1
Evaluate.
1. 6 + (-14) 2. -40 ÷ 5 3. -8 ∙ -7 4. -3 – 19
5. -17 + (-12) 6. 4 – 28 7. -9 ∙ 6 8.
Operation Rule Examples
Addition
Same signs: Add and keep the sign
Different signs: Subtract and keep
the sign of the number with the
larger absolute value.
-7 + (-9) = -16
7 + 9 = 16
-8 + 10 = 2
8 + (-10) = -2
Subtraction
Use Keep, Change, Change
Keep the first integer the same
Change the operation to addition
Change the sign of the second
integer to its opposite
Then follow the rules of addition.
-9 – 4 = -9 + (-4) = -13
10 – (-6) = 10 + 6 = 16
-2 – (-11) = -2 + 11 = 9
-13 – (-5) = -13 + 5 = -8
Multiplication
Positive • Positive = Positive
Negative • Negative = Positive
Positive • Negative = Negative
Negative • Positive = Negative
5 • 5 = 25
-5 • -5 = 25
6 • -5 = -30
-6 • 5 = -30
Division
The rules for division are the same
as the rules for multiplication.
27 ÷ 9 = 3
-27 ÷ -9 = 3
-4 ÷ 2 = -2
4 ÷ -2 = -2
Integer Rules Review
2
Translating Expressions with Multiple Operations: (When order DOES NOT matter, always write the term with the VARIABLE 1
st and write the
CONSTANT LAST)
1) Eight less than four times a number z.
2) Four times the quantity six less a number n.
3) Three times the sum of a number x and nine.
4) The quotient of twice a number x and ten is three.
5) Two more than the product of nine and a number is less than forty-seven.
Operation Verbal Phrase Expression
Addition:
Sum, Plus, Total, More than,
Increased by
The sum of two and a number x
A number n plus seven
Subtraction:
Difference, Less, Less than,
Minus, Decreased by
The difference of a number n and
six.
Six less a number y
Six less than a number y
Multiplication:
Times, Product,
Multiplied by, Of
The product of twelve and
a number y
One third of a number x
Division:
Quotient, Divided by,
Divided into
The quotient of a number k and 2
Learning Target: SOL A.1: The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables.
Translating and Evaluating
3
Evaluating Expressions
6) 7 + 64 ÷ 23 • 4
7)
8
2 – 10a + 7 when a = 6
8)
when x = 5 and y = 2
You Try It! Directions: Translate each mathematical statement below. Then, evaluate each expression if x = -4, y = 8, and z = -2 9) Twelve less a number y. 10) Three more than twice a number x.
11) Seven less than the product of x and z. 12) Five times the quotient of y and x. Evaluate the following if a = 4, b = 1, and c = -5
13)
Order of Operations
P
E
M
D
A
S
4
Property Explanation Addition Multiplication
Closure
When two real numbers are added or multiplied, the result will always be a real number.
8 + 5 = 13 Real # + Real # = Real #
6 ⋅ 9 = 54 Real # ⋅ Real # = Real #
Associative
Numbers may be grouped differently without affecting the final value.
Commutative
The numbers on each side of an operation sign may be commuted (switched) around without affecting the final value.
Identity
Find a number that will produce an identical answer to the original number.
Additive Identity Multiplicative Identity
Inverse
Find a number that will turn a number into the identity element.
Additive Inverse Multiplicative Inverse
Multiplicative Property of Zero
Any number times zero is zero, and zero times any number is zero.
Distributive
This property is used to multiply a number by the sum or difference of two numbers in parentheses. The first number is “distributed” across the parentheses by multiplying it with both of the numbers inside the parentheses.
Properties of Real Numbers Learning Target: SOL A.4b: The student will justify steps used in simplifying expressions and solving equations.
5
Using the Distributive Property The distributive property is often used to simplify expressions that cannot be simplified using the order of operations. Simplify each expression below. Combine like terms when necessary. 3(4x + 7) -5(3x – 6) -7(2x + 8) 9(x – 7) – (4x + 5) Division and the Distributive Property The distributive property can also be used to simplify fractions with a monomial expression in the denominator. Instead of multiplying each coefficient, divide each coefficient by the number in the denominator.
Try it!
Simplify the following expressions using the distributive property. Combine like terms when necessary. 1. 5(6a + 7b – c) 2. -3(12x – 8y) 3. -5(2x + 1) – 3(4x – 9)
4.
5.
6
Properties of Equality
Property Explanation Arithmetic Example Algebra Example
Reflexive
A quantity is equal to itself.
x = x
Symmetric
Quantities on each side of an equal sign may be switched.
If x = y,
then y = x.
Transitive
If a first quantity equals a second, and the second equals a third, then the first and third quantities are also equal.
If a = b and b = c,
Then a = c.
Substitution
Equal quantities may be substituted for each other.
If y = 3,
then 2y = 2(3).
Try it! Identify the property displayed in each example. 1. –3 + (2 + 6) = (-3 + 2) + 6 2. (x • 4) • 9 = (4 • x) • 9 3. 11 = x so x = 11 4. 1y = y 5. x = z and z = 5, so x = 5 6. –5.5 + 0 = -5.5
7. 7 + (8 – 3) = 7 + 5 8.
9. (-6 + 6) + 19 = 0 + 19
10. If x = 2, 3x = 3(2) 11. 3(5x + 4) = 15x + 12 12. 0 • a = 0 13. 9 = 3(4x + 1) 14. A = A 15. (-6 • 4) • 9 = -6 • (4 • 9) so 3(4x + 1) = 9
7
What does it mean to solve something?
Equations with Addition
x + 12 = 23 42 = n + 5
7 + x = 19 y + (-12) = 33
Equations with Subtraction
x – 12 = 23 n – 5 = 42
z – (-7) = 19 33 = y - 12
Equations with Multiplication
5x = 35 -3x = 27
68 = 4x -8x = -44
Equations with Division
65
x 10
6
x
328
x 7
3
1x
Solve by Combining Like Terms
7x – 4x = 21 -16 = 5d – 9d
Learning Target: SOL A.4d: The student will solve multistep linear equations algebraically.
One-Step & Two-Step Equations
8
The Basics of Solving Two Step Equations
8x + 3 = 33 -47 = 3x – 50
5143
m
1034
x
You Try It! Solve the following equations. Show all of your work. Any non-integer answers should be
expressed as fractions.
1) -12x = -60 2) 28 = 5x + 8 3)
4) 16 – x = 12 5)
6) 60x + 4 – 15x = 16
7)
Backwards
PEMDAS! 1.
2.
9
Practice:
Identify the property displayed in each example.
1. (2 + x) + 5 = 2 + (x + 5) ____________________________________
2. 2x 1 = 2x ____________________________________
3. 2(x - 7) = 2x - 14 ____________________________________
4. If 2 = x, then x = 2. ____________________________________
5. 4.12 + 0 = 4.12 ____________________________________
6. (x + 4) + 6 = (4 + x) + 6 ____________________________________
7. 0 • a = 0 ____________________________________
8. (4 + - 4) + 7 = 0 + 7 ____________________________________
Properties Explanations Algebraic Example
Addition Property of Equality
Adding the same number to each
side of an equation produces an
equivalent equation.
If a = b
Then a + x = b + x
Subtraction Property of Equality
Subtracting the same number
from each side of an equation
produces an equivalent equation.
If a = b
Then a - x = b - x
Multiplication Property
of Equality
Multiplying both sides of an
equation by the same number
produces an equivalent equation.
If a = b
Then
Division Property
of Equality
Dividing both sides of an
equation by the same number
produces an equivalent equation.
If a = b
Then
What property is being used when simplifying the expression or equation below?
1. 5x + 2 = 12 2. 2 • a • (7 • b) 3.
5x + 2 - 2 = 12 - 2 2 • (a • 7) • b
Learning Target: SOL A.4b: The student will justify steps used in simplifying expressions and solving equations.
Proofs for Equations
10
Proofs Solve each two-step equation below. Justify each step in the boxes provided.
Solution Justification
15y + 31 = 61 Given
15y + 31 + (-31) = 61 + (-31)
15y + 0 = 30
15y = 30
1y = 2
y = 2
Find the three mistakes in the following proof. Identify and correct these
mistakes.
Given
Symmetric Property of Equality
Inverse Property of Addition
Addition Property of Equality
Additive Identity Property
⋅
⋅ Division Property of Equality
1x =
Inverse Property of Multiplication
x =
Multiplicative Identity Property
Solve the given problem. Justify each step.
11
You Try It!
1. Justify the steps for proof in the boxes provided.
2. Find the three mistakes in the following proof. Identify and correct these mistakes.
Given
Subtraction Property of Equality
Additive Inverse
Zero Product Property
Division Property of Equality
1x = Multiplicative Identity
x = Multiplicative Inverse
3. What property is being used when simplifying the expression or equation below?
a. 7x = 21 b. 15 = 3x + 2 c. 18z + 0 = 2634 d. 1x = 5
3x + 2 = 15 18z = 2634 x = 5
Solution Justification
35 = 7 ( x +3) Given
7( x + 3) = 35
7x + 21 = 35
7x + 21 - 21 = 35 - 21
7x + 0 =14
7x = 14
1x = 2
x = 2
12
Solve each equation. Leave any non-integer answers as fractions unless decimals were used in the original problem. Round any decimals according to the decimals in the problem. 1. 6x + 22 = -3x + 31 2. 7.23x + 16.51 = 47.89 – 2.55x
3.
(3x + 9) = 5x + 3x + 2 4. 4x – 3(x – 2) = 21
5.
(x – 4) = 66 6.
+ 16 = 2
7.
8.
Solving Multi-Step
Equations
Learning Target: SOL A.4d: The student will solve multistep linear equations in two variables including solving multistep linear equations algebraically and graphically.
13
Solve and discuss with your neighbor. 9. 3(x – 3) = 3x + 10 10. 2(y – 7) = 2y – 14
Proofs
11. Using the given proof, justify each step in the boxes provided.
Steps Justification
5x + 3 = y if y = 23
Given
5x + 3 = 23
5x = 20
x = 4
12. Using the given proof, justify each step in the spaces provided.
Solution Justification
3x + 5 + 7x = 43 Given
3x + 7x + 5 = 43
10x + 5 = 43
10x + 5 + (-5) = 43 + (-5)
10x + 0 = 38
10x = 38
1x =
x =
14
You try it! Solve each equation. Leave any non-integer answers as fractions unless decimals were used in the original problem. Round any decimals according to the decimals in the problem. 1. 17 – 2x = 14 + 4x 2. 4(1 – y) + 3y = -2(y + 1)
3. -2(2x – 6) +
(12x + 8) = 5 – 3(2x + 1) 4.
(10x + 15) = 18 – 4(x – 3)
5.
6.
7. Hermione, Harry, and Ron were solving three different math problems. The last step of their work is given below. Determine which person’s problem has a solution that is all real numbers, whose problem has no solution and whose problem has the solution x = 0.
Hermione Harry Ron 6x = 0 0x = 5 5 = 5
15
8. Jerri wrote these steps when solving an equation:
Steps Justification
17(x + 3) = 6 – 4
Given
17x + 51 = 6 – 4
17x + 51 = 2
17x = -49
x =
Select a property from the box to justify each step. Write your answer in each box. Substitution Property Subtraction Property of Additive Identity Equality Division Property of Distributive Property Associative Property Equality 9. Using the given proof, justify each step in the spaces provided.
Solution Justification
8x + (7x +3) = 78 Given
(8x + 7x) + 3 = 78
15x + 3 = 78
15x + 3 + (-3) = 78 + (-3)
15x + 0 = 75
15x = 75
1x = 5
16
Solving inequalities is very similar to solving equations with one notable difference!
1. 6x – 2 > 4x + 8 2. –x + 4 < 2(x – 8)
3. -4 – x > -6 4.
+ 5 > -8
5. Graph the solution for number #4. Then give two possible values of x that will make the
inequality true.
Addition and
Subtraction Properties of
Inequality
Adding or subtracting the same number to
each side of an inequality produces an
equivalent inequality.
If a > b, then a + c > b + c.
If a < b, then a – c < b – c.
Multiplication and
Division Properties of
Inequality
If you multiply or divide each side of an
inequality by a…..
Positive number you produce an
equivalent inequality.
Negative number you have to flip the
sign to produce an equivalent inequality.
If a < b and c is
positive, then ac < bc.
If a < b and c is
negative, then ac > bc.
Solving Linear Inequalities Learning Target: SOL A.5a: The student will solve multistep linear inequalities in two variables including solving multistep linear inequalities algebraically and graphically.
17
6.
< -12 7. 8(8x – 1) > -8 + 8x
8. 3(x + 1) < 3x + 7 9.
28
Using the axioms of inequality and the properties of real numbers, justify each step in the
solutions given below.
Solution Justification
4 – 2x > -6 Given
-2x + 4 > -6
-2x > -10
x < 5
Solution Justification
Let y = 2x and y – 4 > -18 Given
2x – 4 > -18
2x – 4 + 4 > -18 + 4
2x + 0 > -14
2x > -14
(2x) >(
)(-14)
1x > -7
x > -7
18
Try it! 1. -3(2x + 1) > 1 – 8x 2. 5x < -25
3. x – (-3) > -2 4. -2(x + 3) < 4x – 7
5. 7 + 3x < 16 6.
+ 7 > -12
7. Graph the solution for number #1. Then give two possible values of x that will make the
inequality true.
8. Using the axioms of inequality and the properties of real numbers, justify each step
in the solutions given below.
Solution Justification
3(2x – 1) < 2(4x + 3) Given
6x – 3 < 8x + 6
6x – 3 + 3 < 8x + 6 + 3
6x – 8x < 8x – 8x + 9
-2x < 9
x >
19
sANDwich inequalities OR inequalities
x is at least negative six and at x is either less than negative 2 most five or x is greater than seven
Solve each compound inequality. Solve each compound inequality.
3. -3 < 2x + 1 7 5. 2x – 3 < 5 or 3x + 1 16
4. 2 < -3x + 8 < 17 6. -4x + 2 6 or 2x -6
Solving and Graphing
Compound Inequalities
Learning Target: SOL A.5a: The student will solve multistep linear inequalities in two variables including solving multistep linear inequalities algebraically and graphically.
20
Special Cases Solve and graph each compound inequality.
7. 3x + 1 < 4 or -2x – 5 > 7 8. 5x – 6 -11 or -3x – 7 > -13
Try it! Solve each compound inequality.
1. -4 < 9x – 1 < 5 2. -3x – 7 8 or -2x – 11 - 31
3. 2x + 7 < 3 or 5x + 5 10 4. 7 < -2x + 21 31 5. Graph the answer to #3. 6. Graph the answer to #4. 7. Write an inequality that corresponds to each graph below:
2 3 4 5 6 7 8 9 10 -2 -1 0 1 2 3 4 5 6
21
Solve for the indicated variable.
1. A = ½bh, solve for h 2. C = 2 r, solve for r 3. A = ½h(b1 + b2), solve for h 4. A = ½h(b1 + b2), solve for b1
5. P = 2w + 2l, solve for l 6. F =
C + 32, solve for C
Solving Literal Equations
(Formulas)
Goal: Rearrange a formula (“Solve the Formula”)
so that a new variable is isolated.
Learning Target: SOL A.4a: The student will solve literal equations (formulas) for a given variable.
22
Function Form: A two-variable equation (usually x and y) is written in function form if one of its variables is isolated on one side of the equation.
Rewrite the following equations so y is a function of x.
(Write y in terms of x…in other words, isolate y!) Write all answers in
simplest form! 7. -7x + y = 8 8. 6y – 3x = 12 9. 19 – 3y = 8x – 2x + 10 10. ⅓(y + 2) + 3x = 7x
23
Formulas and Functions Try it! Solve for the indicated variable.
1. I = Prt, solve for r 2. A = ½h(b1 + b2), solve for b2
Rewrite each equation so that y is a function of x. Write your answer in simplest form. 3. 13 = 12x – 2y 4. y – 7 = -2x 5
5.
(25 – 5y) = 4x – 9y + 13 6. 20x = 4y – 4
24
Learning Target: SOL A.4f The student will solve real- world problems involving multi-step linear equations in two variables.
Steps: A. Determine the object/units being compared. B. Write a proportion that represents the situation. C. Solve your proportion. D. Circle your final answer. Include units.
1. The ship model kits sold at a hobby store have a scale of 1ft : 600ft. A completed model of
the Queen Elizabeth II is 1.6 feet long. Estimate the actual length of the Queen Elizabeth II.
2. Mr. Land is trying to decide how my pizzas to buy for the 7th grade picnic. If he usually needs 9 large pizzas to feed his class of 26 students, how many large pizzas should he buy if there are 204 students in the entire 7th grade?
3. Based on the 2000 census, each member of the U.S. House of Representatives represents an average population of 646,952 people. If Virginia currently has 11 representatives, what was the approximate population of Virginia in 2000?
4. The ratio of male students to female students in the freshman class is 2:3. There are 216 girls in the freshman class. Find the number of males.
5. 85% of the books on Ms. Park’s book shelf are mystery novels. If she has 40 books on her bookshelf, how many of them are mystery novels?
Using Proportions to
Solve Word Problems
25
Try it!
Directions: Write a proportion to represent each word problem below. Then, find the solution. SHOW ALL OF YOUR WORK!
1. Tara babysits every Saturday afternoon. She typically gets paid $42 for four hours worth of work. If she is asked to stay late, how much should she be paid for six hours worth of work?
2. Triangle ABC and Triangle DEF are similar. The height of ABC is 4 cm and the base is 7 cm. If the height of DEF is 14 cm, how long is the base?
3. Elizabeth is standing next to a flagpole that is 24 feet high. If the flagpole’s shadow is 13 feet and Elizabeth’s shadow is 3 feet, how tall is Elizabeth? Round to the nearest 100th if necessary.
4. Biologists wanted to know how many fish were in Lake Neterer. Last week, they tagged 220 fish. This week, the biologists counted 15 tagged fish out of a sample of 300 fish from the same lake. Estimate the total number of fish in Lake Neterer.
5. Tyler is preparing for a college entrance exam. On a practice test, he answered 8 problems in 15 minutes. At this rate, will he be able to finish a 90 question exam in 150 minutes?
26
Learning Target: SOL A.4f The student will solve real- world problems involving multi-step linear equations in two variables.
Steps:
A. Define your variables.
B. Use your variables to write an equation that represents the situation.
C. Solve your equation
D. Circle your final answer. Include units.
1. As a lifeguard, you earn $6 per day plus $2.50 per hour. How many hours must you work to
earn $16 in one day?
2. The sum of the ages of three brothers is 59. Jason is twice as old as Brian. Alex is five more
than three times Brian’s age. How old is each brother?
3. The perimeter of a rectangle is 168 feet. Its length is 5 times the width. Find the dimensions
of the rectangle.
Using Equations to
Solve Word Problems
27
4. The sum of three consecutive integers is 270. Find the numbers.
5. Find three consecutive odd integers such that the sum of the third and three times the first is
the same as thirty more than twice the second.
6. In triangle ABC the measure of angle B is fifteen degrees less than the measure of angle A.
The degree of angle C is ten degrees more than the sum of the measures of angles A and B.
How much does each angle measure?
28
Try It!
1. A lifeguard at the community pool makes $9.50 per hour. A lifeguard at the country club
makes $8.25 per hour, but has a weekly bonus of $40. How many hours do the lifeguards
need to work in one week to earn the same amount of money?
2. Tess is hiking the entire length of the Superior Trail in Minnesota. She has already hiked 80
miles. If she continues to hike at a constant rate of 18 miles per day, how many days will it
take her to reach the end of this 275 mile trail along the edge of Lake Superior?
3. Find three consecutive even integers such that the sum of the first and twice the third is the
same as twenty-eight less than four times the second.
29
4. Jared is training for a marathon. His goal is to run a total of 27 miles over the course of this
three-day holiday weekend (Saturday, Sunday, and Monday). If he plans to decrease the
length of his run by 2 miles each day, how many miles will he need to run on Saturday if he
is going to meet his goal?
5. Plumber Joe charges a flat fee of $45 plus $15 per hour for every house call he makes. If he
charges Mr. and Mrs. Centennial $112.50, how many hours did he work at their house?
6. John is five years older than Henry and Henry is 3 years older than Fred. The sum of their
ages is 32. Find their ages.
30
Learning Target: SOL A.5c: The student will solve real- world problems involving multi-step linear inequalities in two variables.
Recognizing Inequalities
x is at most 4 ____________ x is at least 4 ____________
x is no more than 4 _________ x is no less than 4 _________
Steps:
A. Define your variables. B. Use your variables to write an inequality that represents the situation. C. Solve your inequality. D. Circle your final answer. Include units.
1. Sally wants to rent tables for her outdoor wedding. The rental shop in town will charge her $11.25 to rent the long rectangular table for her bridal party that day. Each of the round tables her guests will sit at cost $8.75. If she can spend no more than $160 on tables, what are the possible numbers of round tables she can rent? Represent your answer algebraically and graphically.
2. A blank CD can hold at most 70 minutes of music. So far you have burned 25 minutes of music onto the CD. You estimate that each song lasts 4 minutes. What are the possible numbers of additional songs that you can burn onto the CD?
3. A gym is offering a trial membership for 3 months by discounting the regular monthly rate by $50. You will consider joining the gym if the total cost of the trial membership is less than $100. What must the price of a regular monthly membership be in order for you to take advantage of this deal?
Using Inequalities to
Solve Word Problems
31
4. Your cell phone plan costs $49.99 per month for a given number of minutes. Each additional minute or part of a minute costs $0.40. You budgeted $55 per month for phone costs. What are the possible additional minutes (x) that you can afford each month?
5. Pretty Mountain State Park rents cabins for guests to stay in for $110 per night. If you have purchased a year-long stage parks pass, they will discount your nightly rate by $15. At the time of rental, guests can also opt to pay $55 for an unlimited supply of firewood. You have a state parks pass and you will choose to pay for the unlimited supply of firewood. For how many nights can you rent the cabin if you are determined to spend less than $1,100?
Try It! Directions: Write an inequality to represent each word problem below. Then, find the solution. SHOW ALL OF YOUR WORK!
1. A gas station charges $0.10 less per gallon of gasoline if a customer also gets a car
wash. The price of gas is regularly $2.09 a gallon, and a car wash is $8.00. If you get a car wash, what are the possible amounts (in gallons) of gasoline that you can buy if you can spend at most $20?
2. Tony is a new waiter at the family restaurant in town. He is hoping to earn at least $100 during his 8 hour shift. If he makes $70 in tips, what would his hourly rate need to be to reach this goal?
32
3. To become a member of an ice skating rink, you have to pay a $30 membership fee. The cost of admission to the rink is $5 for members and $7 for nonmembers. After how many visits to the rink is it less expensive to be a member than a nonmember? In other words, at what point is it worth it to get the membership?
4. Bryan’s dog is three years more than twice his cat’s age. Find all possible ages of the cat if the sum of their ages is at most 18. Represent your answer algebraically and graphically.
5. Jacob is training for a marathon and is using a pyramid training pattern for the next five days. He plans to increase the number of miles he runs by a single mile each day from the first to the third day, peaking on the third day, and then decreasing the number of miles run by a single mile per day for the last two days.
a) If x represents the number of miles Jacob runs on Day 1, write expressions for how many miles, in terms of x, he runs from Days 2 through 5.
Day 1 = Day 2 = Day 3 = Day 4 = Day 5 =
b) Find all possible values that Jacob can run on Day 1 such that his total number of miles run over the five days is at least 64. Represent your answer algebraically and graphically.
33
Example 1: How many starbursts did each student take?
Measures of central tendency tell us how the data __________________________.
We can analyze data using three measures of central tendency:
1. Mean –
2. Median –
3. Mode –
Which measure of central tendency most accurately represents the data set? Why?
Example 2:
Which measure of central tendency most accurately represents this data set? Why?
Learning Target: SOL A.9 The student will, given a set of date, interpret variation in real world context and calculate and interpret mean absolute deviation, standard deviation, and z-scores.
Measures of Central
Tendency
34
Example 3: The heights (in feet) of 8 waterfalls in the state of Washington are listed below.
Find the mean, median, and mode. Which measure of central tendency best represents the data?
Why?
1000, 1000, 1181, 1191, 1200, 1268, 1328, 2584
Example 4: Bugs Bunny is participating in two months worth of carrot eating contests every
Saturday morning. His carrot consumption on each of the last seven Saturdays has been as
follows: 84, 102, 95, 69, 77, 83, 90. How many carrots must Bugs eat on the eighth Saturday if
he wants his mean consumption to be exactly 85 carrots?
Example 5: Suppose that Dale has test scores of 75, 95, 82, and 77. He has one test remaining
and his goal is to have a test average of exactly 90. What must Dale score on the final test in
order to meet his goal? Assuming that the scores above are percentage points, is this an
obtainable goal for Dale based on Mercer’s grading system?
Outliers!
35
You Try It!
1. Find the mean, median and mode(s) of the data:
a. 1, 2, 1, 1, 3, 5, 5, 3, 6 b. 300, 320, 341, 348, 360, 333
2. The weights (in pounds) of ten pumpkins are 22, 21, 24, 24, 5, 24, 5, 23, 24, and 24.
a. Find the mean, median, and modes(s) of the pumpkin weights.
b. Which measure of central tendency best represents the data? Explain.
3. The Mississippi River discharges an average of 230 million tons of sediment per year. The
average sediment discharges (in millions of tons per year) of the seven U.S. rivers with the
greatest discharges are 230, 80, 65, 40, 25, 15, and 11.
a. Find the mean and median of the data. Which measure represents the data better?
b. Find the mean of the data for the other six rivers, excluding the Mississippi River. Does
this mean represent the data better than the mean you found in part (a)? Explain.
4. So far you have score 84, 92, 76, 88, and 76 on five of the six tests you will take in a
particular class. Your goal is to finish the year with a test average of 85 or greater. What
score must you achieve on the sixth test in order to reach this goal?
36
Find the Mean and the Median of data sets A, B, and C:
Set A: 10, 10, 10, 10, 10 Set B: 8, 9, 10, 11, 12 Set C: 7, 9, 10, 11, 13
Measures of dispersion tell us about the __________________________of the data.
Range:
Find the Range of Sets A, B, and C.
Mean Absolute Deviation: On average, how far away from the mean is each element?
MAD = Find the Mean Absolute Deviation of Sets A, B, and C.
1. Find the absolute deviation (distance between each data point and the mean.)
2. Find the mean of the new data set.
In which set is the data dispersed furthest from the mean?
The greater the value of the mean deviation, the further the data tends to be dispersed from the
mean.
Set A
Absolute Deviation
Mean Absolute
Deviation:
Set B
Absolute Deviation
Mean Absolute
Deviation:
Set C
Absolute Deviation
Mean Absolute
Deviation:
Measures of
Dispersion
Learning Target: SOL A.9 The student, given a set of data, will interpret variation in real-world contexts and calculate and interpret mean absolute deviation, standard deviation, and z-scores.
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Practice Find the range, the mean, and the mean absolute deviation of the data.
1. 30, 35, 20, 85, 60 2. 111, 135, 115, 120, 145, 130
Standard Deviation A calculation that tells us how dispersed the data is –
the ____________ the number, the _______ dispersed the data.
the ____________ the number, the _______ dispersed the data.
Use the calculator to find the standard deviation of Sets A, B, and C.
Steps:
1. Put the data into a spreadsheet
2. Open a calculator page
3. Menu, ____________, _____________, ____________
Set A : _____ Set B : _____ Set C : _____
What does the standard deviation tell you about these three sets of data?
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Using Standard Deviation Consider Set C:
1. Label the mean value in the middle of the number line.
2. What was the standard deviation of Set C?
3. Use the number line to determine what value is exactly 1 standard deviation away from
the mean.
4. Use the number line to determine what value is exactly 2 standard deviations away from
the mean.
5. The value 13 is how many standard deviations away from the mean?
Questions to Consider:
1. If the mean of a data set is 14 and the standard deviation is 4, what value is two standard
deviations below the mean?
2. If the mean of a data set is 20 and the value 26 is three standard deviations above the
mean, what is the standard deviation?
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You Try It!
1. The average scores of the bowlers on two different bowling teams are given. Compare the
spreads of data using (a) the range and (b) the mean absolute deviation.
Team 1: 162, 150, 173, 202 Team 2: 140, 153, 187, 196
2. The sizes of e-mails (in kilobytes) in your inbox are 1, 2, 2, 7, 4, 1, 10, 3, 6.
a. Find the standard deviation of the data.
b. Which number below represents an email that is approximately two standard deviations
above the mean?
A. 7 B. 10 C. 1 D. 6
3. In 2000 the numbers (in thousands) of households in the 13 states with Atlantic Ocean
coastline are given. Use a graphing calculator to find the standard deviation of the data.
299, 6338, 3006, 518, 1981, 2444, 475, 3065, 7057, 3132,
408, 1534, 2699
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4. The heights (in feet) of 9 pecan trees are 72, 84, 81, 78, 80, 86, 70, 80, 88. For parts (a)-(c)
below, round your answers to the nearest tenth.
a. Find the standard deviation of the data.
b. What is the approximate height of a pecan tree that is one standard deviation below the
mean?
c. What is the approximate height of a pecan tree that is two standard deviations above the
mean?
d. Suppose you include a pecan tree with a height of 136 feet. Predict the effect of the
additional data on the standard deviation of the data set.
e. Find the standard deviation of the new data set in part (d). Compare the results to your
prediction in part (d).
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The Normal Curve (Bell Curve): Data is said to have a normal distribution if most of the
elements in the data set are close to the “average” while relatively few elements tend to
one extreme or the other.
Remember Standard Deviation?
The standard deviation of a given data set told you how tightly all the various elements
were clustered around the mean.
Approximately 68% of the data will be within 1 standard deviation of the mean.
Approximately 95% of the data will be within two standard deviations and approximately
99% of the data will fall within three standard deviations.
Anything that is more than three standard of deviations from the mean is considered an
outlier.
When the standard of deviation is small…. When the standard of deviation is large….
Learning Target: SOL A.9 The student, given a set of data, will interpret variation in real-world contexts and calculate and interpret mean absolute deviation, standard deviation, and z-scores.
Z-Scores
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Z-Scores:
A z-score is used to tell us (in standard deviations) where a piece of data lies on the
normal distribution curve.
If a z-score is _________, then the data value lies ________ the mean.
If a z-score is __________, then the data value lies ________ the mean.
If a z-score is more than 3 or less than -3, the data value is an __________________.
Formula:
Example 1: Example 2: Find the z-score for the data value 14.75 Find the z-score for the data value 8
if the data set has a mean of 12.2 and a if the data set has a mean of 15 and a
standard deviation of 1.75. What does the standard deviation of 4.6. What does
z-score tell you? the z-score tell you?
Example 3: The number of days the House of Representatives spent in session each year from
1996 to 2004 is represented by the data set below:
110, 119, 122, 123, 132, 133, 135, 137, 142
Find the mean and standard of deviation. Then, calculate a z-score for the data value 122. What
does the z-score tell you about the data value?
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Example 4: A group of school children were asked how many hours they watched television
every week. The mean number of hours was ten. The standard deviation was 2.5 hours.
a. If the number of hours Johnny watched television has a z-score of 2, how many hours
does he watch?
b. If Joanne’s z-score was -1.5, how many hours does Joanne watch?
You Try It!
1. Which z-score represents a data value that is further from the mean: -2.4 or 1.6?
2. Which z-score represents a data value that is below the mean: 1.2 .8 or -2.9?
3. Which z-score represents an outlier: 2.8 -1.3 3.4 or 0?
4. Find the z-score for the data value 108 5. Find the z-score for x = 10 if
if the data set has a mean of 103 and a = =10 and
standard deviation of 2.5. What does the = 4.6.
z-score tell you? What does the z-score tell you?
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6. The number of days the Senate spent in session each year from 1996 to 2004 is represented by
the data set below:
132, 133, 141, 143, 149, 153, 162, 167, 173
a. Find the mean and standard of deviation.
b. Calculate a z-score for the data value 141. What does this z-score tell you about the data
value?
c. Calculate a z-score for the data value 173. What does this z-score tell you about the data.
Which z-score is closer to the mean, 141 or 173?
7. Carrie goes for a hike every weekend. The mean length (in miles) of the hikes she has done is
7 miles. The standard deviation is 1.5.
a. If the length of her hike last weekend has a z-score of -3, how many miles was the hike?
b. The length of the hike Carrie plans to complete next weekend has a z-score of 2.5. How
many miles is Carrie planning to hike next weekend?
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A Box-and-Whisker Plot organizes data values into four groups using the following five
number summary:
1. Minimum:
2. Lower (1st) Quartile:
3. Median (2nd
Quartile):
4. Upper (3rd
) Quartile:
5. Maximum:
Example 1: The lengths of songs (in seconds) of a CD are listed below. Make a box-and-
whisker plot of the song lengths.
173, 206, 179, 257, 198, 251, 239, 246, 295, 181, 261
Example 2: Make a box-and-whisker plot of the ages of eight family members:
60, 15, 25, 20, 55, 65, 40, 30
Learning Target: SOL A.10 The student will compare and contrast multiple univariate data sets using box and whisker plots.
Box-and-Whisker
Plots
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Interquartile Range: The interquartile range is another measure of dispersion. It measures the
variation in the middle half of the data and ignores the extreme values, whose variation may not
be representative of the whole.
Find the interquartile range of Examples 1 and 2.
Example 3: The box-and-whisker plots below show the normal precipitation (in inches) each
month in Dallas and in Houston, Texas.
a. For how many months is Houston’s precipitation less than 3.5 inches? For how many
months was the precipitation in Dallas more than 2.6 inches?
b. Compare the medians. In general, which city gets more precipitation?
c. Compare the range and interquartile range. In general, which city has a larger variation
in precipitation?
Houston
Dallas
3.0
3.5 3.8 4.4
5.4
1.9 5.2
2.3 2.6 3.2
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You Try It!
1. Make a box-and-whisker plot of the data. Identify any outliers.
Hours worked per week: 15, 15, 10, 12, 22, 10, 8, 14, 18, 22, 18, 15, 12, 11, 10
2. Consider the box-and-whisker plot below:
0 5 10 15 20 25
a. About what percent of the data are greater than 20?
b. About what percent of the data are less than 15?
3. Two students’ summative test scores are given below.
Student One: 73, 84, 72, 98, 81, 76 Student Two: 92, 81, 63, 74, 86, 62
a. Create a box-and-whisker plot for each students’ scores.
b. Compare the medians. In general, which student has the higher test scores.
c. Compare the range and interquartile range. Which student has the greatest variation in
test scores?