Download 6th Grade Digits Notes 2016-2017

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Table of Contents
Topic 1: Variables and Expressions
1-1: Numerical Expressions (pg 3-5)
1-2: Algebraic Expressions (pg 6-7)
1-3: Writing Algebraic Expressions (pg 7-8)
1-4: Evaluating Algebraic Expressions (pg 8-10)
1-5: Expressions with Exponents (pg 10-11)
Topic 9: Rational Numbers
9-1: Rational Numbers and the Number Line (pg 71)
9-2: Comparing Rational Numbers (pg 72)
9-3: Ordering Rational Numbers (pg 73)
9-4: rational #s & the Coordinate Plane (p 74-75)
9-5: Polygons in the Coordinate Plane (pg 75-78)
Topic 2: Equivalent Expressions
2-1: The Identity and Zero Properties (pg 12)
2-2: The Commutative Properties (pg 13)
2-3: The Associative Properties (pg 14)
Special: Divisibility Rules & Prime Factorization (p 15-6)
2-4: Greatest Common Factor (pg 17-18)
2-5: The Distributive Property (pg 19-20)
2-6: Least Common Multiple (pg 21-22)
Topic 10: Ratios
10-1: Ratios (pg 79-80)
10-2 to 10-4: Equivalent Ratios (pg 80-81)
10-5: Ratios as Decimals (pg 82)
10-6: Problem Solving (pg 83)
Topic 11: Rates
11-1/11-2: Unit Rates and Unit Price (pg 84-86)
11-3: Constant Speed (pg 87-88)
11-4: Measurements and Ratios (pg 89-90)
Topic 3: Equations and Inequalities
3-1: Expressions to Equations (pg 23 )
3-3: Solving Addition & Subtraction Equations (pg 24 )
3-4: Solving Multiplication & Division Equations (pg 25)
3-5: Equations to Inequalities (pg 26-27)
3-6: Solving Inequalities (pg 27-28)
Topic 12: Ratio Reasoning
12-1: Plotting Ratios and Rates (pg 91)
12-2: Recognizing proportionality (pg 92)
12-3: Introducing Percents (pg 93)
Topic 4: Two-Variable Relationships
4-1: 2 Variables to Represent a Relationship (pg 29-30)
Special: Graphing Linear Functions (pg 31-32)
4-2: Analyzing Patterns with Tables & Graphs (pg 33-4)
4-3: Relating Tables & Graphs to Equations (pg 35-36)
Topic 13: Area
13-1/13-3:Rectangles, Squares, & Parallelograms (p 94)
13-2/13-4: Triangles (pg 95)
13-5: Polygons (pg 96)
13-6: Problem Solving (pg 97)
Special: Interior Angles of Triangles & Quads(p 98)
Special: Adding and Subtracting Fractions
Special: Simplifying and Converting Fractions (pg 37 )
Special: Adding and Subtracting Fractions (pg 38-39)
Topic 14: Surface Area and Volume
14-1/14-2: Analyzing 3-D Figures & Nets (pg 99-101)
14-3: Surface Areas of prisms (pg 102-103)
14-4: Surface Areas of Pyramids (pg 104)
14-5: Volumes of Rectangular Prisms (pg 105)
Topic 5: Multiplying Fractions
5-1: Multiplying Fractions and Whole Numbers (pg 40 )
5-2: Multiplying Two Fractions (pg 41-42)
5-3/5-4: Multiplying Mixed Numbers (pg 42-43)
Topic 15: Data Displays
15-1: Statistical Questions (pg 106)
15-2: Dot Plots (pg 107-108)
15-3: Histograms (pg 109-110)
15-4: Box Plots (pg 111-112)
15-5: Choosing an Appropriate Display (pg 113-114)
Topic 6: Dividing Fractions
6-1: Dividing Fractions and Whole Numbers (pg 44-45)
6-2/6-3: Dividing Fractions (pg 46-47)
6-4: Dividing Mixed Numbers (pg 48 )
Special: Solving Equations Fractions & Mixed #s (p 49)
Topic 16: Measures of Center and Variation
16-1/16-2: Median and Mean (pg 115-116)
16-3: Variability (pg 117)
16-4: Interquartile Range (pg 118-119)
16-5: Mean Absolute Deviation (pg 120-121)
Topic 7: Fluency with Decimals
7-1: Adding and Subtracting Decimals (pg 50)
Special: Place Value and Rounding (pg 51)
7-2: Multiplying Decimals (pg 52-53)
7-3: Dividing Multi-Digit Numbers (pg 54)
7-4: Dividing Decimals (pg 55-56)
7-5: Decimals and Fractions (pg 56-58)
7-6: Compare/Order Decimals & Fractions (p 58-9)
Topic 8: Integers
8-1: Integers and the Number Line (pg 60)
8-2: Comparing and Ordering Integers (pg 61)
8-3: Absolute Value (pg 62)
8-4: Integers and the Coordinate Plane (pg 63-64)
8-5: Distance (pg 65-66)
8-6: Problem Solving (pg 67)
Special: Adding Integers (pg 68)
Special: Subtracting Integers (pg 69)
Special: Multiplying and Dividing Integers (pg 70)
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Topic 1: Variables and Expressions
1-1 Numerical Expressions
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Topic 1: Variables and Expressions
1-1 Numerical Expressions
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Topic 1: Variables and Expressions
1-1 Numerical Expressions
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1-2 Algebraic Expressions
Algebraic Expression – an expression with numbers, variables, and operations.
Variable – a symbol usually a letter, that takes the place of a number.
Term – a number, variable, or the product of a number and one or more variables.
Ex –
Constant – a term that only has a number part
Ex –
Coefficient – the number part of a term with a variable
*if there is no number in front of the variable, then the coefficient is _____
Ex –
Variable Quantity – a quantity that varies or changes.
Ex 1
Classify each expression as a numerical expression or an algebraic expression
Ex 2
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1-2 Algebraic Expressions
Ex 3
Identify the terms, constants, and coefficients.
1.) 3x + 4 + x
2.) 10 + 4y + 5y + 8
3.) 7k + 9k + 2k + k
Terms
Constants
Coefficients
1-3 Writing Algebraic Expressions
Ex 1
Let Statement -
Ex 2
A state park has 3 lakes. In the spring, each lake was stocked with the same number of fish. Write an
algebraic expression to represent the total number of fish in the lake.
Ex. 3
Spring Lake is 3 miles shorter than Grand lake. Write an expression that represents the length of Spring
Lake where l is the length of Grand Lake.
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1-3 Writing Algebraic Expressions
Ex 4
Ex 5
A national park charges $25 per bus and $12 per
person for an organized tour group. Write an algebraic
expression to represent the total cost for a one-day
tour with one bus and n people?
A national park charges $26 per adult and
$16 per child for rafting down one of their two
rivers. Write an algebraic expression that can be
used to represent the total cost for a
adults and c children to raft down the Wild
River?
1-4 Evaluating Algebraic Expressions
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1-4 Evaluating Algebraic Expressions
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1-4 Evaluating Algebraic Expressions
Ex. 3
A dog walker charges $10 to walk a large dog and $6 to walk a small dog. Write an expression to
represent how much the dog walker earns for walking b large dogs and s small dogs. How much will he
earn for 8 large dogs and 3 small dogs?
1-5 Expressions with Exponents
Power – a way to write repeated multiplication
Base – the repeated factor
Exponent – the number of times a factor is repeated
Special Powers
Squared –
Cubed –
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1-5 Expressions with Exponents
Ex. 1
Ex. 2
Evaluate the following expressions.
a.) 43 + 6
b.) 2x2 – y2 when x = 12 and y = 8
Example 3
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c.) 52 – 3(4 + 2)
Topic 2: Equivalent Expressions
2-1 The Identity and Zero Properties
Identity Property of Addition:
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Identity Property of Multiplication:
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Zero Property of Multiplication:
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Ex 1
Tell which property each statement represents.
1×a=a
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1 × 99 = 99 _____________________________
1+0=1
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3×0=0
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x+0=x
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w×0=0
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Ex 2
At a carnival, you earn 1 point for each balloon you pop with a dart. You pop b balloons. Write two
equivalent algebraic expressions that show how many points you earned.
Ex 3
Hockey teams earn 3 points for a win, 0 points for a loss, and 1 point for a tie. Use the expression below
to find the total number of points Team A and Team B have earned. Name the properties used to
simplify the expression.
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2-2 The Commutative Property
Commutative Property of Addition:
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Commutative Property of Multiplication:
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Ex 1 Write an equivalent expression to the given expression using the commutative property
a.)
b.)
c.)
d.)
Ex 2
In the first 10 months of the year, California experienced 555 earthquakes. The remaining two months of
the year are predicted to have e earthquakes. Write 2 equivalent expressions to show the total number
of earthquakes that California will have in that year.
Ex 3 While planning dinners, Sam, Carl, Jack, and Mary wrote their initials beneath the meals that they
would make for the family. Use the commutative property of multiplication to write 2 equivalent
expressions that show the total number of meals that will be prepared each week.
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2-3 The Associative Property
Associative Property of Addition:
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Associative Property of Multiplication:
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Ex 1 Use the associative property to write an equivalent expression for the given expression.
a.)
b.)
c.)
d.)
Ex 2
A pilot’s schedule for the month of May is 63 hours in-flight, 75 hours at work but not flying, and 25
hours on-call. Use the associative property of addition to write 2 equivalent expressions to find the total
number of hours the pilot worked during the month of May.
Ex 3
An airline ships 3,000 lb. of salmon each day. How many pounds of salmon are shipped in w weeks? Use
the associative property of multiplication to write 2 equivalent expressions.
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Special: Divisibility Rules/Prime Factorization
Divisibility – the ability to divide a number into another number evenly (NO REMAINDERS!)
Divisibility Rules
A number is divisible by:
4,212
2 if the number is even
3 if the sum of the digits is divisible by 3
4 if the last two digits of the number are divisible by 4
5 if the number ends in five or zero
6 the number is divisible by 2 and 3
9 if the sum of the digits is divisible by 9
10 if the number ends in zero
Ex 1 YES or NO…are these numbers divisible by…..
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99
6, 060
2:
2:
2:
3:
3:
3:
4:
4:
4:
5:
5:
5:
6:
6:
6:
9:
9:
9:
10:
10:
10:
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Special: Divisibility Rules/Prime Factorization
factor – a number that divides into another number without a remainder
prime number: a number larger than 1 that only has one and itself as a factor
2, 3, 5, 7, 11, 13, 17, 19, 23, …..
composite number: a number that has more than one and itself as factors
4, 6, 8, 9, 10, 12, 14 15,……..
Ex 2 Write whether each number below is prime or composite and list all the factors.
17
46
91
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Factor trees are a way to write the prime factorization of a number.
Prime Factorization – showing a number as the product of prime numbers
Ex 3 Let’s make some factor trees for the numbers below.
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36
44
16
100
2-4 Greatest Common Factor
Common Factor: a factor that 2 or more numbers share
Ex –
Greatest Common Factor (GCF): the largest of the common factors
There are 2 ways you can find the GCF of a set of numbers:
1.) Listing all the factors of each
2.) Using Prime factorization
Ex. 1 Find the GCF of 36 and 54 by listing the factors.
Prime Factorization Method
1.) Make a factor tree for each number
2.) Write the prime factorization of each number (do not use exponents)
3.) Circle sets of common factors
4.) Multiply one from each set
Ex. 2
30 and 75
Ex. 3
24 and 15
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Ex. 4
20 and 49
2-4 Greatest Common Factor
Ex. 5
60 and 88
Ex. 6
30, 25, and 90
Ex. 7
I am handing out treats to my classes. I have 150 blue jolly ranchers and 200 pink jolly ranchers. Every
class will get the same number of each color to split. What is the greatest number of each color that I
can give to each class?
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2-5 The Distributive Property
Distributive Property – each term inside a set of parentheses can be multiplied by a factor outside the
parentheses to produce an equivalent expression.
Ex –
Ex 1 Which rocks show the distributive property? Cross out the ones that do not.
Ex 2 Use the Distributive property and common factors to write an equivalent expression.
a.) 15 + 35
b.) 36 – 27
Ex 3
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2-5 The Distributive Property
Ex 4 Use the distributive property to write an equivalent expression.
a.) 5(3x + 10)
b.) 7(4y – 5)
Ex 5 Use the distributive property and common factors to write an equivalent expression.
a.) 20y + 15
b.) 12x – 16y
c.) 17x + 12x
Ex. 6 A family makes a budget for a camping trip to Spring Lake. Write two equivalent expressions that
shows the cost for camping and groceries for d days.
Ex. 7 A camper canoes for 2 hours and hikes for 3 hours each day. The number of hours she hikes and
canoes in d days is represented by the expression d(2 + 3). Use the distributive property to write an
equivalent expression that also shows the number of hours the camper canoes and hikes in d days.
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2-6 Least Common Multiple
multiple: the number multiplied by any other number other than zero
Ex -
LCM Least Common Multiple: the least multiple shared by the numbers
There are 2 ways to find LCM:
1.) Listing the Multiples
2.) Using Prime Factorization
Ex. 1 Find the LCM of 80 and 30 by listing the multiples.
Prime Factorization Method
1. Make factor trees for each number.
2. List the prime factorizations in exponent form.
3. Circle the largest power of a # and cross out any others with the same base
4. Multiply the circled numbers.
Ex. 2
Ex. 3
40 and 60
42 and 72
Ex. 4
49 and 20
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2-6 Least Common Multiple
Ex. 5
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Topic 3: Equations and Inequalities
3-1 Expressions to Equations
Equation – a mathematical sentence that has an equal sign comparing two expressions.
Ex.1 Tell which are expressions and which are equations.
Expressions
Equations
True Equation – has equivalent expressions on each side of the equal sign.
False Equation – has expressions that are not equivalent on each side of the equal sign.
Open Sentence – an equation that includes one or more variables.
Solution – a value of a variable that makes the equation true.
Ex. 2
Which equations have a solution of 12?
a.) 17 – m = 5
b.) 3x = 24
c.) 2g – 4 = 20
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3-3 Solving Addition and Subtraction Equations
Inverse Operations – Operations that undo each other.
Ex Solving 1-step Addition/Subtraction Equations
Addition
Subtraction
Ex. 1
Ex. 2
Ex. 3
Ex. 4
Ex. 5
Ex. 6
Ex 7
How many comic books do you need to add to the 17 comic books you already own to have a total of 30
comic books? Write an equation to model the situation along with a let statement identifying your
variable.
Ex 8
You have a deck of trading cards. You gave 19 cards to a friend and have 5 left for yourself. How many
cards were in the original deck? Write an equation to model the situation along with a let statement
identifying your variable.
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3-4 Solving Multiplication and Division Equations
Steps for Solving
Multiplication Equations
Division Equations
Ex. 1
Ex. 2
Ex. 3
Ex. 4
Ex. 5
Ex. 6
Ex. 7
An octopus’s suction cups are evenly divided among its 8 arms. There are 240 suction cups in all. Write
and solve a multiplication equation with a let statement to find the number of suction cups on each arm.
Ex. 8
A band director wants to make 3 groups of 10 students. Write and solve a division equation with a let
statement to find the total number of students in the band.
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3-5 Equations to Inequalities
Differences Between Equations and Inequalities
Equations –
Inequalities –
Symbol
Meaning
Graphing Inequalities
Open Points
Algebraic
Numerical
Closed Points
Sentence Example
Examples
When shading your number line:
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Ex. 1 Write a let statement and inequality for the situation described.
A.) My age is greater than 36.
B.) The candy left in the bag is 66 pieces or less.
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3-5 Equations to Inequalities
Ex. 2
Decide whether each situation could be represented by an equation or an inequality. Next, write the
equation or inequality. Finally, graph it.
A.) John rode his scooter 2 miles.
B.) The speed limit is 45 miles per hour.
C.) The voting age is 18 and older.
D.) To go on the kiddie rides, you must be 36
inches or less
E.) The movie ticket costs $6.
F.) My mom gave me less than $10 to spend on
snacks.
3-6 Solving Inequalities
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3-6 Solving Inequalities
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Topic 4: Two-Variable Relationships
4-1 Using 2 Variables to Represent a Relationship
Related Unknown Quantities – Quantities that when a change happens in one, it affects the other.
Ex 1
A fruit basket business takes 103 orders in one month. Orders can be placed online or over the phone.
Identify the related and unknown quantities.
Ex 2
The fruit basket business takes 312 orders in one month. Orders can be shipped overnight or standard
delivery. Identify the related and unknown quantities.
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4-1 Using 2 Variables to Represent a Relationship
Ex 3
After-school practice is 90 minutes long. Some of that time is spent doing warm-ups and the rest is spent
doing drills. Write an equation to represent the situation.
Ex 4
The fruit basket company offers gift wrapping for an additional charge. Some customers choose to have
their order gift-wrapped and others do not. One month, 145 people do not want their order giftwrapped. Write an equation to represent the situation.
Dependent Variable – changes in response to another variable.
Independent Variable – affects change on the dependent variable.
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Special: Graphing Linear Functions using a table
Function – a pairing of each number in one set with a number in a second set.
Linear Function – a function whose graph is a straight line.
Input – a number in the first set (the independent variable, the x-value)
Output – the number in the second set that pairs with exactly one input (the dependent variable, the yvalue).
Function Rule – an equation that relates the input to the output (how the independent variable affects
the dependent variable, how the x-value affects the y-value).
How to graph using an input/output table:
1.) Write the values of x in the first column
2.) Use the function rule and plug in the values of x to find the y (SHOW WORK IN THE CENTER
COLUMN)
3.) Write the values found for y in the third column
4.) Write the ordered pairs you want to graph
5.) Graph your ordered pairs (x-tells you how much to move right, y-tells you how much you
move up)
Make an input-output table using the function rule and the input values x = 0, 4, 6, and 8, then graph the
function.
Ex. 1 y = x + 2
x
y
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Special: Graphing Linear Functions using a table
Ex. 2 y = 8 – x
x
y
Ex. 3 𝒚 =
𝒙
𝟐
x
Ex. 4
x
y
𝒚=
𝟏
𝒙+
𝟐
𝟑
y
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4-2 Analyzing Patterns Using Tables and Graphs
Ex. 1
Ex 2
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4-2 Analyzing Patterns Using Tables and Graphs
Ex. 3
Ex 4
Acres
# of Trees
0
0
5
3100
10
6200
15
20
25
Ex 5
Year
Height (ft.)
0
0
1
12
2
24
3
4
5
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4-3 Relating Tables and Graphs to Equations
Ex. 1
Ex. 2
*look for a pattern! (It should be what you do to x to get to a value for y).
Ex. 2
Use the graph to complete the table, then write an equation that represents a relationship between x
and y.
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4-3 Relating Tables and Graphs to Equations
Ex. 3 Write an equation that represents the relationship between x and y. Use the graph to help you
come to your conclusion.
x
1
2
3
5
9
y
10
9
8
6
2
Ex. 3 Complete the table and then write an equation that shows the relationship between the
number of games bowled and the total cost. (It costs $3 to bowl a game and $5 to rent shoes).
Ex. 4
Write an equivalent equation to the given equation.
Original: x + y = 19
Equivalent Equations:
1.) Can be found by subtracting x from both sides.
2.)Can be found by subtracting y from
both sides.
3.) Can be found by using the commutative property.
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Special: Simplifying Fractions and Converting Mixed Numbers and Improper Fractions
Simplifying Fractions – finding a common factor of the numerator and denominator and dividing it out
until they have a denominator of 1 (If you are stuck, refer back to page 13 for divisibility rules).
Ex. 1
Ex. 2
Ex. 3
Converting Mixed Numbers to Improper Fractions
1.) Multiply the denominator by the whole number
2.) Add the numerator to the resulting product
3.) Place that over the original denominator
Ex. 4
Ex. 5
Ex. 6
Converting Improper Fractions to Mixed Numbers
1.) Divide the numerator by the denominator until you get a remainder
2.) The quotient becomes the whole number
3.) The remainder becomes the new numerator
4.) The denominator stays the same
Ex. 7
Ex. 8
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Ex. 9
Special: Adding and Subtracting Fractions
Adding/Subtracting Fractions
1.) Rename the fractions so they have the same denominators if necessary
2.) Add/Subtract the fractions
3.) Add/Subtract the whole numbers
4.) Simplify
Ex. 1
Ex. 2
Ex. 3
Ex. 4
Ex. 5
Ex. 6
Ex. 7
Ex. 8
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Special: Adding and Subtracting Fractions
Steps for Subtracting Fractions if borrowing is necessary
1.) Rewrite fractions to have common denominators
2.) Borrow (look at common denominator and add that to the numerator of the top fraction)
3.) Subtract
4.) Simplify
Ex. 9
Ex. 10
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Ex. 11
Topic 5: Multiplying Fractions
5-1 Multiplying Fractions with Whole Numbers
Ex 1
Ex 2
Multiplying Fractions and Whole Numbers
1.)
2.)
3.)
4.)
Change the Whole number to a fraction (place it over one)
Multiply the numerators
Multiply the denominators
Simplify (as a fraction in simplest form or as a mixed number)
Ex 3
Ex 4
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5-2 Multiplying Fractions 2 Fractions
How to show a model of multiplying a fraction by a fraction:
1.) Draw a rectangle
2.) Draw the number of rows as the number in the denominator of the first fraction (shade the
fraction that is shown)
3.) Draw the number of columns as the number in the denominator of the second fraction (shade
the fraction that is shown)
4.) Count how many total boxes within the rectangle that is the denominator of the product
5.) Count the number of shaded boxes that overlap that is the numerator of the product
6.) Simplify
Ex 1
Multiplying Fractions
1.) Simplify numbers in the numerators with numbers in the denominators (if possible… this will
make your multiplication easier)
2.) Multiply the numerators
3.) Multiply the denominators
4.) Make sure your answer is in simplest form
Ex 2
Ex 3
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Ex 4
5-2 Multiplying Fractions 2 Fractions
Ex 5
5-3/5-4 Multiplying Mixed Numbers
Ex. 1
Ex. 2
1
1
Use the Area Model to model 2 3 × 3 4
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5-3/5-4 Multiplying Mixed Numbers
Multiplying Fractions (the final List)
1.) Change any mixed number to an improper fraction.
2.) Simplify early if possible
3.) Multiply Numerators
4.) Multiply Denominators
5.) Make sure your answer is in simplest form (change any improper fractions back to a mixed number)
Ex 3
Ex. 4
Ex. 5
Ex. 6
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3
An Ipod has 7 2 gigabytes of memory available. If you use 5 of that memory for music, how many
gigabytes are used for music?
Ex. 7 An art club has made a mural for the wall of their school. Using the diagram of the mural, what is
its area?
Ex. 8
The table below shows the trails in Diablo State Park in Walnut Creek, CA. A jogger has decided to run all
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4 trails 3 2 times. How many miles did the jogger run?
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6-1 Dividing Fractions and Whole Numbers
Ex 1
Ex 2
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6-1 Dividing Fractions and Whole Numbers
Reciprocals – If two numbers, when multiplied together, produce a product of 1.
Ex 3
Ex 4
Ex 5
Dividing Fractions, Whole Numbers, and Mixed Numbers
1.)
2.)
3.)
4.)
5.)
6.)
Change any mixed number or whole number to an improper fraction
Change the division sign to a multiplication sign and the SECOND fraction to its reciprocal
Simplify if possible
Multiply the numerators
Multiply the denominators
Make Sure your answer is in simplest form
Ex 6
Ex 7
45
6-2/6-3 Dividing Fractions
Ex 1
Ex 2
46
6-2/6-3 Dividing Fractions
Ex 3
Part 2
Dividing Fractions, Whole Numbers, and Mixed Numbers
1.)
2.)
3.)
4.)
5.)
6.)
Change any mixed number or whole number to an improper fraction
Change the division sign to a multiplication sign and the SECOND fraction to its reciprocal
Simplify if possible
Multiply the numerators
Multiply the denominators
Make Sure your answer is in simplest form
Ex 4
Ex 5
Ex 6
47
6-4 Dividing Mixed Numbers
Dividing Fractions, Whole Numbers, and Mixed Numbers
1.)
2.)
3.)
4.)
5.)
6.)
Ex 1
Change any mixed number or whole number to an improper fraction
Change the division sign to a multiplication sign and the SECOND fraction to its reciprocal
Simplify if possible
Multiply the numerators
Multiply the denominators
Make Sure your answer is in simplest form
Ex 2
Ex 4
Ex 5
48
Ex 3
Special: Solving Equations with Fractions and Mixed Numbers
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Topic 7: Fluency with Decimals
7-1 Adding and Subtracting Decimals
Steps for Adding and Subtracting Decimals
1.)
2.)
3.)
4.)
Line up the decimal points
Add zero place holders when necessary
Add/Subtract as normal
Place decimal point in the answer directly in line with the decimal point in the problem
Ex. 1
Ex. 2
Ex 3
Ex 4
Ex 5
50
Special: Place Value and Rounding
Ex. 1 State the place value of the given digit.
a.)
b.)
c.)
Rounding Rules:
1.) Underline the place value you want to round to
2.) Look at the digit behind it: 4 or below leave it alone
5 or above add one
3.) Drop all digits after the underlined digit
Ex. 2
Round to the indicated place value
a.)
b.)
d.)
e.)
c.)
51
7-2 Multiplying Decimals
Steps for Multiplying Decimals
1.)
2.)
3.)
4.)

Place the number with the most nonzero digits on top
DO NOT line up the decimal points
Multiply as normal
Count the number of decimal points in the problem and place the same amount in the answer
If you do not have enough digits in your answer for the decimal places, place additional zeros IN
FRONT of the digits you got from multiplying
Ex 1
Ex 2
Ex 3
Ex 4
0.7 × 0.08
4.29(5.03)
4.8 × 3.235
0.066 × 0.05
Ex 5
Find the area of the mini-tv shown below.
52
7-2 Multiplying Decimals
Ex 6
* Rounding to the nearest cent
means rounding to the nearest
hundredth (because you can’t
have less than a whole penny)
Ex 7
53
7-3 Dividing Multi-Digit Numbers
1.)
2.)
3.)
4.)
Divide
Multiply
Subtract
Bring down
Remember IOU (Inner divided by the outer gives U (you) the answer)
Ex. 1
Ex. 2
Ex. 3
Ex. 5
54
Ex. 4
7-4 Dividing Decimals
Remember your IOU!
-Be sure to place your decimal point in your answer directly in line with the one in the dividend (inner
number).
Ex 1
Ex 2
Steps for Dividing a Decimal by a Decimal
1.) Move the decimal point in the divisor (second number) so it becomes a whole number
2.) Move the decimal point in the dividend (first number) the same number of times
3.) Divide as normal (IOU) adding zeros to bring down when necessary until you get a remainder of
0, the digits begin to repeat, or the problem asks you to round to a certain place value.
4.) Place decimal point in answer directly in line with the decimal point in the dividend (inner
number)
Ex 3
Ex 4
55
Ex 5
7-4 Dividing Decimals
Ex 6
Last week, a 6th grade class received $10.32 for aluminum cans they recycled. The scrap yard paid them
$0.48 for each pound. How many pounds of cans did the class recycle?
Ex 7
A shelf used to store DVDs is 60.96 cm long. If each DVD is 1.5 cm wide, what is the maximum number of
DVDs that can be stored on the shelf?
7-5 Decimals and Fractions
Common Fractions and Decimal Equivalence
Fraction
Decimal
Fraction
𝟐
𝟓
𝟑
𝟓
𝟒
𝟓
𝟏
𝟑
𝟐
𝟑
𝟏
𝟒
𝟏
𝟐
𝟑
𝟒
𝟏 𝟐 𝟑
,
,
, 𝒆𝒕𝒄.
𝟏𝟎 𝟏𝟎 𝟏𝟎
𝟏
𝟓
56
Decimal
7-5 Decimals and Fractions
Decimals  Fractions
1.) The denominator will be the same as the place value of the last digit of the decimal (ex if the last
digit is in the tenths place the denominator is a 10, hundredths place the denominator is 100).
2.) Place anything left of the decimal point as a whole number and anything to the right of the
decimal in the numerator.
3.) Simplify
Ex. 1
Ex. 2
Fractions  Decimal
𝐼
1.) Divide the numerator by the denominator (your IOU looks like this 𝑂 = 𝑈)

If you see a number begin to repeat, round to the indicated place value (this means divide to
one place PAST the indicated place value, and then round.)
Ex. 3
Ex. 4
Ex. 5
Round to the nearest hundredth
Ex. 6
Round to the nearest tenth
57
7-5 Decimals and Fractions
Changing a fraction to a decimal without dividing
Factors of 100 – 1, 2, 4, 5, 10, 20, 25, 50, 100
- If you have these numbers as a denominator, you can find the decimal equivalence by using equivalent
fractions.
Ex. 7
a.)
b.)
7-6 Comparing and Ordering Decimals and Fractions
-When comparing decimals you must compare the digits from left to right (once the digits stop being the
same compare to see which is greater in that place value)
Ex 1
* BE CAREFUL TO SEE IF YOU ARE ORDERING FROM LEAST TO GREATEST OR GREATEST TO LEAST!
Ex 2
Compare
58
7-6 Comparing and Ordering Decimals and Fractions
In order to compare fractions and decimals, you need to change the numbers so that they are all
fractions or all decimals (I don’t care which you choose)
For Examples 3 – 5 Compare the numbers
Ex 3
Ex 4
Ex 6
59
Ex 5
Topic 8: Integers
8-1 Integers and the Number Line
Integers:
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Positive Numbers:
_____________________________________________________________________________________
Negative Numbers:
_____________________________________________________________________________________
Opposites:
_____________________________________________________________________________________
Draw a number line from -10 to 10. Label the area where there are positive numbers, negative numbers,
and show one set of opposite numbers.
Ex. 1
What is the opposite of 9?
Ex. 2
What is the opposite of -3?
Ex. 3
Ex. 4
The highest point as an integer: ___________
The lowest point as an integer: ___________
60
8-2 Comparing and Ordering Integers
On a number line, the more right a number is when you graph it, the larger the number is.
Ex
Ex
Ex 1
Ex 2
61
8-3 Absolute Value
Absolute Value – The distance a number is from zero on a number line. (Always positive)
Symbol
Ex 1 Find the absolute value.
a.)
b.)
c.)
Ex 2
Ex 3
Depth will always be measured as an absolute value. The larger the absolute value the deeper a diver is.
62
8-4 The Coordinate Plane
Graphing Ordered Pairs (points on a coordinate plane)
Ordered Pair (x, y)
(1st # right (+) or left (-), 2nd # up (+) or down (-))
Ex 1
Graph the following ordered pairs and give its location
A (2, -3)
_____________________
B (0, -2)
_____________________
C (-1, 4)
_____________________
D (3, 0)
_____________________
63
8-4 The Coordinate Plane
Ex 2
State the ordered pairs for the locations on the graph.
Hospital:
________________
School:
________________
Grocery Store: ________________
Library:
________________
Transformation – when a figure changes size, position, or shape on a coordinate plane.
Reflection – a transformation that “flips” over a line called the line of reflection to produce a mirror
image.
Reflection on the x-axis
(x, y)  (x, -y)
Reflection on the y-axis
(x, y)  (-x, y)
Ex 3
a.)
b.)
c.)
64
8-5 Distance
To find distance between 2 numbers on a number line:
1.) Same Signs (both positive or both negative) – Subtract the smaller absolute value from the
larger absolute value.
2.) Different Signs ( one positive and one negative) – Add the absolute values
Ex 1
Ex 2
Distance on a coordinate plane
Vertical Distance – add/sub the absolute values of the y-coordinate depending on the sign.
Ex 3
Ex 4
65
8-5 Distance
Horizontal Distance – add/subtract the absolute values of the x-coordinate depending on the sign.
Ex 5
*To double check you can always count the spaces between the points on a number line.
Ex6
Ex 7
66
8-6 Problem Solving
Ex. 1
Ex 2
Ex 3
Ex.4
67
Special: Adding Integers
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Special: Subtracting Integers
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Special: Multiplying and Dividing Integers
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70
Topic 9: Rational Numbers
9-1 Rational Numbers and the Number Line
Rational Numbers – Numbers that can be written in the form
𝑎
𝑏
𝑎
𝑏
𝑜𝑟 − where a is a whole number and b
is a positive whole number.
Ex 1
Prove that the number is a rational number.
a.)
b.)
c.)
Ex 2
Ex 3
Explain what the number 0 represents in each situation. Then write a rational number to represent the
situation.
2
The pant leg shrunk 3 inch after washing.
_______________________________________
Dad gained 1 pounds during vacation.
1
4
_______________________________________
It is 10 seconds to blast off.
_______________________________________
71
9-2 Comparing Rational Numbers
*In order to compare rational numbers, they must be the same type of numbers – all fractions or all
decimals.
Ex 1
<, >, or =
Ex 2 <, >, or =
Ex 3
72
9-3 Ordering Rational Numbers
Ex 1: Order the numbers from least to greatest
Ex 2: Order the numbers from greatest to least
Ex 3: Order the animals depths from the least depth to the greatest depth
Ex 4:
Accountants use parentheses as negative signs. (Example ($209) means -209 dollars). Whenever you see
an accounting number in parentheses, you can write is as a negative number, showing a
______________ or __________________ balance.
**** Outside of accounting, however, you should NOT interpret a number in parentheses to be a
negative number****
Use rational numbers to write an expression that compares an income of ($41.16) to an income of
($14.61), interpreted as size of loss.
73
9-4 Rational Numbers and the Coordinate Plane
Ex 1
Give the ordered pair of each point on the coordinate plane then tell which quadrant the point is
located in.
Ex 2
74
9-4 Rational Numbers and the Coordinate Plane
Ex 3
*Remember:
Reflection on the x-axis
(x, y)  (x, -y)
Reflection on the y-axis
(x, y)  (-x, y)
9-5 Polygons and the Coordinate Plane
Polygon – a closed figure formed by 3 or more line segments that do not cross.
Regular Polygon – a polygon whose sides and angles are all equal in measure.
Examples
Non-examples
Vertex – a point where any 2 sides of a polygon meet.
Ex –
75
9-5 Polygons and the Coordinate Plane
Types of Polygons
Name
Number of Sides
What it looks like
76
9-5 Polygons and the Coordinate Plane
Ex 1
Ex 2
Find the length of segments AC and BC.
AC =
BC =
77
9-5 Polygons and the Coordinate Plane
Ex 3
Plot the following points and connect them to complete a polygon.
(-4, 5) (-6,2)
(-2,2)
Now, reflect the original polygon over the y-axis.
The new coordinates are:
Finally, reflect the original polygon over the x-axis.
The new coordinates are:
Ex 4
78
Topic 10: Ratios
10-1 Ratios
Ratio:
_____________________________________________________________________________________
Terms of a Ratio:
_____________________________________________________________________________________
Ex 1
Ex 2
Write 2 different ratios to compare the number of headphones to the number of MP3 players.
Ex 3
You toss a coin 12 times and get heads 7 times. Write the ratio of the number of heads to the number of
tails.
79
10-1 Ratios
Ex 3
Use the table below to state whether the ratios are true or false. If the statement is false, correct the
statement.
a. The ratio of votes for Jordyn to total votes is
10:35
b. The ratio of votes for Devin to votes for
Jordyn is 10 to 20.
c. The ratio of votes for Micah to total votes is
5:30.
d. The ratio of votes for Devin to votes for
Micah is 20:5.
e. The ratio of total votes to votes for Devin is
35 to 20.
10-2 to 10-4 Equivalent Ratios
Equivalent Ratios – ratios that express the same relationship
Lesser Terms – terms that are smaller than the original terms
Greater Terms – terms that are larger than the original terms
Ex 1
80
10-2 to 10-4 Equivalent Ratios
Ex 2
Write one ratio using lesser terms and one ratio using greater terms for the given ratio.
Ex 3
In one class, 3 out of 8 students have braces. There are 32 students in the class. How many students
have braces?
Ex 4
A volunteer at an animal shelter recorded the ratio of the number of dogs adopted to the number of
cats adopted each month. She finds that all ratios are equivalent. Complete the table.
Ex 5
a.) the number of students who play
the saxophone to the number of
students who play the drums.
b.)The number of students who play the
flute to the total number of students
Ex 6 Write 2 equivalent ratios to
10
.
25
Ex 7 Simplify the ratio 12 to 40 as a fraction.
81
10-5 Ratios as Decimals
Steps for writing the ratio as a decimal
1.) Write the ratio as a fraction
2.) Divide the numerator by the denominator to change that fraction to a decimal
Ex 1
Ratio
Fraction
Decimal
3:8
Ex 2
Write the ratio as a fraction in simplest form.
0.45
Changing a Decimal Ratio to a fraction
1.) Count the number of decimal places
in the number
2.) Place a 1 plus 0s for the number of
decimal places in the problem in the
denominator (2 decimal places 100, 3
decimal places 1000, etc.)
3.) Place everything else in the
numerator
Ex 3
A batter’s average is 0.350 in 60 at bats. How many hits did the batter have?
1.) Change the decimal to a
fraction
2.) Simplify
3.) Set up 2 equivalent ratios to
find the missing piece.
82
10-6 Problem Solving
Ex. 1
Fill in the table using equivalent ratios.
Ex 2
*** Make a table to help you solve this.
83
Topic 11: Rates
11-1/11-2 Unit Rate and Unit Price
Ex. 3
1.) Set up the ratios for each class (or item you
are looking for in other types of problems)
2.) Write equivalent ratios to find the different
combinations of fish that add up to the total
number of fish.
Ex. 4
A gym teacher has 15 soccer balls and 16 footballs. The teacher wants to put twice as many footballs as
soccer balls in one bin. The teacher wants to put 5 soccer balls for every 3 footballs in another bin. How
many soccer balls and footballs should the teacher put in each bin?
84
11-1/11-2 Unit Rate and Unit Price
Rate – A ratio that compares 2 quantities that are measured in different units.
Ex –
Unit Rate – A rate for one unit of a given quantity. When a unit rate is written as a fraction the
denominator is 1.
Unit Price – The price of an item per unit.
Ex –
Unit Rate
Unit Price
Ex. 1
A box of crackers contains 84 crackers and has a total of 7 servings. How many crackers are there per
serving?
Write the rate
Divide both the numerator and the denominator by the denominator to find the unit rate.
Ex. 2
In 16 years the trunk of a tree grew approximately 4 inches. How much did the tree trunk grow per
year?
Ex. 3
Your dishwasher uses 11 gallons of water to wash 2 loads of dishes. How many gallons of water will your
dishwasher use to wash 7 loads of dishes?
85
11-1/11-2 Unit Rate and Unit Price
Ex. 4
A 5-minute shower uses approximately 12 gallons of water. Each minute the shower is running, the rate
of water used is the same. Use this rate to complete the table.
*Make a unit rate first!
Ex. 5
86
11-3 Constant Speed
Constant Speed – Comparing something’s distance traveled to time traveled.
Ex –
Equation
d = rt
where d = distance, r = rate of travel (constant speed), and t = time
Steps:
1.) Write the formula
2.) Substitute in for what you know
3.) Solve the resulting equation (show balance if necessary)
 When finding distance this means – multiplying the rate and time
 When finding rate this means – dividing both sides of the equation by the time
 When finding time this means – multiply both sides of the equation by the reciprocal of the rate
Ex 1
Your aunt drives at a constant speed of 45 miles per hour. How far will your aunt travel in 20 minutes?
*watch for time not
being in the same unit!
Ex 2
Lisa bikes at a constant speed of 8 miles per hour. If she bikes for 30 minutes, how far does she travel?
87
11-3 Constant Speed
Ex 3
Ex. 4
A bus travels 70 miles in 2 hours. What is the speed of the bus in miles per hour?
Ex. 5
Marie and her brother are driving from city A to city B. The two cities are 318 miles apart. Marie drives
53 miles per hour. How long does it take them to make the trip?
Ex. 6
On a busy road, cars can travel only 5 miles in 10 minutes.
a.) At this speed, how far will a car travel in 15 minutes?
88
*Find the constant speed first!
b.) At this speed, how long will it take
the car to travel 20 miles?
11-4 Measurements and Ratios
Conversion Factor – A relationship between 2 equivalent measurements of different units. It will always
equal 1.
Ex –
1 pound = 16 ounces
1 ton = 2000 pounds
1 foot = 12 inches
1 yard = 3 feet = 36 inches
1 mile = 1760 yards = 5280 feet
1 cup = 8 fluid ounces
1 pint = 2 cups
1 quart = 2 pints
4 quarts = 1 gallon
Steps for converting units
1.) Write the unit you have as a fraction
2.) Multiply that by a conversion factor from the chart (a second fraction that is
𝑢𝑛𝑖𝑡 𝑦𝑜𝑢 𝑤𝑎𝑛𝑡 𝑡𝑜 𝑐ℎ𝑎𝑛𝑔𝑒 𝑡𝑜
)
𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑢𝑛𝑖𝑡
3.) Simplify your answer making sure you label the new unit
Ex 1
Ex 2
Ex 3
4 pints to cups
35 oz to pounds
Ex 4
89
3
5
2 𝑇 𝑡𝑜 𝑝𝑜𝑢𝑛𝑑𝑠
11-4 Measurements and Ratios
Is a rider 3 feet 10 inches tall allowed to ride this roller coaster? Use 1 in. = 2.54 cm.
Ex. 5
A bread recipe calls for 500 grams of flour. About how many pounds of flour do you need? Use 1 oz =
28.4 g
Ex. 6
You have 2 gallons of bubble solution. About how many liters of bubble solution do you have?
Use 1 qt. = 0.95 L
90
Topic 12: Ratio Reasoning
12-1 Plotting Ratios and Rates
Ex 1
𝑦
Ratios are always plotted as 𝑥 where y is the y-coordinate and x is the x-coordinate. This is always
2
constant. You will use this fact to write equivalent ratios (Ex 3 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑎𝑠
Ex 2
Ex 3
91
4
6
)
12-2 Understanding Proportionality
Equivalent ratios form a proportional relationship.
Ex. 1 Tell whether the ratios form a proportional relationship.
a.)
b.)
c.)
d.)
Proportional Relationships can be shown with a table, graph, or equation.
Table
Graph
Ex. 2 Does each table show a proportional relationship?
Equation
Ex. 3 Do the equations
show proportional
relationship?
4
a.) 𝑦 = 3 𝑥
4
b.) 𝑦 = 3 𝑥 + 3
92
12-3 Introducing Percent
Percent – a ratio that compares a number to 100.
Ex. 1
Ex. 2 Complete the table
Ex. 3
2 out of 5 students have braces. What percentage of students have braces?
93
Topic 13: Area
13-1 and 13-3 Rectangles, Squares, and Parallelograms
Area – the number of square units a figure encloses
Area Formula
Rectangle
A=lw
Area = length × width
Square
A=s2
Area = (side length)2
Parallelogram
A=bh
Area = base × height
*When labeling area you use units2 (ft2, mm2, in2, ect).
Steps for solving for all 2D and 3D figures
1.) Write the appropriate formula
2.) Substitute in the values you know
3.) Solve the resulting equation (showing balance if necessary)
Find the area.
Ex. 1
Ex. 2
Ex. 3
Find the missing dimension.
Ex 4
A rectangle with an area of 72 mm2 and a length of 9 mm.
Find the width.
Decomposing – Breaking up a shape into other shapes.
94
Ex 5
A parallelogram with an area of 15 in2
1
and a base of 2 in. Find the height.
2
13-2 and 13-4 Area of Triangles
Area of a Triangle Formula
𝟏
𝑨 = 𝟐 𝒃𝒉 𝒐𝒓 𝑨 = 𝟎. 𝟓𝒃𝒉
Formula comes from 
Composed Rectangle and Parallelogram
Find the area.
Ex. 1
Ex. 2
Ex 4.
Ex. 3
Ex. 5
You have a triangle with an area of 34
cm2 and a base of 4 cm. What is the
height of the triangle?
95
13-5 Polygons
Formulas
Trapezoid
A = ½h(b1 + b2)
Area = ½height(base1 + base2)
Hexagon
A= 6(½bh)
A= 6(area of 1 triangle)
Octagon
A=8(½bh)
A= 8(area of 1 triangle)
Ex 1
Ex 2
Ex 3
Ex 4
96
13-6 Problem Solving
Find the area of the figure.
Ex. 1
Ex. 2
Find the area of the shaded region.
Ex. 3
Ex. 4
The area of the rectangle is 48 m2. Find
the value of x then the area of the
trapezoid.
97
Special: Interior Angles of Triangles and Quadrilaterals
Classifying Triangles by Sides
Equilateral
Isosceles
Scalene
Obtuse
Right
Acute Angle –
Obtuse Angle –
Right Angle –
Classifying Triangles by Angles
Acute
Find the value of x.
Ex. 1
Ex. 2
Are the angle measures that of a triangle. Justify your answer. If yes, classify the triangle by angles.
Ex. 3
Ex. 4
98
Topic 14: Surface Area and Volume
14-1/14-2 Analyzing 3D Figures and Nets
Face – a flat surface of a 3D figure shaped like a polygon.
Edge – a segment formed by the intersection of 2 faces.
Vertex – a point where 3 or more edges intersect on a 3D figure.
Ex. 1
Prism – a 3D figure with 2 parallel polygonal faces that are the same size and shape.
99
14-1/14-2 Analyzing 3D Figures and Nets
Ex 2
Pyramid – a 3D figure with a base that is a polygon and triangular faces that meet at a vertex.
Ex 3
100
14-1/14-2 Analyzing 3D Figures and Nets
Net – a 2D pattern that you can fold to form a 3D figure.
Ex. 4 Match the figure with its net.
Ex. 5 Which net is the net of a cube?
Ex. 6
Ex. 7 Draw the net for the given triangular prism.
101
14-3 Surface Area of Right Prisms
Surface Area – the sum of the areas of a 3D figures faces.
To find surface area:
1.) Make a net of the solid
2.) Find the area of each polygon
3.) Add all the areas together
Rectangular Prism Formula
Cube Formula
Ex. 1
102
14-3 Surface Area of Right Prisms
Ex. 2
Ex. 3
Ex. 4
103
14-4 Surface Area of Right Pyramids
Ex. 1
Ex. 2
Ex. 3
104
14-5 Volume of Rectangular Prisms
Rectangular Prism Formula
Cube Formula
Labeled as __________
Find the volume of the prism or cube.
Ex. 1
Ex. 2
Ex. 3
105
Topic 15: Data Displays
15-1 Statistical Questions
Statistical Question – a question that investigates an aspect of the world and can have more than one
possible response.
Ex. 1
Ex. 2 Which questions are statistical questions and which questions are not statistical questions?
Data – are numbers or other pieces of information collected by asking questions, measuring, or making
observations about the real world.
Ex. 3
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15-2 Dot Plots
Dot Plots show the shape of a
corresponding value on a number line.
by representing each
point as a dot over its
Frequency describes how often a _________________ occurs.
Ex 1
Your friends hold a basket-ball shooting contest. The person who makes the most baskets in one minute
wins. Use the dot plot to answer the questions.
a. How many people made 8 baskets?
b. How many baskets did the most people make?
c. What is the least number of baskets that a person made?
Ex 2
The data show the age of the dancers on a dance team. Make a dot plot of the data to find out which
age is most common on the team.
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15-2 Dot Plots
Distribution: The way data is
over all values.
Cluster: Area where
dots are stacked
Gaps: Area where there are significantly
number of dots.
Values that stray: Dots that are located
from the main set of data.
Ex 3
The dot plot shows the heights of plants in a research laboratory. Identify the clusters, the gaps, and
any data values that stray. What do they tell you about the heights of the plants?
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15-3 Histograms
Histograms show the shape of a
values on a number line.
with
above intervals of
Ex 1
You are helping a new social networking website company analyze data. Use a histogram to answer the
questions.
How many users were surveyed?
of the users surveyed have 150 friends or more
How many of the users surveyed have 0 and 49 friends ____
50 is in interval __________
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15-3 Histograms
Ex 2
The participants in a ski race are divided into four groups of six skiers each. The table shows the results
of the races. Make a histogram to show how the race times are distributed among all the participants.
Use the intervals 86 – 87.9, 88 – 89.9, 90 – 91.9, 92 – 93.9, 96 – 97.9, 98 – 99.9, 100 – 101.9
Ex 3
Three wind turbines were constructed in your town. Each is expected to generate 3,000 kilowatts (kW)
per day. During the testing phase, engineers recorded the daily amount of energy produced by each
turbine. What does the histogram show about the test results?
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15-4 Box Plots
A box plot shows five boundary values. To find these values you MUST put the data values in numerical
order.
Minimum – the least value in a data set
Maximum – the greatest value in a data set
Middle of the Data (Median) – the middle value in an ordered data set. If there are 2 middle values, you
add the numbers and divide by 2.
Middle of the Lower Half (Lower Quartile or 1st Quartile) – the middle value of the lower half of the
ordered data set. If there are 2 middle values, you add the numbers and divide by 2.
Middle of the Upper Half (Upper Quartile or 3rd Quartile) – the middle value of the upper half of the
ordered data set. If there are 2 middle values, you add the numbers and divide by 2.
Ex. 1
Order the data set. Then identify the 5 boundary values.
30, 16, 68, 35, 57, 5, 27, 76, 21, 91, 44
Box Plot
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15-4 Box Plots
Ex 2
Ex 3
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15-5 Choosing Appropriate Data Displays
Dot Plots are helpful to see:




A
A
A
A
Histograms are helpful to see:



A
A
A
Box Plots are helpful to see:


A
A
Ex. 1
Ex. 2
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15-5 Choosing Appropriate Data Displays
Ex. 3
Ex. 4
You need to be in the top 25% to move on to the second round in your town’s golf tournament. Which
data display would you use if you wanted to see what the lowest possible score is so that you can move
on? Explain.
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Topic 16: Measures of Center and Variation
16-1/16-2 Median and Mean
Measures of Center- A measure that is a value that represents the middle of a data set. There may be
more than one measure of center for a data set (The ______________________ and the
____________________ are both measures of center).
To find the median you must FIRST
__________________________________________________________
Odd number of data values
Even number of data values
Mean – the sum of all the data values divided by the total number of data values in the data set. Also
can be called the “average.”
Ex 1 Find the median of each data set
a.) 7, 8, 9, 12, 14, 16, 17, 19, 20, 50
b.) -132, -105, -19, 16, 17, 17, 22, 25
Ex. 2
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16-1/16-2 Median and Mean
Ex. 3 Find the mean.
24, 27, 30, 31, 33, 35
Ex. 4 Find the value of x.
16.5, 17.5, x
1.) Set up the problem like in example 3
2.) Add the known data values
3.) Multiply each side by the denominator
4.) Subtract that sum from both sides to
Find the value of x.
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16-3 Measure of Variability
Variability – how spread out or not spread out data is
3 Types of Variability
No Variability
High Variability
Low Variability
Measures of Variability – a value that describes the amount of variability in a data set.
Range =
Ex. 1
Find the range and identify any stray value(s).
33, 10, 34, 33, 35
Ex. 2
For a project, the teacher asked a group of seven students to each count the number of pedestrians that
cross the street where they live on a certain day. The data collected has a range of 41. Six of the seven
data values are shown below. Find the missing data value.
29, 20, 25, 11, 34, 20
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16-4 Interquartile Range
Minimum: the LEAST value
Range: maximum - minimum
Maximum: the GREATEST value
First Quartile (lower quartile): the LOWER MIDDLE value
Median: the MIDDLE value
of the data
Interquartile Range (IQR): the middle 50%
Third Quartile (upper quartile): the UPPER MIDDLE value
Ex 1
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16-4 Interquartile Range
Ex 2
Find the range and the IQR of the data set.
110, 123, 3, 8, 64, 2, 35, 45, 12, 3, 33, 23, 91, 64, 12, 8, 150, 42, 40, 76
Ex 3
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16-5 Mean Absolute Deviation
Deviation of Data Values from the Mean – how far away a number is from the mean. If the number is
less than the mean the deviation is negative. If the number is greater than the mean the deviation is
positive.
Ex.
Ex. 1
Absolute Deviation - the distance a data value is away from the mean of the data set (it is the absolute
value of the deviation).
Mean Absolute Deviation (MAD) – the mean of the absolute deviations.
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13-5 Polygons
To find MAD
1.) Find the mean of the data set (if not given)
2.) Find the Deviation of each of the data points (the mean’s deviation is 0)
3.) Change them to their absolute deviations by finding the absolute value
4.) Add all the absolute deviations and divide by the total number of numbers in the data set.
Ex. 2
Ex. 3 Find the MAD of the data set.
1, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7, 8, 8, 9, 9, 9, 10
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