1

Name: ________________________ Teacher: _______________

FINAL REVIEW PACKET MATH 7A

Unit 1: Basic Skills

The Real Number System

ALL the numbers we worked with this year are REAL NUMBERS. That means every number we

worked with was either RATIONAL or IRRATIONAL.

RATIONAL Numbers are numbers that CAN be written as fractions.

COUNTING NUMBERS 1, 2, 3… also known as NATURAL NUMBERS

WHOLE NUMBERS 0, 1, 2, 3…

INTEGERS …, -5,-4,-3,-2,-1, 0, 1, 2, 3…

FRACTIONS ALREADY A FRACTION!!

TERMINATING DECIMALS 0.13

REPEATING DECIMALS 0.333…

PERFECT SQUARES 7

IRRATIONAL Numbers are numbers that CANNOT be written as fractions.

PI 3.1415926…

NON-PERFECT SQUARES

NON-TERMINATING NON-REPEATING DECIMALS 0.12112111211112…

WHAT ARE PERFECT SQUARES? A number is a perfect square if its square root is a whole

number. That is, the number is equal to a number times itself. FOR EXAMPLE: 25 = 5 5 AND

25 = -5 -5 therefore, 25 IS A PERFECT SQUARE.

Name ALL the sets of numbers to which each number belongs.

Real, Irrational, Rational, Integer, Whole, Counting/Natural

1) -8 __________, __________, ____________

2) __________, __________, __________, __________, __________

3) __________, __________

RA

TIO

NA

L

NU

MB

ER

S

IR

RA

TIO

NA

L

NU

MB

ER

S

4) __________, __________

5) __________, __________

6) - __________, __________, __________

7) 0.25 __________, __________

8) __________, __________

9) 5 __________, __________, __________, __________, __________

Answer each of the following with ALWAYS, SOMETIMES or NEVER true.

10) Integers are ______________________ rational numbers.

11) Real Numbers are ______________________ irrational numbers.

12) Whole numbers are ______________________ integers.

13) Rational numbers are ______________________ irrational numbers.

14) Name the largest negative integer. __________

15) Name the smallest positive integer. __________

Absolute Value measures the distance a number is from zero on the number line.

The symbol for absolute value is “| |.”

Evaluate.

16) |102| - |-2| 17) |102 - -2| 18) -|10| - |-2| 19) |-36 - 4|

20) -12 -20 21) -18 + 30 22) 0 - -14 23) -21 - (-14)

24) -9(-11) 25) (-15)(7) 26) 27) -90 ÷ -15

28) -(32) 29) -3

230) -(-3

2) 31) (-3)

2

Write a number sentence and evaluate.

32) A dolphin swam to a depth of 110 feet below sea level. Then, it rose 85 feet. What was

the dolphin’s final depth?

33) The temperature outside was 22˚F. The wind chill made it feel like -8˚F. Find the

difference between the real temperature and the apparent temperature.

34) The temperature one morning in was –16oF. By the afternoon, the temperature had

risen 9oF. What was the temperature in the afternoon?

Evaluate each expression, using the correct order of operations.

35) 6 + 9 ÷ 3 10 36) (15 - 7) 6 + 2 37)

Evaluate each expression if: a = 3, b = 6, and c = -5.

85) -a - c 39) -b2 + c

340) 5b – 2c 41)

State whether the following answers will be zero or undefined.

42) 43) 44) 0 ÷ 22 45) 22 ÷ 0

Term – a part of an expression that is separated by a "plus" or "minus" sign.

Ex: 3x + 4y → 3x is a term & 4y is a term

Coefficient – a number in front of a variable

Ex: 4n → 4 is the coefficient and n is the variable

Constant Term – a term that has a number but no variable. Ex: 5, 7, 100, 2,000

Like Terms – terms with the EXACT same variables and EXACT same exponents

Examples: 5y and 6y 5x2 and 6x

2 10 and -2

Non-examples: 5x and 3y 2x and 3 -4x and 3x2

List the terms, like terms, coefficient(s), and constant(s) for the following expressions.

Remember, the sign in front of the number goes with the number.

46) 5x + 2y – x + 3y – 7 47) -4a – 10c + 8 – 2a + 7

Terms: ____, ____, ____, ____, ____ Terms: ____, ____, ____, ____, ____

Like Terms: ____ and ____; ____ and ____ Like Terms: ____ and ____; ____ and ____

Coefficient(s): ____, ____, ____, ____ Coefficient(s): ____, ____, ____

Constant(s): ____ Constant(s): ____, ____

Distributive Property states: a (b + c) = ab + ac or a (b - c) = ab - ac

Two steps in simplifying an expression:

Step 1: Get rid of parentheses by using the Distributive Property.

Step 2: Combine like terms.

Simplify each expression.

48) -7 (3 + 4x) + 2 (4 + 5x) 49) 10 - 6 (3x + 2) + 9x

50) (-19x + 24) + (9x - 13) 51) (12x - 17) - (-7x + 9)

52) (10x - 20) - 19x - 5 53) 0.5 (-30x - 24y) + 34 - 16

54) x + 5 + x – 7 55) x - y - x - y

Unit 2: The Real Number System

WHAT ARE PERFECT SQUARES? A number is a perfect square if its square root is a whole

number. That is, the number is equal to a number times itself.

FOR EXAMPLE: 25 = 5 5 AND 25 = -5 -5 therefore, 25 IS A PERFECT SQUARE.

First 15 Perfect Squares.

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225

Remember there are positive roots and negative roots. Be sure you know which root you

are looking for. When solving for a variable there will ALWAYS be 2 solutions. Read

carefully to see if you need to reject the negative root.

indicates the positive, or principal square root of 64. Therefore, = 8.

- indicates the negative square root of 121. Therefore, - = -11.

± indicates BOTH positive and negative square roots of 225. Therefore, ± = ±15

The opposite of cubing a number is taking its cube root. The symbol for the cube root is .

When written this way: means “the cube root of 8” or find the number that when cubed is

equal to 8. List the first 8 perfect cubes.

1, 8, 27, 64, 125, 216, 343, 512

Find each square root.

1) = _____ 2) ±= _____ 3) -= _____

Find each cube root.

4) 273 = _____ 5) 1253 = _____ 6) 643 = _____

Find the 2 consecutive whole numbers each non-perfect square is between.

7) is between _____ and _____ 8) is between _____ and _____.

9) 115 is between _____ and _____ 10) 48 is between _____ and _____

An expression is in SIMPLEST RADICAL FORM if all of the following statements are true:

The radicand has no perfect square factors OTHER THAN 1. The radicand contains NO FRACTIONS. No radicals appear in the DENOMINATOR of a fraction.

Simplified Not Simplified

53 x9

4

10 123

2

x

7

5

Simplify each radical expression:

11) 1125 12) 121-3 13) 4

63x

14) 5

150xx 15) 5

45y3 16) 8

136b2b

Simplify each product or quotient:

17) 306 18) 9632 19) 4274-

20) 30n2n

2

21)

556x240x3

22)

3320x12x

4

3

23) 49

36

24) 9

8

25) 2

5

25x

49x

26) 4

7

64x

225y

When simplifying sums and differences you can combine the coefficients of the LIKE radical

expressions.

3 5 + 7 5 = 10 5 3 5 - 7 5 = - 4 5

If the radicals are UNLIKE then try to simplify (make them the same).

27) 8 3 + 34 28) 2 11 + 11 29) 747

30) 48238 31) 452203 32) 75712

Unit 3: Laws of Exponents & Scientific Notation

LAWS OF EXPONENTS

(Remember, these shortcuts only work if the bases are the same (x is the base)

Multiplication of Exponents: x2 x

5 = x

2+5 = x

7 5 54 = 5

1+4 = 5

5

If the bases are the same: KEEP the base and ADD the exponents.

Division of Exponents: = 89-5

= 84

= x2-5

= x-3 = (remember no negative exponents)

If the bases are the same: KEEP the base and SUBTRACT the exponents.

Power to a Power: (x2)5 = x

(2)(5) = x

10

A power raised to another power: KEEP the base and MULTIPLY the exponents.

Write each expression using exponents.

1) 2 2 2 2 2) s s s s s s s 3) a a b a b a a

Simplify using the Laws of Exponents. Express your answer using POSITIVE EXPONENTS.

Ex. -4x3(7x

5) = -4 x

3 7 x

5 = -4 7 x

3 x

5 = -28x

8

4) 3-2

5) 4-3

42

6) x-5

x-3

7) 27 2

28) 4

2 4

49) 10

2 10

3

10) k8 k 11) a

4c

6(a

2c) 12) 2w

2x 5w

3x

4

13) 3x3 7x

314) 4y

4(-4y

3) 15) (-6x

7)(5x

2)

16) 7y3 6y 17) (-2w

7z

4)(-8w

3 z

2) 18) (-3x

2y

3)(2xy

4)

19) (a2c)

420) (x

3y

4)2

21) (m2n

3)3

22) (2xy)4

23) (-3x4y

7)3

24) (10xy5)2

25) 26) 27) 28)

29) 30) 31) 32)

33) 34)

Scientific Notation

Scientific Notation is when we rewrite a number as a PRODUCT of 2 factors.

Factor #1:

Must be greater than or equal to 1 AND less than 10.

Factor #2:

Must be a power of 10 . Numbers greater than 1 have positive exponents, numbers less than one

have negative exponents.

Ex: Write 24,000 in scientific notation. EX: Write 0.00045 in scientific notation

Scientific notation is 2.4 x 104

Scientific notation is 4.5 x 10-4

Standard Form:

Remember the exponent tells you: How many places to move the decimal.

Positive exponents are numbers greater than or equal to 1 .

Negative exponents are small numbers, numbers less than 1, decimals.

Ex: Write 2.03 106 in standard form.

The exponent is a positive 6 so you move the decimal 6 places to the right.

Standard form is 2,030,000

Ex: Write 3.2 10-8 in standard form.

The exponent is a negative 8 so you move the decimal 8 places to the left.

Standard form is 0.000000032

Write each number in scientific notation.

35) 6,590 36) 4,733,800 37) 2,204,000,000

38) 0.29 39) 0.00000571 40) 0.0008331

Write each number in standard form.

41) 6.7 101

42) 6.1 104

43) 1.6 103

63) 2.91 10-5

64) 8.651 10-7

65) 3.35 10-1

Compare using or .

66) 3.7

107 _____ 8.5

10

467) 7.5

10

3 _____ 9.42

10

3

68) 9.5

10-6

_____ 3.7

10-2

69) 9.75

10

-4 _____ 3.5

10

-6

Find the product. Write your answer in scientific notation.

70) (2

106)

(3

10-4) 71) (4 x 10

6) (12 x 10

3)

Find the quotient. Write your answer in scientific notation.

72) ( 8.5 x 104) ÷ (1.7 x 10

2) 73) (44

10

5)

(4

10-7

)

Unit 4: Polynomials

Polynomials – a monomial or the sum/difference of monomials. Each monomial in a polynomial is

called a term.

Types of Polynomials:

Monomials – one term Binomial – two terms Trinomial – three terms

If a polynomial has more than three terms, it is simply called a polynomial.

Standard Form – a polynomial in one variable with no like terms, and having exponents of the

variables arranged in descending order. Constant terms are always last in standard form.

Ex: 5x3 – 2x

2 + 3x + 7

To find the DEGREE of a monomial you add the exponents of its variables. A constant has a

degree of 0.

To find the degree of a polynomial you find the degree of each term and choose the largest.

EX. 4 + 3a – 8a3 The degree of the polynomial is 3. The last term, -8a

3, had the largest sum of

exponents.

Classify the polynomial as a monomial, a binomial, a trinomial, or a polynomial. Then write

each polynomial in standard form and state its degree.

Classify Standard Form Degree

1) 14x + 2 – 3x2 + 5x

3

2) – 2z + 9z3

3) 2y – 7y5 + 3y

2 + 2

4) x3 – 2x

2 + 7x

5 + 4

5) 2x + 5x2 – 7

6) 3x2

Simplify (distribute and combine like terms) each polynomial, write each in standard form.

7) x2 + 5x – 7 – 12x – 13 8) 5(x

4 – 3x

3 + 2x

2) 9) -3(2y – 7y

2 + 8) – 5y

10) 4(x + 4) – 3x4 + 9x

2 -2x + 5 11) n(n + 5) + 6n 12) x(2x + 3) + 4x

2 – x + 1

To add polynomials:

Distribute the positive sign to each term in parentheses.

o This does not change the sign of each term.

Combine like terms following the rules for adding integers. Remember to keep the sign

with the term.

To subtract polynomials:

Distribute the negative sign to each term in parentheses.

o This changes the sign of each term to its opposite.

Combine like terms following rules for adding integers. Remember to keep the sign with

the term.

Find the sum or difference:

13) (2x + 4y – 1) + (-x – 7 – 6y) 14) (5x2 + 12x – 17) – (13 - 4x

2 + 8x)

15) (3x + 10) + (6x – 15) 16) (x2 + 9x) – (5x

2 – 3x)

17) (–5x2 + 8x + 12) + (3x

2 – 4x + 8) 18) (6x

2 – 11x – 17) – (9x

2 – 8x – 12)

19) 20)

9y - 5

21) Find the perimeter of a rectangle if the length is 23xx2

and the width is 7)(3x

22) Find the perimeter of an equilateral triangle if each side is 5x3 + 3y.

Simplify the following expressions by multiplying or dividing. Be sure write in standard form if

necessary. (Remember your Laws of Exponents!)

23. x2y (x

3 – y) 24. -8mn (14m – 8n + 3) 25. x

3z

2(z

2 – 5)

26. xa4(x

2a – 8c) 27. 4x

2y (2x

3y + 15z - 7) 28. 3x

3 (y

2 + 2x + z)

29) George has a rectangular garden measuring (2x) feet by (5x -2) feet. Find the area AND

perimeter of the garden.

30) (x + 5) (x + 1) 31) (y + 4) (y – 7) 32) (x – 10)(x + 3)

33) (a – 4) (a-5) 34) (2x + 6) (x – 1) 35) (2x + 2)(6x + 1)

36) (5x + 7) (5x – 7) 37) (2x + 4) (3x + 9) 38) (4x + 1) (x2 + 3x – 2)

39) (x + 6)2 40) (x – 6)

2 41) (x + 3) (x

2 + 4x + 2)

42) 4x

4x12x2

43) a3

ab6ba3a1523

44) xy2

xy6xyz12xyz422

45) uv2

uvw8wuv10uv82

46) yx

zyxyx2yx

2

3334

Factoring

The first step to factoring is ALWAYS to look for a GCF!

Factoring an expression by finding the GCF is reversing the Distributive Property.

To factor out a GCF:

1) Find the GCF of ALL terms in the expression.

2) Divide each term of the expression by the GCF.

3) Rewrite the expression as the product of the GCF and the remaining factors.

Find the Greatest Common Factor (GCF) of each pair of terms.

47) 25x and 30y 48) 3x

and 21xy 49) 4y

and 16 50) 12y and 28xy

Factor each expression (using the GCF)

51) x – xy 52) –15m + 50 53) 18n + 24 54) 21xy – 28y

55) 4x4 – 22x

2 + 18x 56) 21x

5 + 35x

3 + 49x

2

57) 2c5d

4 – 3c

4 + 4c

3 58) 23y

10 – 46y

7 + 68y

2 + 10y

To find the factors of a trinomial we REVERSE the multiplication process. We look for

factors of the constant term that have a SUM of the coefficient of the linear term.

59) x2 + 14x + 40 60) x

2 + 5x - 24 61) x

2 - 9x + 20

62) x2 + 16x + 15 63) x

2 - 9x + 14 64) x

2 - 11x + 24

65) Marcus is creating a dog pen for his grandmother.

The diagram to the right is his diagram for the project.

A) Write an expression (in simplest form) to show the amount

of fencing it will take to create the pen?

B) Write an expression (in simplest form) to show the amount space his grandmother’s

puppy will have to play?

C) If x = 2, find the amount of space (in square feet) that the puppy will have to play.

Unit 5: Equations

Step 1: SIMPLIFY

Get rid of any parentheses by using the Distributive Property FIRST

THEN, Combine Like-Terms on the same side of the equal sign.

(Same Side Use Same Operation)

Ex. -5x + 2x + 12 = -10x +16 + 17

-3x +12 = -10x + 33

Step 2: Get all variables on one side & constants on the other side.

(Opposite Sides Use Opposite Operations)

Ex. -3x + 12 = -10x + 33

+10x = +10x

7x + 12 = 33

-12 = -12

7x = 21

Step 4: Solve for the Variable

Ex. 7x = 21

7 7

x = 3

Solve the following equations algebraically.

1) 4x – 7 = 2x + 15 2) 11(2c + 8) = -22 3) 4(-5x + 4) – 2x = -7(2x + 4)

4) 7

3

x - 12 = -27 5) 3

5x2

= 5 6) 4

1

(-16x + 32) = 2

1

(4x – 32)

7) 3.2x – 7.1 = 6.8 – 1.8x 8) 0.4(y – 9) = 0.3(y + 4) 9) 7

3

= 2x

2x

Solve each equation algebraically if possible. State whether the equation is an Identity or a

Contradiction and explain what that means in terms of the solution.

10) 3x + 7 = 3(x + 2) 11) 2x + 9 = 2(x + 4) + 1

** 3-Step Check:

1) Rewrite the equation

2) Replace the variable

3) PROVE (Do the math!)

Solve for x algebraically. Be sure to state ALL solutions

12) x2 = 49 13) x

2 + 10 = 91 14) -5 + 2x

2 = 45

Solve each literal equation algebraically for x.

15) 3x – q = 5q 16) 9x - 24a = 6a + 4x

17) r = 5(x + 2y) 18) y = 2x + z

Solve Algebraically. Don’t forget your let statement.

19) It costs $7.50 to enter a petting zoo. Each cup of food to feed the animals is $2.50. If

you have $12.50, how many cups of food can you buy?

20) You are selling chocolates that you have made for $3 each. You spent $45 on materials.

How many chocolates must you sell to make a profit of $105?

21) Together two items cost $130. One item costs $8 more than the other. Find the cost of

each item.

22) You want to buy a bicycle that costs $280. Your parents agree to pay $100 and you have

to pay for the rest. You can save $20 a week. How many weeks will it take you to save

enough money?

23) The sum of two consecutive odd integers is -84. Find the lesser of the two integers.

Unit 6: INEQUALITIES

“Greater Than” “Greater Than or Equal To”

“Less Than” “Less Than or Equal To” “Not Equal To”

True or False:

4 4 False 4 is not greater than 4

4 4 True 4 is not greater than 4, however, 4 is equal to 4.

GRAPHING INEQUALITIES

You can only graph an inequality on a number line if the VARIABLE is BY ITSELF.

You solve inequalities the same way you solve equations. Remember, whatever you do to one side

of the inequality you must do the same thing to the other side.

*When you multiply/divide both sides of the inequality by a negative number

you need to FLIP the sign.*

Use an Open Circle “ “ for “Greater than” or “Less than”

Use a Closed Circle “ “ for “Greater than or equal to” or “Less than or equal to”

Solve and graph on a number line.

1) 0.5(x + 12) 9 2) -2x + 7 17

3) -10x – 2 8x -20 4) –8x + 12x – 3 21

When translating. . .

● Identify the key words

● Translated in the exact order they are read

● Switch the order ONLY when you read: “less than”, “more than”, “fewer than” ,

“subtracted from” and “taken away from”

● Place parentheses around sums and differences

Translate the following statements into INEQUALITIES.

5) Four less than twice a number, x is greater than 17. ____________________

6) Three more than 5 times a number, n is no less than 38. ____________________

7) The quotient of a number, x and 2 increased by 7 is at most 40. ___________________

8) The elevators, e, in an office building have been approved for no more than 3600 pounds.

______________________

9) An assignment, a, requires at least 45 minutes. ______________________

10) While shopping, Abby, a, can spend at most $50. ______________________

Solve Algebraically. Don’t forget your let statement.

11) Maria and Dan are raising money to build a playground for their neighborhood. They

already raised $2157 from 2 carwashes and a carnival. The total cost of the playground is

$4500. To raise the rest of the money they are going to put on a talent show and sell

tickets for $12 each. What is the minimum number of tickets they must sell to raise

enough money to pay for the new playground?

12) Kaylee and Michelle are going to the movies. They have $38 to spend and movie tickets

are $12.75 each. If boxes of candy cost $3 each, how many boxes of candy can the girls

buy?

Unit 7: Ratios, Rates & Proportionality

We use proportions to solve for missing information, to solve word problems, conversion

problems, scale problems and so much more. The key to using proportions correctly is to

BE CONSISTENT!!!!

Express each rate as a unit rate.

1) 6 pounds lost in 12 weeks 2) $800 for 40 tickets

Solve albebraically.

3) 4)

5) The distance between two cities on a map is 3.2 cm. If the scale on the map is

1 cm = 50 km, find the actual distance between the two cities.

6) A 14-ounce energy drink contains 10 teaspoons of sugar. How much sugar is on one

ounce of the drink?

7) In 4 minutes Benny swims 6 laps. What is this rate in minutes per lap?

8) Find the UNIT PRICE to tell which is the better buy. Show all work and explain.

20 pounds of pet food for $14.99 OR 50 pounds of pet food for $37.99

9) Find the UNIT PRICE to tell which is the better buy. Show all work and explain.

2 DVDs for $26.50 OR 3 DVDs for $40.00

10)

PROPORTIONAL RELATIONSHIPS

What is a proportional relationship? A Proportional relationship can be recognized in a table, on

a graph, or by using an equation.

On a Table:

On a Graph:

# of tickets 1 2 4 6 8

Cost ($) 8 16

11) You can buy 4 tickets for $16. This is a proportional relationship. Fill in the missing

values in the table.

12) What is the constant of proportionality in the table? ______

13) Write an equation that relates the cost, c, to the number of tickets, t. _____________

14) Graph the data on the grid:

Remember your LABELS!!!

Use the following graph about the number of cups to the number of brownies to answer the

following questions.

Looking at

the Graph:

The Equation:

15) Does the graph represent a proportional or non-proportional relationship? Explain.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

16) What is the constant of proportionality? ___________

17) How many brownies will 4 cups make? ___________

18) If b represents the number of brownies and c represents the number of cups write an

equation to show the relationship between the number of brownies and the number of cups

used.

19) Using the above equation, how many brownies will 18 cups make? ___________

20) You can buy 3 apps on your iPhone for $2.97 or 5 apps for $4.95. Write an equation to

show the relationship between the number of apps, a and the total cost, c.

21) If 5 out of 21 people have green eyes. How many people in a room of 512 would you expect

to have green eyes?

22) A car company found that 15 out of every 1230 cars have a problem with their headlights.

If a dealership bought 375 cars from the car company, how many cars should they expect

to have headlight problems?

Each of the following is a pair of similar figures. Solve for the missing side.

23) 24) y

6 ft15 ft

9 ft

Unit 8: PERCENTS

= or =

Percent Increase or Decrease Percent Error

amount of increase/decrease r amount of error r

original amount 100 actual amount 100

Simple Interest: Interest = Principal x Rate x Time (I=PRT)

I = Interest, The ($) amount of interest that is owed or earned.

P = Principal, the amount of money that was borrow, saved or invested.

R = Rate, the percent of interest. *Make sure to convert your % to a decimal

T = Time, time is always in years.

Complete the following chart.

Fraction Decimal Percent

1) 2)

3) 2.15 4)

5) 6) 4%

7) 8)

9) 0.02 10)

11) 12) 28%

Solve Algebraically. One way to solve each of the following word problems is using the

percent proportion. There are quicker methods, but beware, they also require a higher

level of understanding.

13) Mary sold $192 worth of greeting cards in February. If she receives a base pay of $100

per month plus 20% commission on her sales, how much money did she earn in February?

14) Jenny bought a pair of boots priced at $85. If the boots were on sale for 15% off the

regular price, how much did Jenny pay for the boots?

15) The cost of a bicycle is $99.50 including sales tax. If sales tax was 7.5%, how much was

the bicycle before sales tax?

16) There are 350 people at a luncheon. If 12% of the people will win a door prize, how many

door people will not win a door prize?

17) Jen’s bill at a restaurant before tax and tip is $22. If tax is 5.25% and she wants to

leave 15% of the bill including the tax for a tip, how much will she spend in total?

18) A $300 mountain bike is discounted by 30%, and there is a 8% sales tax. Find the final

cost of the mountain bike.

19) Kevin paid $65 for a backpack that was on sale for 30% off. What was the original

price of the backpack?

20) Michael borrowed $1750 from his brother to buy a computer. He agreed to repay the

money in 2½ years at 8.75% interest. How much interest will he pay? What is the total

amount of money Michael will have to repay his brother?

21) Michelle started a bank account that earns 12.25% interest. After 1½ years, she earned

$147 in interest. How much money did Michelle start her bank account with?

22) When John got his puppy, she weighed 8 pounds. Now that she is one year old, her weight

is 60 pounds. What is the percent increase in the puppy’s weight?

23) Irene thinks she has the space for a 45-inch-wide bookcase. It turns out that she only

has space for a 40-inch-wide bookcase. What is the percent error in Irene’s

measurement?

24) The planners of a school carnival estimate that they will sell 500 hot dogs. They only sell

400. What is the percent error in their estimate?

25) If the original price of a sweater was $75.99. The sweater is now on sale for $62.50,

what was the percent decrease in the price of the sweater? (Round to the nearest 0.1%)

Unit 9: Statistical Analysis & Probability

PROBABILITY

There are 2 kinds of events: Independent Events and Dependent Events.

Independent Events – 1st event does not affect the 2

nd event

Dependent Events – 1st event does affect the 2

nd event

Probability is a ratio, a comparison of 2 numbers.

EXPERIMENTAL PROBABILITY = NUMBER OF FAVORABLE OUTCOMES

NUMBER OF TRIALS

A box contains 5 green pens, 3 blue pens, 8 black pens, and 4 red pens. A pen is picked

at random. Answer questions 1-9 using the above information.

1) P (blue or red) ________ 2) P (gold) ________ 3) P (green) ____

4) P (green, blue, black, or red) _______________

5) P (blue and green) with replacement _______________

6) P (green and red) with replacement _________________

7) P (green and red) without replacement _________________

8) P (blue, black, and green) without replacement _______________

9) P (black and blue) with replacement ____________________

10) A spinner has four sections: a tree, a flower, a cloud and a sun. Jordan spins the pointer

on the spinner 20 times. The pointer lands on the tree 3 times, the flower 9 times, the

cloud 2 times and the sun 6 times.

A) Based on the experiment, what is the probability the spinner lands on the sun?

B) If Jordan spins the spinner 150 times, how many times should she expect it to land on

the sun?

11)

12) Whitney has a choice of a floral, plaid, or striped blouse to wear with a choice of tan,

black, navy, or white skirt. How many different outfits can she make?

13) You flip 3 coins. How many possible outcomes are there?

STATISTICS

14) Find the MEAN, MEDIAN, MODE, and RANGE of the following set of data: Show work.

15, 12, 21, 18, 25, 11, 17, 19, 20

15) Find the median, upper and lower quartiles and the extremes, and then create a box plot

to display the following data set. Remember your labels!!

1, 3, 5, 4, 3, 6, 2, 7, 6, 4

16) The double box plot shows the costs of MP3 players at two different stores.

On average which department store charges more for an MP3 player?

The following box whiskers diagrams represents how many minutes it take a cleaning service to

clean the same house. Answer questions 17 – 20 below based on the data from the diagram.

17) On average, what service appears to clean the house faster? __________ explain why?

______________________________________________________________________

18) Which service would you say is more consistent?__________ explain why? ____________

_______________________________________________________________________

19) What percent of the time did service B finish under 41 mins? ________________

20) What percent of the time did service A clean at least 34 mins? _________________

Unit 10: Geometry

CIRCLES

Area: A = r2 Circumference: C = d or C = 2r (d is diameter & r is radius)

Exact means to leave your answer in terms of .

Approximate means use the button on your calculator.

1) Approximately what is the area AND circumference of the following circle. Round answer

to nearest tenth.

2) Find the EXACT CIRCUMFERENCE of a circle whose diameter is 10 m.

3) Find the EXACT CIRCUMFERENCE of a circle with a radius of 10 in.

4) Find the EXACT AREA of a circle with a diameter of 12 in.

5) If the circumference of a circle is 12 m, find its radius.

6) If the area of a circle is 36 in2, find its diameter.

7) If the circumference of a circle is 18 cm, find its EXACT area.

8) Find the area of the irregular figure:

9 yd

4 yd

5 yd

2 yd

2 yd

SURFACE AREA & VOLUME

You should know the area formulas for 2-dimensional figures. (square, rectangle, rhombus,

parallelogram, triangle and trapezoid)

Rectangular Prism: Cylinder:

Cube: Rectangular Prism: V= lwh Triangular prism:

Cylinder: Pyramid: , where B is the area of the shape’s base

** Remember your units!

Area and Surface area are units2

Volume is units 3

9) Find the surface area AND volume of the rectangular prism below.

10 m

5 m

15 m

10) First, find the EXACT surface area AND EXACT volume of the following cylinder, then

round your answers to the nearest tenth.

5 in

4 in

11) Find the EXACT volume of the cone, then round your answer to the nearest tenth.

12) Maxwell wants to know how much space his globe occupies. Should he find its surface area

or volume? Explain your answer.

13) A toy maker will paint four sides of this toy chest. He will not paint the bottom

or top surface. How many square feet of the chest will the toymaker paint?

8 ft

25 ft

** Remember: ESTIMATE means to

round the numbers FIRST, before

you do any calculations!!!

14) A store keeps about 240 boxes of crayons in its inventory. If each box of crayons

Measures S 6 inches by 2.5 inches by 4 inches, how many cubic inches is needed to store

ALL of the boxes in its inventory?

15) If the volume of a cylinder with a radius of 3m is 81π m3, what is the height?

16) The volume of a rectangular prism is 40 m3. If the width is 2m and the length is 4m, what

would the height be for the rectangular prism?

Unit 11: Angles

ANGLE RELATIONSHIPS

Complementary angles – Two angles are complementary if the sum of their angle measures is 90

Supplementary angles –Two angles are supplementary if the sum of their angle measures is 180

Vertical Angles – congruent angles formed by 2 intersecting lines. They are opposite each other

Alternate interior angles – interior angles on opposite sides of the transversal. ( if lines are parallel)

Alternate exterior angles – exterior angles on opposite sides of the transversal. ( if lines are parallel)

Corresponding angles – hold the same position on 2 different lines. (congruent if lines are parallel)

The sum of the angles of a triangle is 180.

The sum of the angles of a quadrilateral is 360.

1) Find the complement of a 40 angle. __________

2) Find the supplement of a 55 angle. __________

2 ft

1 ft

3 ft

3) Two angles are vertical angles. If one of the angles measure 135, what is the

measure of the other angle?

4) Could 48°, 37°, 111° be measures of angles of a triangle? __________

For #s 5- 7: Solve for x ALGEGBRAICALLY, and then find the measure of each angle.

5) 6) 7)

8) In a triangle the measure of one angle is 68, the measure of the second angle is 38.

Find the measure of the third angle, x.

9) Given a right triangle with an angle that measures 36, and another angle that is 2x.

Find x.

10) Given quadrilateral WXYZ. If mX = 45, mY = 110, mZ = 65 find the mW.

11) Given quadrilateral ABCD, if mA = x, mB = x + 5, mC = x + 15, and mD = 2x find

the measure of each angle.

Use the diagram below and the given information to answer questions 12 – 21.

Given:

Line m || Line n

t is a transversal

m 1 = 2x – 20

m 4 = x + 5

In a right triangle, the sides that form the

right angle are called the legs. (sides a and b)

The side opposite the right angle is called the

hypotenuse. (Side c)

The hypotenuse is always the longest side of a

right triangle

12) Find the value of x. ______

13) Find the m 4. ______

14) Find the m 2. ______ Justify ___________________________________________

15) Find the m 3. ______ Justify __________________________________________

16) Find the m 5. ______ Justify __________________________________________

17) Find the m 8. ______ Justify __________________________________________

18) 1

8. Justify _____________________________________________________

19) 3

6. Justify _____________________________________________________

20) 1

5. Justify _____________________________________________________

21) Find the m 9. __________

TRIANGLES

The Triangle Inequality states that the sum of any 2 sides of a triangle must be greater than

the third side.

The 3rd side of a triangle must be less than the sum of the two given side lengths but greater

than the difference of the given side lengths.

IMPORTANT FACTS

The number of congruent sides is equal to the number of congruent angles and vice versa.

Largest angle is opposite the longest side and vice versa.

Smallest side is opposite smallest angle and vice versa.

Right Triangles

C

Use the following triangle to answer questions 22-27.

(Triangle Not Drawn To Scale.)

B 22) Name the smallest side = _______

23) Name the smallest angle = _______

24) Name the largest side = _________

60o

25) Name the largest angle = ________

26) Classify by it sides. __________________

27) Classify by it angles. _________________

Use ΔABC to answer questions 28-33. (Triangle Not Drawn To Scale.)

28) Name the smallest side. _____________

29) Name the smallest angle. ____________

30) Name the longest side. ______________

31) Name the largest angle. ______________

32) Classify by its sides. ________________

33) Classify by its angles. ______________

Use the triangle inequality to answer questions 34 & 35.

34) State whether 8, 6, 9 could be the three sides of a triangle. Show work to support your

answer.

35) The lengths of two sides of a triangle are 6 in. and 9 in. What can you say about the length

A

of the third side?

Use the Pythagorean Theorem to answer questions 36 & 37. State your final answer in

simplest radical form.

36)

12 ft 10 ft

16 m 30m

?

?

38) A 50-foot tree fell towards a house. The base of the tree was 40 feet from the house.

How high up did the tree hit the house? Draw a picture.

Unit 12: Linear Equations & Transformations

GRAPHING – LINEAR EQUATIONS

Slope is generally represented by the variable “m”.

Slope = =

m = , when you are finding the slope you should be CONSISTENT!

There are different methods for graphing linear equations:

T-Chart: If you choose this method use at least 4 values for x.

**** Slope-Intercept: y = mx + b, where m is slope and b is the y-intercept

x and y intercepts: to find the x-intercept let y = 0 and solve for x.

to find the y-intercept let x = 0 and solve for y.

y = -2x + 3 y = -2x + 3 y = -2x + 3

x -2x + 3 y (x, y) m = y-intercept x-intercept

2 -2(2) +3 -1 (2, -1) y-intercept = 3 y = -2(0) + 3 0 = -2x + 3

1 -2(1) +3 1 (1, 1) y = 3 -3 = -2x

0 -2(0) +3 3 (0, 3) (0, 3) = x

-2 -2(-2) +3 7 (-2, 7) ( , 0)

Find each slope. Be sure to show all work and be consistent!

1) A (-3, 1) and B (4, 5)

2) C (2, 6) and D (3, 5)

Name the slope and y-intercept for each linear equation.

3) y = -2x + 7

4) y = ½x – 5

5) y = 5x + 1

Find the x-intercept and y-intercept ALGEBRAICALLY for each linear equation.

6) y = -6x + 12 7) y - 4x = 3

Write an equation in slope-intercept form.

8) m = b = 5 9) m = 5 b = -3 10) m = 1 b = -6

11) Graph the equation y = -2x + 3 12) Graph the inequality y ≥ -2x - 4

Use ANY Method. Remember to label. Use ANY Method. Remember to label.

13) Solve the system of equations by graphing. y = x – 1 AND y = x + 1

What is the solution? __________

Using the graphs shown below, state the equation of the line.

14) 15)

_________________ _________________

Unit 13: Functions

Relation— Any pairing of the elements in one set (domain) with the elements in another set (range). A

relation is usually represented by a set of ordered pairs (x, y) where the domain is x and the range is y.

Function— A special type of relation in which each value of the domain is paired with exactly one value of

the range

State whether the following relations are functions. EXPLAIN YOUR ANSWER!

1) (-2, 4), (2, 4) (4, 2) (-2, -4) ______________________________________________

______________________________________________________________________

2) (3, -2), (6, 1) (-3, 5) (4, 1) ______________________________________________

______________________________________________________________________

3)

Input -

4

0 4 8

Outpu

t

8 0 8 3

2

State the ordered pairs of the relation graphed below.

______ _______ _______ _______ _______

State the domain. __________________________

State the range. __________________________

Is the relation a function? Why?

____________________________________________

____________________________________________

4) Bryce is deciding whether a graph is a function. What feature of the graph assures that

the graph is a function?

A) The graph has a vertical line of symmetry.

B) The graph has a horizontal line of symmetry.

C) A horizontal line can be drawn that will intersect the graph at only one point.

D) Every possible vertical line that can be drawn will intersect the graph at only one point.

5) Which set of ordered pairs does NOT represent a function?

A) {(3, -2) (4, -3) (5, -4) (6, -5)} B) {(3, -2) (3, -4) (4, -1) (4, -3)}

C) {(3, -2) (5, -2) (4, -2) (-1, -2)} D) {(3, -2) (-2, 3) (4, -1) (-1, 4)}

6) The amount of orange juice in a bottle is modeled by the equation y = 128 – 8x. If y is the

number of ounces of juice left and x is the number of servings of juice poured from the

bottle, how many ounces of juice are in an unopened bottle?

A) 208 ounces B) 136 ounces

C) 128 ounces D) 120 ounces

7) In the graph below, how could the behavior of the graph between D and E be described?

A) linear and increasing B) Linear and decreasing

C) nonlinear and increasing D) nonlinear and decreasing

8) This relationship represented in this table names a function where x is the independent

variable and y is the dependent variable:

x y

0 5

1 5

2 6

3 6

Is this relation a function? Explain? ____________________

________________________________________________

9) Tyler and Jason were discussing the relation shown by the table below. The boys decided

that that if x is the input and y is the output that the relation would not be a function,

however, if they were allowed to reverse the input and output, the relation would be a

function. Do you agree or disagree with the boys claim? Support your argument with

details about functions.

x y

3 -3

2 -2

1 -1

0 0

1 1

2 2

3 3

10) Two relations are drawn on the grids below. Which graph represents a function? _______

Explain how you can use the graphs to decide.

A) y B) y

x

Write the function rule (an equation in slope-intercept form) to express the relationship

between x and y.

x y

1 -

4

5 0

7 2

1

0

5

x y

2 5

4 9

0 1

1

0

21

11) 12)

Unit 14: Transformations

1) Which transformation is a slide (shift) in the coordinate plane? ______________

2) Which transformation is a flip in the coordinate plane? ______________

3) Which transformation is a turn in the coordinate plane? ______________

4) In which transformation does the figure get larger or smaller? ___________

5) In a dilation if the scale factor is greater than one then the image will get __________

and if the scale factor is less than one then the image will get __________.

6) Which transformations are examples of an isometry? __________________________

Transformation Rules

Reflection over the x – axis: (x, y) (x, -y)

Reflection over y – axis: (x, y) (-x, y)

Rotate 90˚ clockwise: (x, y) (y, -x)

Rotate 90˚ counter-clockwise: (x, y) (-y, x)

Rotate 180˚ (x, y) (-x, -y)

Dilation: (x, y) (Kx, Ky), where K is the scale factor

Translations P(x, y) P ’ (x + a, y + b)

*Remember, negate means to take the opposite*

7) Find the coordinates of point A’ after point A (-3, 5) is reflected over the y-axis.

8) Find the coordinates of A’ after point A (-3, 5) is reflected over the x-axis.

9) Find the coordinates of A’ after point A (-3, 5) is translated according the rule T-2, 6

10) Find the coordinates of B’, the image of point B (2, -8) after a dilation of 3.

11) Find the coordinates of point R’, the image of point R (4, -7) after a 90° Clockwise

Rotation.

12) Find the coordinates of point T’, the image of point T (3, -9) after a 180° rotation.

13) Given trapezoid MATH, graph and label M’A’T’H’ after a dilation with a scale factor of 2.

14) Given ABC, graph A’B’C’ after a reflection over the x-axis.

15) Given ABC, graph A’B’C’ after a 90 counterclockwise rotation, then translate A’B’C’

using the translation rule T7,-4

16) Graph A(-5, 5) B(-2, 5) C(-5, 2) and D(-2, 2). Then rotate ABCD 180 degrees and

label it A’B’C’D’

A final note from your teacher:

This review packet is a resource to help you when studying for your final exam, it

is NOT a complete study guide. In addition, you have all the tools you will need to

answer each question on the final exam . . . that does not mean you have seen an

example of every question. Finally, please remember that your final exam grade

will affect your placement for next year so study and prepare accordingly . . . no

placement is guaranteed!

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