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?