Final Exam Review:Final Exam Review: Part II (Chapters 9+) Part II (Chapters 9+)

55thth Grade Advanced Math Grade Advanced Math

First topic!First topic!

Chapter 9Chapter 9Integers Integers

DefinitionDefinition Positive integer – a number Positive integer – a number

greater thangreater than zero. zero.

0 1 2 3 4 5 6

DefinitionDefinition Negative number – a Negative number – a

number number less thanless than zero. zero.

0 1 2 3 4 5 6-1-2-3-4-5-6

Place the following Place the following integers in order from integers in order from

least to greatestleast to greatest::

297 -56 39 -125 78 297 -56 39 -125 78 00

Place the following Place the following integers in order from integers in order from

least to greatestleast to greatest::

297 -56 39 -125 78 297 -56 39 -125 78 00

-125 -56 0 39 78 -125 -56 0 39 78 297297

DefinitionDefinition

Absolute Value – The Absolute Value – The distancedistance a number is from a number is from zero on the number linezero on the number line

The absolute value of 9 or of –9 is 9.

HintHint

If you don’t see a negative or If you don’t see a negative or positive sign in front of a positive sign in front of a number, it is number, it is ALWAYSALWAYS positive.positive.

9+

Integer Addition Integer Addition RulesRules

Rule #1 – When adding two Rule #1 – When adding two integers with the integers with the same signsame sign, , ADDADD the numbers and keep the sign.the numbers and keep the sign.

9 + 5 = 14-9 + -5 = -14

SolveSolve the Following the Following Problems:Problems:

-3 + -5 =-3 + -5 = 4 + 7 =4 + 7 = (+3) + (+4) =(+3) + (+4) = -6 + -7 = -6 + -7 = 5 + 9 =5 + 9 = -9 + -9 = -9 + -9 =

Check Your Check Your AnswersAnswers::

-3 + -5 =-3 + -5 = 4 + 7 =4 + 7 = (+3) + (+4) =(+3) + (+4) = -6 + -7 = -6 + -7 = 5 + 9 =5 + 9 = -9 + -9 = -9 + -9 =

-8

-18

14-13

+7

+11

Solve the following:

1. 8 + 13 =2. –22 + -11 =3. 55 + 17 =4. –14 + -35 =

Check Your Answers

1. 8 + 13 = +21

2. –22 + -11 = -33

3. 55 + 17 = +72

4. –14 + -35 = -49

Integer Addition Integer Addition RulesRules

Rule #2 – When adding two Rule #2 – When adding two integers with integers with different signsdifferent signs, find , find the the differencedifference ( (SUBTRACTSUBTRACT) and ) and take the sign of the larger number. take the sign of the larger number.

-9 + +5 =9 - 5 = 4

Larger absolute value:

Answer = - 4

Solve These ProblemsSolve These Problems

3 + -5 =3 + -5 = -4 + 7 =-4 + 7 = (+3) + (-4) =(+3) + (-4) = -6 + 7 = -6 + 7 = 5 + -9 =5 + -9 = -9 + 9 = -9 + 9 =

-2 5 – 3 = 2

0 -4

+1-1

+3

9 – 9 = 0

9 – 5 = 4

7 – 6 = 1

4 – 3 = 1

7 – 4 = 3

Solve the following:

1. –12 + 22 =2. –20 + 5 = 3. 14 + (-7) =4. –70 + 15 =

Check Your Answers

1. –12 + 22 = +102. –20 + 5 = -153. 14 + (-7) = +74. –70 + 15 = -55

Integer Subtraction RuleSubtracting a negative number is the

same as adding a positive one. Change the sign and add.“Keep, change, change.”

2 – (-7) is the same as 2 + (+7)

2 + 7 = 9

Here are some more examples.

12 – (-8)

12 + (+8)

12 + 8 = 20

-3 – (-11)

-3 + (+11)

-3 + 11 = 8

Solve the following:

1. 8 – (-12) =

2. 22 – (-30) =

3. – 17 – (-3) =

4. –52 – 5 =

Check Your Answers

1. 8 – (-12) = 8 + 12 = +20

2. 22 – (-30) = 22 + 30 = +52

3. – 17 – (-3) = -17 + 3 = -14

4. –52 – 5 = -52 + (-5) = -57

Integer Multiplication Integer Multiplication RulesRules

Rule #1 Rule #1 When When multiplyingmultiplying two two integersintegers with the with the same signsame sign, ,

the the productproduct is always is always positivepositive..

Rule #2Rule #2 When When multiplyingmultiplying two two integersintegers with with different signsdifferent signs,, the the productproduct is always is always negativenegative..

Rule #3Rule #3 If the number of If the number of negative signsnegative signs is is eveneven, , the the productproduct is always is always positivepositive. .

Rule #4Rule #4 If the number of If the number of negative signsnegative signs is is oddodd, , the the productproduct is always is always negativenegative. .

Solve the following:

1. +8 x (-12) =

2. -20 x +30 =

3. – 17 x (-3) =

4. +50 x +5 =

Check Your Answers:

1. +8 x (-12) = -96

2. -20 x +30 = -600

3. – 17 x (-3) = +51

4. +50 x +5 = +250

Integer Division Integer Division RulesRules

Rule #1 Rule #1 When When dividingdividing two two integersintegers with the with the same same signsign, ,

the the quotientquotient is always is always positivepositive..

Rule #2Rule #2 When When dividingdividing two two integersintegers with with different different signssigns,, the the quotientquotient is always is always negativenegative..

Solve the following:

1. (-36) ÷ 4 =

2. 200 ÷ -5 =

3. – 18 ÷ (-9) =

4. +50 ÷ +5 =

Check Your Work:

1. (-36) ÷ 4 = -9

2. 200 ÷ -5 = -40

3. – 18 ÷ (-9) = +2

4. +50 ÷ +5 = +10

Evaluate the following:

1. -45 + 10 3² - 2

2. 7 + -4(9 – 4) =

3. -50 ÷ 5² + (3 - 6) =

Check Your Work:

1. -45 + 10 -35 = -5 3² - 2 7

2. 7 + -4(9 – 4) = -13

3. -50 ÷ 5² + (3 - 7) = -6

Evaluate the following if n = -2 :

1. -5 (2n – 2)²

2. -48n - 6

Check Your Work:

1. -5 (2n – 2)² -180

2. -48 6n - 6

Next chapter…Next chapter…

Chapter 11Chapter 11Expressions & Expressions &

Equations Equations

Write an algebraic expression for Write an algebraic expression for the following. Tell what the the following. Tell what the

variable represents.variable represents.

Ben has 12 pencils. He Ben has 12 pencils. He lost 3 and bought some lost 3 and bought some

more.more.

Write an algebraic expression for the Write an algebraic expression for the following. Tell what the variable following. Tell what the variable

represents.represents.

Ben has 12 pencils. He lost Ben has 12 pencils. He lost 3 and bought some more.3 and bought some more.

12 – 3 + p 12 – 3 + p (p = pencils that Ben (p = pencils that Ben

bought)bought)

Write each algebraic expression in Write each algebraic expression in words words

s - 47s - 47

Write each algebraic expression Write each algebraic expression in words in words

s - 47s - 47

47 less than some number47 less than some number

Write each algebraic equation in Write each algebraic equation in words words

½ n = 16½ n = 16

Write each algebraic equation in Write each algebraic equation in words words

½ n = 16½ n = 16

one half of some number is 16one half of some number is 16

Evaluate the following Evaluate the following algebraic expressions for algebraic expressions for

the given value of the the given value of the variable:variable:(n = 11) (n = 11)

(n + 25)(n + 25) 99

Evaluate the following Evaluate the following algebraic expressions for algebraic expressions for

the given value of the the given value of the variable:variable:(n = 11) (n = 11)

(n + 25)(n + 25) 9 9 44

Simplify the following expressions Simplify the following expressions (combine like terms). Then (combine like terms). Then

evaluate the expression for the evaluate the expression for the

given value of the variable:given value of the variable: (if (if a = 3)a = 3)

3a + 10 - a 3a + 10 - a

Simplify the following expressions (combine Simplify the following expressions (combine like terms). Then evaluate the expression like terms). Then evaluate the expression

for the given value of the variable: (if a = for the given value of the variable: (if a = 3) 3)

3a + 10 - a3a + 10 - a2a + 102a + 10

2(3) + 10 2(3) + 10

1616

Write an algebraic equation for Write an algebraic equation for the following and evaluate. Tell the following and evaluate. Tell what the variable represents.what the variable represents.

Sam had fish in his fish tank. 6 Sam had fish in his fish tank. 6 of them died. There were 12 of them died. There were 12

left swimming in the tank.left swimming in the tank. How many fish did Sam have How many fish did Sam have

originally?originally?

Sam had fish in his fish tank. 6 of Sam had fish in his fish tank. 6 of them died. There were 12 left them died. There were 12 left

swimming in the tank.swimming in the tank. How many fish did Sam have How many fish did Sam have

originally?originally?

f – 6 = 12f – 6 = 12f = 18f = 18

f is the number of fish Sam had f is the number of fish Sam had originally in his tank.originally in his tank.

Evaluate the following algebraic Evaluate the following algebraic equations. equations.

Show your work.Show your work.

30 = 5y 30 = 5y ¼n = 7¼n = 78 = x ÷ 9 3 = 18 + s 8 = x ÷ 9 3 = 18 + s t - 12 = 23 y ÷ 6 = t - 12 = 23 y ÷ 6 = 77

Evaluate the following algebraic Evaluate the following algebraic equations. equations.

Show your work.Show your work.

30 = 5y 30 = 5y 6 6 ¼n = 7 ¼n = 7 2828 8 = x ÷ 9 8 = x ÷ 9 72 72 3 = 18 + s 3 = 18 + s --

1515 t - 12 = 23 t - 12 = 23 3535 y ÷ 6 = y ÷ 6 = 7 7

4242

For word problem For word problem practice, review textbook practice, review textbook

pages pages 331, 340 and 341. 331, 340 and 341.

Next topic…Next topic…

Chapter 12:Chapter 12:Patterns Patterns

Guess What’s Next

A. What is the Rule?A. What is the Rule?B. What are the next 3 numbers in the B. What are the next 3 numbers in the sequence? sequence?

20 30 45

44 36 28 _______ _______ _______

_______ _______ _______

A. What is the Rule?A. What is the Rule?B. What are the next 3 numbers in the B. What are the next 3 numbers in the sequence? sequence?

Rule is: Subtract 8

Rule is: Multiply by 1.5

20 12 4

67.5 101.25 151.875

A. What equation shows the function?A. What equation shows the function?B. Find the missing term.B. Find the missing term.

xx yy

11 11

22 88

33 2727

44

55 125125

A. What equation shows the function?A. What equation shows the function?B. Find the missing term.B. Find the missing term.

xx yy

11 11

22 88

33 2727

44 6464

55 125125

Equation: x³ = y

Missing Term: 64

A. What equation shows the function?A. What equation shows the function?B. Find the missing terms.B. Find the missing terms.

ww 3030 3535 4040 4545 5050

tt 66 77 1010

A. What equation shows the function?A. What equation shows the function?B. Find the missing terms.B. Find the missing terms.

Equation: w ÷ 5 = t Missing Term: 8, 9

ww 3030 3535 4040 4545 5050

tt 66 77 1010

Draw the seventh possible figure in the pattern.How many squares will it have?

Draw the seventh possible figure in the pattern.How many squares will it have? 7 x 7 49 small squares

Find the Find the 99thth term term in the sequence. in the sequence.

20, 40, 60, 80………… 20, 40, 60, 80…………

What is the rule?What is the rule?Write the rule as an algebraic Write the rule as an algebraic

expression.expression.

Find the Find the 99thth term term in the sequence: in the sequence: 180180

20, 40, 60, 80………… 20, 40, 60, 80…………

What is the rule?What is the rule? Multiply the Multiply the positionposition of the term by of the term by 20.20.

Write the rule as an algebraic expression: Write the rule as an algebraic expression: 20n20n

Next chapter…Next chapter…

Chapter 13Chapter 13Graph Relationships Graph Relationships

Inequalities Inequalities

Inequality: is an algebraic sentence Inequality: is an algebraic sentence that contains the symbol: that contains the symbol:

> (greater than)> (greater than)

< (less than)< (less than)

≥ ≥ (greater than or equal to)(greater than or equal to)

≤ ≤ (less than or equal to)(less than or equal to)

≠ ≠ (not equal to)(not equal to)

***Inequalities can be graphed on a ***Inequalities can be graphed on a number line***number line***

Graphing Inequalities Graphing Inequalities on a Number Line on a Number Line

Graphing Functions Graphing Functions

Function: a relationship between Function: a relationship between two numbers or variables. One two numbers or variables. One quantity depends uniquely on the quantity depends uniquely on the otherother

Remember your Remember your Quadrants!Quadrants!

Plotting CoordinatesPlotting Coordinates

(x,y)(x,y)

Find the point on the x-axis first Find the point on the x-axis first

(horizontal / left to right)(horizontal / left to right)

Then find the point on the y-axis Then find the point on the y-axis and graph (vertical / up and down)and graph (vertical / up and down)

Linear EquationsLinear Equations

When graphing a function, some When graphing a function, some functions form a straight linefunctions form a straight line

Equations thatEquations that

are straight linesare straight lines

when graphed arewhen graphed are

called called linear linear

equationsequations

Next topic…

Chapter 22: Chapter 22: Ratio and ProportionRatio and Proportion

Ratios, rates, unit rates, Ratios, rates, unit rates, maps & scales, maps & scales,

solving proportionssolving proportions

Use the picture to write the ratios. Tell whether the ratio compares

part to part, part to whole, or whole to part.

All shapes to triangles.

Rectangles to ovals.

Ovals to all shapes.

Use the picture to write the ratios. Tell whether the ratio compares

part to part, part to whole, or whole to part.

All shapes to triangles.

18 : 9 whole to part

Rectangles to ovals.

3 : 6 part to part

Ovals to all shapes.

6 : 18 part to whole

Which of the following shows two equivalent ratios?

a. 7 : 9 and 14 : 16

b. 7 : 9 and 14 : 18

Which of the following shows two equivalent ratios?

b. 7 : 9 and 14 : 18

7 = 14

9 18

Write two equivalent ratios for each of the following.

a. 12 : 15

b. 1

3

Write two equivalent ratios for each of the following.

a. 12 : 15 24 : 30 4 : 5

b. 1 2 3

3 6 9

*Note: There is more than 1 right answer.

Tell whether the ratios form a proportion.

Write yes or no.

4 and 26 24 and

27

10 65 6 9

Tell whether the ratios form a proportion.

Write yes or no.

4 and 26 24 and

27

10 65 6 9

Yes No

Solve the following proportions using Cross Products. Show your work!!

8 = x 9 =12

36 54 x20

Solve the following proportions using Cross Products. Show your work!!

8 = x 9 =12

36 54 x20

36x = 8(54) 12x = 9(20)

36x = 432 12x = 180

36 36 12 12

x = 12 x = 15

Find the Find the %% of the number. of the number.

75% of 12075% of 120

Find the Find the %% of the number. of the number.

75% of 12075% of 120

.75 x 120 = 90.75 x 120 = 90

Find the Find the %% of the number. of the number.

30% of 5030% of 50

Find the Find the %% of the number. of the number.

30% of 5030% of 50

.30 x 50 = 15.30 x 50 = 15

Find the Find the %% of the number. of the number.

6% of 3006% of 300

Find the Find the %% of the number. of the number.

6% of 3006% of 300

.06 x 300 = 18.06 x 300 = 18

What is the What is the unit rateunit rate ? ? Show your work!!Show your work!!

a. Earn $56 for an 8 hour day

b. Score 120 points in 15 games

What is the What is the unit rateunit rate ? ? Show your work!!Show your work!!

a. $$ $56 = x

hours 8 1 x = $7 per hour

b. points 120 = x

games 15 1 x = 8 points per game

If the If the map scalemap scale is 1 in. = 15 is 1 in. = 15 miles, what is the miles, what is the map distancemap distance if if

the the actual distance is 60 miles? actual distance is 60 miles?

If the If the map scalemap scale is 1 in. = 15 is 1 in. = 15 miles, what is the miles, what is the map distancemap distance if if

the the actual distance is 60 miles? actual distance is 60 miles?

InchInch 1 1 = = xxMiles 15 60Miles 15 60

15x = 1(60)15x = 1(60)15x15x = = 606015 1515 15

x = 4 inchesx = 4 inches

It takes Kenny 25 minutes to It takes Kenny 25 minutes to inflate the tires of 50 inflate the tires of 50

bicycles. How long will it bicycles. How long will it take him to inflate the tires take him to inflate the tires

of 120 bicycles?of 120 bicycles?

It takes Kenny 25 minutes to It takes Kenny 25 minutes to inflate the tires of 50 bicycles. inflate the tires of 50 bicycles.

How long will it take him to How long will it take him to inflate the tires of 120 bicycles?inflate the tires of 120 bicycles?

minutes minutes 25 25 = = xx bicycles 50 120 bicycles 50 120

50x = 25 (120) 50x = 25 (120) 50x50x = = 3,0003,000

50 50 50 50

x = 60 minutes x = 60 minutes

How many pizzas do you need for How many pizzas do you need for a party of 135 people a party of 135 people

if at the last party, if at the last party, 90 people ate 52 pizzas?90 people ate 52 pizzas?

How many pizzas do you need for a party How many pizzas do you need for a party of 135 people if at the last party, 90 of 135 people if at the last party, 90

people ate 52 pizzas?people ate 52 pizzas?

pizzas pizzas 52 52 = = xx people 90 135 people 90 135

90x = 52 (135) 90x = 52 (135) 90x90x = = 7,0207,020

90 90 90 90

x = 78 pizzas x = 78 pizzas

Next chapter….Next chapter….

Chapter 18: Chapter 18: Measurement Measurement Customary measurement of Customary measurement of length, mass and volume length, mass and volume

Metric measurement of Metric measurement of

length, mass and volumelength, mass and volume

Customary Customary MeasurementsMeasurements A system of measurement used in A system of measurement used in

the United States used to the United States used to describe how long, how heavy, or describe how long, how heavy, or how big something ishow big something is

Examples: inches, feet, yards, Examples: inches, feet, yards, milesmiles

Customary Customary Measurement of Measurement of lengthlength12 inches = 1 foot12 inches = 1 foot

3 feet = 1 yard3 feet = 1 yard

36 inches = 1 yard36 inches = 1 yard

5,280 feet = 1 mile5,280 feet = 1 mile

Customary Customary Measurements of Measurements of weight/massweight/mass

16 ounces (0z) = 1 pound (lb)

2000 pounds (lbs) = 1 ton (T)

Customary Customary Measurement of Measurement of Capacity/ VolumeCapacity/ Volume Capacity/volume: how much a Capacity/volume: how much a

container can holdcontainer can hold

8 fl oz = 1 cup2 cups = 1 pint2 pints = 1 quart2 quarts = 1/2 gallon4 quarts = 1 gallon

Metric Measurements Metric Measurements

A system of measurement used in A system of measurement used in most other countries to measure most other countries to measure how long, how heavy, or how big how long, how heavy, or how big something issomething is

Metric Measurements Metric Measurements of Length of Length 10 millimeters (mm) = 1

centimeter (cm)

100 centimeters = 1 meter (m)

1,000 meters = 1 kilometer (km)

Metric Measurements Metric Measurements of Weight/Massof Weight/Mass 1,000 milligrams (mg) = 1 gram

(g)

1,000 grams = 1 kilogram (kg)

Metric Measurements Metric Measurements of Capacity/ Volumeof Capacity/ Volume The milliliter (mL) is a metric unit The milliliter (mL) is a metric unit

used to measure the capacities of used to measure the capacities of small containers. Example= a small containers. Example= a dropperdropper

The liter (L) is equal to 1,000 mL, so The liter (L) is equal to 1,000 mL, so it is used to measure the capacities it is used to measure the capacities of larger containers. Example= a of larger containers. Example= a bottle of sodabottle of soda

Remember…Remember…KKinging HHenry’senry’s DDaffyaffy UncleUncle DDrinksrinks CChochoc

MMilkilk

*This can help you with conversions………*This can help you with conversions………

Next topic…Next topic…

Geometry Geometry

Quadrilaterals, Quadrilaterals,

Plotting coordinates on a gridPlotting coordinates on a grid

Perimeter and AreaPerimeter and Area

Volume of rectangular prisms Volume of rectangular prisms

QuadrilateralsQuadrilaterals

Quadrilaterals are any Quadrilaterals are any four-sided four-sided shapes. They must have shapes. They must have straight straight lineslines and be and be two-dimensionaltwo-dimensional..

Examples: squares, rectangles, Examples: squares, rectangles, rhombuses, parallelograms, rhombuses, parallelograms, trapezoids, kitestrapezoids, kites

More about More about quadrilateralsquadrilaterals

The SquareThe Square

The square has four equal sides. The square has four equal sides. All angles of a square equal 90 All angles of a square equal 90

degrees. degrees.

The RectangleThe Rectangle

The Rectangle has four right The Rectangle has four right angles and two sets of parallel angles and two sets of parallel lines. lines.

Not all sides are equal to each Not all sides are equal to each other. other.

The RhombusThe Rhombus A rhombus is a four-sided shape

where all sides have equal length. Also opposite sides are

parallel and opposite angles are equal.

A rhombus is sometimes called a diamond.

The ParallelogramThe Parallelogram

A parallelogram has opposite sides parallel and equal in length.

Also opposite angles are equal.

Plotting Coordinates Plotting Coordinates

Plotting Coordinates Plotting Coordinates (continued) (continued) (x,y)(x,y)

Find the point on the x-axis first Find the point on the x-axis first

(horizontal / left to right)(horizontal / left to right)

Then find the point on the y-axis Then find the point on the y-axis and graph (vertical / up and down)and graph (vertical / up and down)

Finding the Perimeter Finding the Perimeter

To find the To find the perimeterperimeter of most of most

two-dimensional shapes, two-dimensional shapes,

just just add upadd up the sides the sides

AreaArea

Area is the measurement of a Area is the measurement of a shape’s surface.shape’s surface.

Remember that units are squared Remember that units are squared for area!!for area!!

Finding the Area of a Finding the Area of a SquareSquare To find the To find the areaarea of a square, of a square,

multiplymultiply the length times the the length times the widthwidth

A= (l)(w)A= (l)(w) A = 2 x 2A = 2 x 2 A = A = 4 cm4 cm²²

Finding the area of Finding the area of rectanglesrectangles To find theTo find the area area of a rectangle, of a rectangle,

just multiply the length and the just multiply the length and the width. width.

A= (l)(w)A= (l)(w)

VolumeVolume

VolumeVolume is is the amount of space that a substance or object occupies, or that is enclosed within a container

Remember that the units of volume are cubed (example: inches^3) because it measures the capacity of a 3-dimensional figure!

Finding the Volume of Finding the Volume of Rectangular Prisms Rectangular Prisms To find the To find the volumevolume of a rectangular of a rectangular

prism, multiply the length by the prism, multiply the length by the width and by the height of the width and by the height of the figurefigure

V = (l)(w)(h)V = (l)(w)(h) V = 6 x 3 x 4V = 6 x 3 x 4 V = V = 72 cm72 cm³³

Practice, Practice,

Practice, Practice,

Practice! Practice!