of Addition and Multiplication
Commutative Property
Associative Property
Distributive Property
It works both ways!3 x 4 = 4 x 38 + 5 = 5 + 8
Numbers like to be in groups!
3 x (4 x 2) = (4 x 3) x 28 + (5 + 2) = (8 + 5) + 2
8 (5 + 2) = (8 * 5) + (8 * 2) = 40 + 16 = 56
7(3 + 5) = (7 * 3) + (7 * 5) = 21 + 35 = 56
Share the wealth!
2 (x + 3) = (2 * x) + (2 * 3) = 2x + 64 (x 8) = (4 * x) (4 * 8) = 4x 32
2(x + 3) = (2 * x) + (2 * 3) = 2x 6
When there is a variable - you can NOT solve, only simplify
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29 Expressions with Algebra Tiles
x2 x2
x2 x2 x2
x x 1 1
1
x2 x2 x x x x
x
111
x2
x2 x2
x x
x x
x2x
1 11 11 11 1
1x2
3x2 + 2x + 1
4x2 - x2+ 2x + 2-1 =Simplify
-2x2 + 3x - 3
-x2 + x + 7
Opposites cancel
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30 Simplifying Expressions (Addition and Subtraction)
(7x2 – x – 4) + (x2 – 2x – 3)
(2x + 5y) + (3x – 2y)
(7x2 – x – 4) (x2 – 2x + 3)+ (x2 + 2x 3)
(2x + 5y) (3x – 2y)
(2x2 2x 2)
Subtraction is adding the opposite!!!
2x2 + 4x + 7
+ (2x2 2x 2)
6x2 + 3
Stack to ADD
+ (x2 – 2x – 3)8x2 - 3x - 7 6x2 + x - 7
+ (3x – 2y)5x + 3y
+ (3x + 2y)-x + 7y
(4x2 + 2x + 5) + (2x2 2x 2)
(4x2 + 2x + 5) +(2x2 + 2x + 2)
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1. Draw “the river” to separate the equation into 2 sides
2. Find the variable so you can undo the operation that is happening to the variable.
3. Whatever you do to one side of "the river" you have to do to the other.
4. Simplify vertically
5. Check your answer by substituting your answer back into the problem
The goal is to "isolate" the variable.
Be sure to check your work!! There is no reason why you should miss a problem!
Solving OneStep Equations
Undo division with multiplication
Undo
multiplicati
on
with
division
Undo
additio
n
with
subtra
ction
Undo subtraction with addition
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32a Solving Multi Step Equations
The goal is to "isolate" the variable.
1. Check for distributive property. 2(x + 6) + 4x = 24
2. Combine Like Terms. 2x + 12 + 4x = 24
3. Circle the variable and the operation "attached" to it.
4. Undo the operation not attached to the variable: usually Addition/Subtraction first
5. Undo the operation attached to the variable:
usually Multiplication/Division
6x + 12 = 24Now you have a two step equation
6x + 12 = 2412 12
6x = 126 6 x = 2
2(x + 6) + 4x = 24
Circle the variable and the operation "attached" to it.
2(x + 3) + 5x = 15
2x + 6 + 5x = 15
7x + 6 = 156 67x = 217 7x = 3
6x 22 = 405
If the whole equation is a fraction, get rid of the fraction.
5 5
6x 22 = 200+22 +226x = 222
x = 376 6
Circle the variable and the operation "attached" to it.
6x 22 = 405
2(x + 3) + 5x = 15
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The value is included in the solution set.
The value is not included in the solution set.
x > 35
x > 35
Closed circle
Open circle
x < 35
x < 35
If the variable in the equation is a negative number, you must reverse the direction
of the inequality symbol.
Inequalities
14 < 3x + 43x > 18 14 < 3x + 4 x 7
< 3
3 3÷ ÷X < 6
7 7
X > 21
If the variable is on the right of the inequality - be careful when you graph it.
35 ≤ x
It says that 35 is less than or equal to the x which means the x values are getting bigger.
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34 Unit Rates
A rate is a ratio that compares two quantities having different units of measure.
for example: miles per hour
cost per ounce
Determine the unit you are looking for before dividing.The 1st word is the numerator or the number that goes in the house for division.
pay per hour
237.50 25
= $9.50 per hour
students per teacher
84 6 = 14 students per teacher
miles hour mileshour
cost oz costoz
= 25 237.50
= 6 84
A Unit Rate is when a rate is simplified to the denominator of 1.
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35 Proportional Relationships
A proportion is an equation stating that two ratios are equal.
x
use unit rate or cross multiplication to solve(Hudgen's circle - Martin's line)
2x = 4 x = 2
2
x
5x = 265 x = 53
53
Ex. 1 pizza cost $8 1 2 3 4 8 16 24 32
= = =
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Constant of Proportionality
Unit 3 Vocabulary
Unit Rate Rate of Change
y = 2x + 1
x y
0 1
1 3
2 5
Equation the number in front of x
Table the constant rate of change
Graph how much the data increases per unit of measure
+2
+2
Where do I find the constant of proportionality in a:
as x increases by 1, y increases by 2
Equation
TableGraph
Direct Variation
A relation is direct if it has a constant of proportionality and passes through the origin.
y = 2x + 1y = 2x
y = 8 x y = 4 x + 3
nothing added means direct
Direct Variation Indirect Variation
something added means indirect
originmeans direct not origin
means indirect
originmeans direct
not originmeans indirect
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36a
or CorrelationPositive Negative
Unit 3 Vocabulary Continued
Positive Negative
as x increases, y increasesor as x decreases, y decreases
as x increases, y decreasesor as x decreases, y increases
y = 9x + 3 y = 5x + 6x y0 81 162 243 32
x y0 401 302 203 10
+8+8
+8
1010
10
positive negative
Discrete or Continuous DataDiscrete Data is COUNTED - can only take certain values (do NOT connect the dots).
Continuous Data is MEASURED - can take any value within a range (DO connect the dots).
can not rent ½ of
a game dis
tance is measured:
can go parts of a
centimeter in part of a
second.
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Scale Drawings show a real object proportionally smaller or larger than the real object. (reduced or enlarged)
Scale Drawings
reduction
normal size
enlargement
Scale Factor: Ratio of the lengths of two corresponding parts.
1cm=1000m
The map scale was 1cm = 10 m. Or . Use a ruler to measure the distance and then set up the equivalent ratio to find actual distance.
1cm10m
3.5 cm1
10 X=
so the distance between Charlotte and Fayetteville is ≈ 3500 meters
Not to scale (used only as an example)1000m 1000m
1000
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38Congruent figures: have the same shape and the same size (corresponding sides and corresponding angles are congruent).
DEF ≅ ABC Sides: DE ≅ ABEF ≅ BCDF ≅ AC
Angles: D ≅ A E ≅ B F ≅ C
Similar Figures: figures that have the same shape with corresponding sides that are proportional and corresponding angles that are congruent. (same shape different size)Figure 1 ABC ~ DEF
Sides: AB ~ DEBC ~ EFAC ~ DF
Angles: A ≅ D B ≅ E C ≅ F
Scale Factor: Ratio of the lengths of two corresponding sides of similar figures.
Figure 2 is ½ the size of Figure 1: Scale Factor is ½
Figure 1 is 2 times the size of Figure 2: Scale Factor is 2
or
ab debc ef~ ab bc
de ef~adjace
nt
(side by side)
correspon
ding
(same pla
ce
on other
figure)
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39 Lines, Angles, and Triangles
point a dot that represents a location in a plane, labeled with a letter
A
line any two points determine a line, continues on through space indefinitely
AB
line segment consists of endpoints, a portion of a line
AB
A B
A B
ray having an endpoint and continuing on indefinitely
AB
angle consists of two different rays joined at a common endpoint (vertex)
D
CDF
FDC
2
D
F
C
A
B
2
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39a Lines, Angles, and Triangles
60O, 60O, 60O
Every Δ has three angles that add up to 180O
Greater
Straight Angle
Exactly 180O
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40 Types of Angle Pairs
Complementary two angles whose sum measures 90 degrees
Supplementary two angles whose sum measures 180 degrees
Linear Pair two angles who share a common ray and vertex with a sum of 180 degrees
45 45 3060
50 130
30150
60120
Adjacent Angles
A
B
40O
30O C
D
A
B
C
D
across from each other
:
Vertical Angles
cS
use the "C" in complementary and the "S" in Supplementary to help you remember which is which.
use the "V" in Vertex and the "V" in Vertical to help you remember across from each other.
VV
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41 Exterior Angle Theorem
is 135O
You could find the other interior angle and then use supplementary angles to find angle 4, but the above theorem is quicker!
180 - (85+50) = 45
45O
180 - 45 = 135
Every Δ has three angles that add up to 180O
Supplementary angles that add up to 180O
135O
= 135O
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Side 1 Side 2 Side 3Acute,
Obtuse, Right, or Equal
Equalateral, Isosceles, or Scalene
Line Segments 1 5 1/2 5 1/2 5 1/2 Equal Equalateral
Line Segments 2 5 1/2 5 1/2 2 1/2 Acute IsoscelesLine Segments 3 5 4 3 Right ScaleneLine Segments 4 4 1/2 3 2 1/2 Obtuse ScaleneLine Segments 5 5 1/2 3 2 NO NO
One does not make a triangle: Which one?
Why does this one not make a triangle?
Use your description above to answer the following:
Can you draw a triangle with sides: 6", 3", and 2"?
Explain why you can or can not.
Can you draw a triangle with sides: 4", 3", and 2"?
Explain why you can or can not.
Can you draw a triangle with sides: 13 cm, 5 cm, and 6 cm?
Explain why you can or can not.
line segments 5
Two shorter sides when added together are not longer than the longest side.
yes
Two shorter sides are longer than the longest side.
no
Two shorter sides are not longer than the longest side.
no
Two shorter sides are not longer than the longest side.
Line segments and Triangles
In order for line segments to create a Triangle: • all 3 segments must be equal (equilateral triangle)• 2 of the segments must be equal lengths (isosceles)• the two shorter legs when added together MUST BE more than the value of the longest side (scalene)
example: 5, 9, 12 (5 + 9 = 14 and 14 is more than 12 therefore it creates a scalene triangle) nonexample: 5, 6, 12(5 + 6 = 11 and 11 is less than 12 therefore, no triangle can be created)
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43
Add up all the sides
(Plug x back in: two sides are 16 and one side is 12)
Area of the triangle is 44
Perimeter of a Triangle
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44 Area of a Triangle
Area = 1/2(5)(6)Area = 15 square inches
Area = b x h12
Area = 1/2(11)(6)Area = 33 square cm
Area = 1/2(14)(10)Area = 70 square m
or
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45 Volume of a Triangular Prism
b
h
Area = ½ bh or bh 2
Note: Lowercase "h" is the height of the triangle.
Volume = Area of the Base(triangle) x H (prism height)
or V = bhH 2
V = bhH or V = 1/2 bhH 2
V = (6)(4)(9) = 216 = 108cm3
2 2
or V = ½(6)(4)(9)(3)(4)(9) = 12(9) = 108cm3
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46 Surface Area of a Triangular Prism
Picture the Net All of the shapes that make up the original shape
8cm
5cm
10cm5cm
9cm
8cm
9cm
5cm 9cm
8cm
Find the area of each shape and add them all together:
2(Area of Triangular Base) + Area of Rectangle + Area of Rectangle + Area of Rectangle
= 2(½bh) + (LW) + (LW) + (LW)
2(½bh) + (LW) + (LW) + (LW)
2(½ 9 8) + (9 5) + (8 5) + (10 5)
(72) + (45) + (40) + (50)207 cm3
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midpoint The midpoint is the point in the center of the circle.
diameter The diameter is the distance across the circle. (equal to twice the radius)
radius
The radius is the distance from the midpoint to the edge of the circle. (equal to half the diameter)
Area and Circumference of Circles
diameter
circumference
radius
The circumference of a circle is the outside perimeter of the circle's shape.
C = 2πr C = πd
diameter
radius
The area of a circle is the entire shape of the circle.
A = πr2
PI is the circle‛s circumference ÷ its diameter.
Approximated as 3.14 or 22
73.14159265358979323846...
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48Collecting Data
Data a collection of facts or numbers that are collected for information
Survey process of collecting data; question or set of questions used to gather data
Population the group of interest in a survey, the (entire) group you want to find out information about
Census a survey of the entire population
Sample a small group of people within the population
Representative Sample a portion of the population that is similar to the entire population
Biased Sample a portion of the population in which some of the members have a greater chance of being selected for the sample than others; the sample does not fairly represent the population
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Always put your data in numerical order first.
The median - is the middle number (if there are two; add them together and divide by two).
The mode - is the number that appears most often (if there is a tie, then they are both/all the mode).
The mean - add all of the numbers together and divide by how many numbers are in your data set.
The range - subtract the lowest number from the highest number in the data set.
Measures of Central Tendancy
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60
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61
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