1-1 Variables and Expressions



1-1 Variables and Expressions

Variable –

Variable Expression –

Ex:1 Identify each expression as a numerical expression or a variable expression. For a variable expression, name the variable.

a) [pic] b) [pic]

Ex:2 Write a variable expression for the cost of p pens priced at $0.29 each.

|Addition |Subtraction |

|Plus |Minus |

|More than |*Less than |

|The sum of |The difference of |

|Increased by |Decreased by |

|Added to |*Subtracted from |

| |(* means the order is backwards) |

|Multiplication |Division |

|The product of |The quotient of |

|Multiplied by |Divided by |

|Times |The ratio of |

|Twice |Half |

|Exponents | |

|Squared | |

|Cubed | |

|To the ___________ power | |

Write a variable expression for each verbal expression.

a) x minus 5

b) x less than 5

c) y cubed

d) half of x increased by 8

e) the quotient of y and 7

f) the quotient of 2 and x

1-2 The Order of Operations

Order of operations:

1. Work inside grouping symbols. ( ), { }, [ ], or

numerator/denominator

2. Simplify any exponents.

3. Multiply and divide in order from left to right.

4. Add and subtract in order from left to right.

Simplify –

Ex:1 Simplify [pic]

Ex:2 Simplify [pic]

Ex:3 Simplify [pic]

1-3 Evaluating Expressions

Evaluate –

Ex:1 Evaluate [pic] for [pic]

Ex:2 Evaluate [pic] for [pic], [pic], and [pic]

Ex:3 The Omelet Café buys cartons of 36 eggs.

a) Write a variable expression for the number of cartons needed for [pic] eggs.

b) Evaluate the expression for 180 eggs.

Ex:4 The One Pizza restaurant makes one kind of pizza, which costs $16. The delivery charge is $2. Write a variable expression for the cost of having pizzas delivered.

Evaluate the expression to find the cost of having two pizzas delivered.

1-4 Integers and Absolute Value

Ex:1 Write a number to represent the temperature shown by the thermometer.

****Label the temperatures on the thermometer and shade it to match Mrs. Cafin’s picture.

[pic]

Opposites –

Integers –

Absolute value –

Ex:2 Graph 2, -2, and -3 on a number line. Order the numbers from least to greatest.

Ex:3 Use a number line to find [pic] and [pic]

1-5 Adding Integers

Additive inverse –

Addition of Opposites:

Whenever you add 2 opposite numbers, you always get an answer of 0.

Ex:1 Use tiles to find [pic]

Ex:2 From the surface, a diver goes down 20 feet and then comes back up 4 feet. Find [pic] to find where the diver is.

Adding Same Signs:

Add the two numbers, ignoring their signs, then give the answer the same sign as the original numbers.

Positive # + Positive # = Positive #

Negative # + Negative # = Negative #

Adding Different Signs:

Take the difference (subtract) of the absolute values and keep the sign of the original number that was farthest from 0 on a number line.

Ex: 3 Find each sum.

a) [pic] b) [pic]

Ex:4 A player scores 22 points. He then gets a penalty of 30 points. What is the player’s score after the penalty?

Ex:5 Find [pic]

1-6 Subtracting Integers

When subtracting, change – to + and change the sign of the number that was being subtracted to its opposite.

Ex:1 Find [pic]

Ex:2 Find [pic]

Ex:3 An airplane left Houston, Texas, where the temperature was [pic]F. When the airplane landed in Anchorage, Alaska, the temperature was [pic]F lower. What was the temperature in Anchorage?

Ex:4 Find [pic]

1-7 Inductive Reasoning

Inductive Reasoning –

Conjecture –

Counterexample –

Ex:1 Use inductive reasoning. Make a conjecture about the next figure in the pattern. Then draw the figure.

**Copy the figure from Mrs. Cafin**

Ex:2 Write a rule for each number pattern.

a) 0, -4, -8, -12, …

b) 4, -4, 4, -4, …

c) 1, 2, 4, 8, 10, …

Ex:3 Write a rule for the number pattern 110, 100, 90, 80, … Find the next two numbers in the pattern.

Ex:4 A child grows an inch for three years in a row. Is it a reasonable conjecture that this child will grow an inch in the year 2015?

Ex:5 Is each conjecture correct or incorrect? If it is incorrect, give a counterexample.

a) Every triangle has three sides of equal lengths.

b) The opposite of a number is negative.

c) The next figure in the pattern below has 16 dots.

***Copy the figures from Mrs. Cafin***

1-8 Look for a Pattern

Ex:1 Each student on a committee of five students shakes hands with every other committee member. How many handshakes will there be in all?

a) How many handshakes will the first student participate in?

b) How many new handshakes will the next student have?

c) Make a table to extend the pattern to 5 students.

1-9 Multiplying and Dividing Integers

When multiplying 2 numbers with the same signs,

The product is positive.

When multiplying 2 numbers with different signs,

The product is negative.

Positive X Positive = Positive

Negative X Negative = Positive

Positive X Negative = Negative

Negative X Positive = Negative

Ex:1 A diver is descending from the surface of the water at a rate of 5 ft/s. Write an expression with repeated addition to show how far the diver is from the surface of the water after four seconds.

Ex:2 Use a pattern to find each product.

a) [pic] b) [pic]

Ex:3 Multiply [pic]

When dividing 2 numbers with same signs,

The quotient is positive.

When dividing 2 numbers with different signs,

The quotient is negative.

Positive [pic]Positive = Positive

Negative [pic]Negative = Positive

Positive [pic]Negative = Negative

Negative [pic]Positive = Negative

Ex:4 Find the average of the differences in the values of a Canadian dollar and a U.S. dollar for 1994-1998.

Value of Dollar (U.S. Cents)

Year Canadian Dollar U.S. Dollar Difference

1994 73 100 -27

1995 73 100 -27

1996 74 100 -26

1997 72 100 -28

1998 68 100 -32

Ex:5 Simplify each quotient.

a) [pic] b) [pic] c) [pic]

d) Find the average of 4, -3, -5, 2, and -8.

1-10 The Coordinate Plane

Coordinate Plane –

X-axis –

Y-axis –

Quadrants –

Origin –

Ordered Pair –

X-Coordinate –

Y-Coordinate –

Draw and label the coordinate plane below here.

Ex:1 Graph the following ordered pairs and identify where they are located.

a) P (2, 4) b) Q (-2, 0) c) R (0, -3) d) S (-4 , -1)

2-1 Properties of Numbers

Commutative Properties: (Comm.)

Arithmetic Algebra

[pic] [pic]

Associative Properties: (Assoc.)

Arithmetic Algebra

[pic] [pic]

Identity Properties: (I.D.)

Arithmetic Algebra

[pic] [pic]

Ex:1 You spend $6 for dinner, $8 for a movie, and $4 for popcorn. Find your total cost. Explain which properties you used.

Ex:2 Name each property shown.

a) [pic] b) [pic] c) [pic]

Ex:3 Use mental math to simplify each expression.

a) [pic] b) [pic]

c) [pic] d) [pic]

Ex:4 Use mental math to find the cost of the groceries.

1 gallon milk $2.30

Bread $1.80

Apples $2.20

Ex:5 Use mental math to simplify each expression.

a) [pic] b) [pic]

c) [pic] d) [pic]

2-2 The Distributive Property

Distributive Property: (Dist.)

Arithmetic Algebra

[pic] [pic]

Ex:1 Find each product mentally.

a) [pic] b) [pic] c) [pic]

Ex:2 Your club sold calendars for $7. Club members sold 204 calendars. How much money did they raise?

Ex:3 Simplify each expression.

a) [pic] b) [pic] c) [pic]

*****Ask Mrs. Cafin for white boards & markers.*****

Ex:4 Use algebra tiles to multiply.

a) [pic] b) [pic] c) [pic]

Ex:5 Multiply.

a) [pic] b) [pic] c) [pic]

2-3 Simplifying Variable Expressions

Term –

Constant –

Like terms –

Coefficient –

Simplify a variable expression -

Deductive reasoning –

Ex:1 Name the coefficients, like terms, and constants.

a) [pic] b) [pic] c) [pic]

Ex:2 Simplify [pic]

Ex:3 Simplify each expression.

a) [pic] b) [pic] c) [pic]

Ex:4 Simplify each expression. Justify each step.

a) [pic]

b) [pic]

2-4 Variables and Equations

Equation –

Open Sentence –

Solution of an equation –

Ex:1 State whether each equation is true, false, or an open sentence. Explain.

a) [pic] b) [pic] c) [pic]

Ex:2 Write an equation for twenty minus x is three. Is the equation true, false, or an open sentence? Explain.

Ex:3 Is the given number a solution of the equation?

a) [pic] ; 1 b) [pic] ; 6

Ex:4 A tent weighs 6 lbs. Your backpack and the tent together weigh 33 lbs. Use an equation to find whether the backpack weighs 27 lbs.

2-5 Solving Equations by Adding or Subtracting

Inverse operations –

Ex:1 Solve each equation.

a) [pic] b) [pic] c) [pic]

Ex:2 Cora measures her heart rate at 123 beats per minute. This is 55 beats per minute more than her resting heart rate, r. Write and solve an equation to find Cora’s resting heart rate.

Ex:3 Solve each equation.

a) [pic] b) [pic] c) [pic]

Ex:4 A soft cover book costs $17 less than its hard cover edition. The soft cover costs $5. Write and solve an equation to find the cost h of the hard cover book.

2-6 Solving Equations by Multiplying or Dividing

Ex:1 Solve each equation.

a) [pic] b) [pic] c) [pic]

Ex:2 Solve each equation.

a) [pic] b) [pic] c) [pic]

Ex:3 Solve each equation.

a) [pic] b) [pic] c) [pic]

2-7 Try, Test, Revise

Take your own notes.

2-8 Inequalities and Their Graphs

Inequality-

Solution of an inequality-

Ex:1 Graph the solutions of each inequality.

a) [pic] b) [pic] c) [pic] d) [pic]

Ex:2 Write an inequality for the graphs.

(Copy graphs from Mrs. Cafin)

a) b)

c) d)

[pic] [pic] [pic] [pic]

less than greater than less than or greater than or

equal to equal to

fewer than more than

no more than no less than

exceeds

at most at least

2-9 Solving One-Step Inequalities by Adding or Subtracting

Ex:1 Solve each inequality. Graph the solutions.

a) [pic] b) [pic] c) [pic]

Ex:2 An airline lets you check up to 65 lbs. of luggage. One suitcase weighs 37 lbs. How much can another suitcase weigh?

Ex:3 Solve each inequality.

a) [pic] b) [pic] c) [pic]

2-10 Solving One-Step Inequalities by Multiplying or Dividing

Ex:1 Solve each inequality.

a) [pic] b) [pic] c) [pic]

**When you divide by a negative number, the inequality symbol must be reversed.

Ex: 2 < 10 but [pic]

[pic]

which is false

[pic]

Ex:2 Solve each inequality.

a) [pic] b) [pic] c) [pic]

Ex:3 Solve each inequality.

a) [pic] b) [pic] c) [pic]

Ex:4 Solve each inequality.

a) [pic] b) [pic] c) [pic]

**When you multiply by a negative number, you must reverse the inequality.

Ex: 5 < 7 but [pic]

[pic]

this is false

[pic]

Ex:5 Solve each inequality.

a) [pic] b) [pic] c) [pic]

Ex:6 Solve each inequality.

a) [pic] b) [pic] c) [pic]

3-1 Rounding and Estimating

**When rounding if digit is below 5, keep the same number. When the digit is 5 or above, round up to the next number.

Ex: 1 Identify the underlined place. Then round each number to that place.

a) 38.41 b) 0.7772 c) 7,098.56

d) 274.9434 e) 5.025 f) 9.851

Ex:2 Estimate by rounding.

a) 355.302 + 204.889 b) 453.56 – 230.07

Front-end estimate –

Ex:3 Estimate using front-end estimation.

a) 6.75 + 2.2 + 9.58

b) $1.07 + $2.49 + $7.40

Clustering-

Ex:4 Estimate using clustering.

a) $4.50 + $5.20 +$5.50

b) 26.7 + 26.2 + 24.52 + 25.25 + 23.9

3-2 Estimating Decimal Products and Quotients

3-3 Mean, Median, and Mode

3-4 Using Formulas

Formula –

Ex:1 Use the formula d = rt . Find d, r, or t.

a) d = 273 miles, t = 9.75 hours

b) d= 540.75 inches, r = 10.5 inches/ year

Ex:2 Use the formula [pic] to estimate the temperature in degrees Fahrenheit for each situation. (n is the number of cricket chirps per minute)

a) 96 chirps/min b) 88 chirps/min c) 66 chirps/min

Perimeter –

Ex:3 Find the perimeter of each rectangle.

a) [pic]

b) [pic]

3-5 Solving Equations by Adding or Subtracting Decimals

Ex:1 Solve each equation.

a) [pic] b) [pic]

Ex:2 A store’s cost plus markup is the price you pay for an item. Suppose a pair of shoes costs a store $35.48. You pay $70. Write and solve an equation to find the store’s markup.

Ex:3 Solve each equation.

a) [pic] b) [pic]

Ex:4 You spent $14.95 for a new shirt. You now have $12.48. Write and solve an equation to find how much money you had before you bought the shirt.

3-6 Solving Equations by Multiplying or Dividing Decimals

Ex:1 Solve each equation.

a) [pic] b) [pic] c) [pic]

Ex:2 You paid $7.70 to mail a package that weighed 5.5 lbs. Write and solve an equation to find the cost per pound.

Ex:3 Solve each equation.

a) [pic] b) [pic] c) [pic]

Ex:4 The formula for batting average is [pic] where a is the rounded batting average, h is the number of hits, and n is the number of times at bat. Suppose your batting average is .222 and you have batted 54 times. How many hits do you have?

3-7 Using the Metric System

3-8 Simplify the Problem

Ex:1 A snail is trying to escape from a well 10 feet deep. The snail can climb 2 feet each day, but at night it slides back 1 foot. How many days will the snail take to climb out of the well?

a) How far up the well will the snail be after the first day and night?

b) How far up the well will the snail be after the second day?

c) How far up after the second day and night?

d) Create a chart with a simpler problem of a 3 ft. and 4 ft. well and try to find a pattern.

e) How many days will it take the snail to escape the 10 ft. well?

4-1 Division and Factors

Divisible-

Divisibility Rules:

Divisible by … If…

2

3

4

5

6

9

10

Ex:1 Is the first number divisible by the second? Explain.

a) 160 by 5 b) 56 by 10 c) 53 by 2

d) 1,118 by 2 e) 64 by 9 f) 472 by 3

g) 174 by 3 h) 43,542 by 9

Ex:2 List the positive factors of each integer.

a) 10 b) 21

c) 24 d) 31

4-2 Exponents

exponent

base ([pic]

power

[pic] is a to the first power

[pic] is a squared

[pic] is a cubed

[pic] is a to the fourth power

Ex:1 Write using exponents.

a) [pic] b) [pic] c) [pic]

Ex:2

a) Simplify [pic]

b) Evaluate [pic] and [pic] for a = 2

Order of operations:

Please Parenthesis (groupings)

Excuse Exponents

My Dear Multiplication/Division from left to right

Aunt Sally Add/Subtract from left to right

Ex:3

a) Simplify [pic]

b) Evaluate [pic] for [pic]

4-3 Prime Factorization and Greatest Common Factor

Prime number –

Composite number –

*The number 1 is neither prime nor composite*

Ex:1 Which numbers 2 to 20 are prime?

Which are composite?

Prime factorization –

Ex:2 Write the prime factorization of each number.

a) 72 b) 121 c) 225 d) 236

Greatest Common Factor –

Ex:3 Use prime factorization to find each G.C.F..

a) 8, 20 b) 12, 87

c) [pic] , [pic] d) [pic] , [pic]

Ex:4 Ladder method for G.C.F.

12 and 8

4-4 Simplifying Fractions

Equivalent fractions –

Simplest form –

Ex:1 Find two fractions equivalent to each fraction.

a) [pic] b) [pic] c) [pic]

Ex:2 Write each fraction in simplest form.

a) [pic] b) [pic] c) [pic]

Ex:3 Write in simplest form.

a) [pic] b) [pic] c) [pic]

4-5 Account for All Possibilities

Ex:1 Aaron, Chris, Maria, Sonia, and Ling are on a class committee. They want to choose two members to present their conclusions to the class. How many different groups of two can they form?

4-6

4-7 Exponents and Multiplication

Multiplying powers with the same base:

Keep the same base and add the exponents.

Arithmetic Algebra

[pic] [pic]

(for positive m and n)

Ex:1 Simplify each expression.

a) [pic] b) [pic] c) [pic]

Ex:2 Simplify each expression.

a) [pic] b) [pic] c) [pic]

Finding the power of a power:

Keep the same base and multiply the exponents.

Arithmetic Algebra

[pic] [pic]

Ex:3 Simplify each expression.

a) [pic] b) [pic] c) [pic]

4-8 Exponents and Division

Dividing Powers with the same base:

When dividing same bases, keep the same base and subtract the exponents.

Arithmetic Algebra

[pic] [pic]

where [pic] and positive integers m and n

Ex:1 Simplify each expression.

a) [pic] b) [pic]

Zero as an exponent:

It makes the answer equal 1

Arithmetic Algebra

[pic] [pic]

for [pic]

Ex:2 Simplify each expression.

a) [pic] b) [pic]

Negative Exponents:

Means the reciprocal of the base.

Arithmetic Algebra

[pic] [pic]

Ex:3 Simplify each expression.

a) [pic] b) [pic]

Ex:4 Write [pic] without a fraction bar.

4-9 Scientific Notation

Scientific notation –

Standard form –

Ex:1 About 6,300,000 people visited the Eiffel Tower in the year 2000. Write this number in scientific notation.

Ex:2 Write 0.00037 in scientific notation.

Ex:3 Write each number in standard notation.

a) [pic] b) [pic]

Ex:4 Write each number in scientific notation.

a) [pic] b) [pic]

Ex:5 Order [pic]and [pic] from least to greatest.

Ex:6 Multiply [pic] and [pic]. Express the result in scientific notation.

Ex:7 In chemistry, one mole of any element contains approximately [pic] atoms. If each hydrogen atom weighs approximately [pic] kg, approximately how much does one mole of hydrogen atoms weigh?

5-1 Comparing and Ordering Fractions

Multiple –

Least Common Multiple (LCM) –

Least Common Denominator (LCD) –

Ex:1 Today, the school’s baseball and soccer teams had games. The baseball team plays every 7 days. The soccer team plays every 3 days. When will the teams have games on the same day again?

Ex:2 Find the LCM of 16 and 36

Ex:3 Find the LCM of [pic] and [pic]

Ex:4 Graph and compare the fractions in each pair.

a) [pic] , [pic] b) [pic] , [pic]

Ex:5 The softball team won [pic] of its games and the hockey team won [pic] of its games. Which team won the greater fraction of its games?

Ex:6 Order [pic] , [pic] , and [pic] from least to greatest.

5-2 Fractions and Decimals

Terminating decimal –

Repeating decimal –

Ex:1 The fuel tank of Scott’s new lawn mower holds [pic] gal. of gasoline. Scott poured 0.4 gal. into the tank. Did Scott fill the tank?

Ex:2 Write each fraction as a decimal. State the block of digits that repeats.

a) [pic]

b) [pic]

Ex:3 Write the numbers in order from least to greatest.

- 0.8, [pic], [pic], 0.125

Ex:4 Write 1.72 as a mixed number in simplest form.

Ex:5 Write [pic] as a fraction in simplest form.

5-3 Adding and Subtracting Fractions

*** You can ONLY add or subtract fractions when they have the SAME denominator (bottom number).

Ex:1 Find each sum or difference. Simplify if possible.

a) [pic]

b) [pic]

Ex:2 Simplify each difference.

a) [pic]

b) [pic]

Ex:3 Suppose one day you rode a bicycle for [pic] hours, and jogged for [pic] hours. How many hours did you exercise?

5-4 Multiplying and Dividing Fractions

Reciprocal –

When multiplying fractions:

1. Change mixed numbers to improper fractions

2. Multiply numerators (top #) for the new numerator

3. Multiply denominators (bottom #) for the new denominator

4. Check to make sure fractions are reduced all the way and to change

improper fractions to mixed numbers.

Ex:1 Find [pic]

Ex:2

a) Find [pic] b) Find [pic]

Ex:3 Keesha’s desktop is a rectangle [pic] ft long and [pic] ft wide. What is the area of her desktop?

When dividing fractions:

1. Change any mixed numbers to improper fractions

2. Change division into multiplication

3. Change the number that was after the [pic] into the reciprocal

4. Follow the steps for multiplying fractions

Ex:4

a) Find [pic] b) [pic]

Ex:5 Find [pic]

5-5 Using Customary Units of Measurement

*******page 776 in Pre-Algebra book has units listed********

Dimensional Analysis –

Ex:1 Choose an appropriate unit of measure. Explain your choice.

a) Weight of a hummingbird

b) Length of a soccer field

Ex:2 Use dimensional analysis to convert 68 fluid ounces to cups.

Ex:3 Fred’s fruit stand sells homemade lemonade in [pic] pint bottles for $1.99. Jill’s fruit stand sells homemade lemonade in [pic] quart containers for the same price. At which stand do you get more lemonade for your money?

5-6 Work Backward

** Copy notes from Mrs. Cafin’s overhead **

5-7 Solving Equations by Adding or Subtracting Fractions

Ex:1 Solve and check each equation.

a) [pic] b) [pic] c) [pic]

Ex:2 Solve and check each equation.

a) [pic] b) [pic]

5-8 Solving Equations by Multiplying Fractions

Ex:1 Solve each equation.

a) [pic] b) [pic] c) [pic]

Ex:2 Solve each equation.

a) [pic] b) [pic] c) [pic]

Ex:3 Solve each equation.

a) [pic] b) [pic] c) [pic]

Ex:4 Solve each equation.

a) [pic] b) [pic] c) [pic]

5-9 Powers of Products and Quotients

Rule for Raising a Product to a Power

To raise a product to a power, each factor must be raised to that power.

Arithmetic Algebra

[pic] [pic] for any positive integer m

Ex:1 Simplify [pic]

Try these on your own.

[pic] [pic] [pic] [pic]

Ex:2

a) Simplify [pic] b) Simplify [pic]

Try these on your own.

[pic] [pic] [pic]

Rule for Raising a Quotient to a Power

Raise the numerator AND the denominator to that power.

Arithmetic Algebra

[pic] [pic]

where b[pic]0 and m is a positive integer.

Ex:3 Simplify each expression.

a) [pic] b) [pic] c) [pic]

Try these on your own.

[pic] [pic]

6-1

6-2

6-3

6-4

6-5 Fractions, Decimals, and Percents

Ex:1 Write each percent as a fraction or a mixed number.

a) 30% b) 175%

Ex:2 Express 7.3% as a decimal.

Ex:3 Express 0.412 as a percent.

Ex:4 Four out of seven members of the chess club are boys. What percent of the chess club members are boys?

7-1 Solving Two-Step Equations

When solving equations undo operations in the following order:

1st addition or subtraction

2nd multiplication or division

3rd exponents

4th groupings or parenthesis

Ex: 1 Solve each equation.

a) [pic] b) [pic]

c) [pic] d) [pic]

Ex: 2 Solve each equation.

a) [pic] b) [pic] c) [pic]

Ex:3 You borrow $350 to buy a bicycle. You agree to pay $100 the first week, and then $25 each week until the balance is paid off. To find how many weeks w it will take you to pay for the bicycle, solve [pic] .

7-2 Solving Multi-Step Equations

**Combine any like terms on either side of the equal sign BEFORE you start solving.

Steps for solving a multi-step equation:

1. Use the Distributive Property if necessary.

2. Combine like terms.

3. Undo addition or subtraction.

4. Undo multiplication or division.

Ex:1 Solve [pic]

Ex:2 The sum of three consecutive integers is 42. (Consecutive integers are integers in order like 4,5,6 or –2,-1,0) Find the integers.

Ex:3 Solve each equation.

a) [pic] b) [pic]

7-3 Multi-Step Equations With Fractions and Decimals

Ex:1 Solve [pic]

Ex:2 Solve [pic]

Ex:3 Solve [pic]

Ex:4 Solve [pic]

4. Write an Equation

Ex:1 A moving van rents for $29.95 a day plus $0.12 a mile. Mr. Reynold’s bill was $137.80 and he drove the van 150 miles. For how many days did he drive the van?

5. Solving Equations With Variables on Both Sides

Steps to solving:

1. Change all subtraction to addition.

2. Get rid of any parenthesis (may have to use distributive property).

3. Combine like terms on the left side of the =

4. Combine like terms on the right side of the =

5. Move variables from right side to the left side by ADDING the opposite

to BOTH side.

6. Simplify.

7. Undo addition.

8. Undo multiplication or division.

Copy examples from the overhead projector below:

7-6 Solving Two-Step Inequalities

Review:

Graph [pic]

Review:

Solve each:

[pic] [pic] [pic] [pic]

[pic] [pic] [pic] [pic]

Ex:1 Solve and graph [pic]

Ex:2 Solve and graph [pic]

Ex:3 A stereo salesperson earns a salary of $1200 per month, plus a commission of 4% sales. The salesperson wants to maintain a monthly income of at least $1500. How much must the salesperson sell each month?

7-7 Transforming Formulas

Ex:1 Solve for the indicated variable.

a) [pic] Solve for s b) [pic] Solve for k

c) [pic] Solve for p

Ex:2 Solve for the indicated variable.

a) [pic] Solve for a b) [pic] Solve for w

c) [pic] Solve for x

Ex:3 You plan a 600-mile trip to New York City. You estimate your trip will take about 10 hours. To estimate your average speed, solve the distance formula [pic] for r. Then substitute to find the average speed.

Ex:4 The high temperature one day in San Diego was 32ºC. Solve [pic] for F. Then substitute to find temperature in degrees Fahrenheit.

Algebra Notes

1-7 The Distributive Property

Distributive Property:

For every real number a, b, and c,

[pic] [pic]

Term –

Constant –

Coefficient –

Like Terms –

Ex:1 Simplify each expression.

a) 13(103) b) 21(101) c) 24(98)

Ex:2 Find the total cost of 6 pairs of socks that cost $2.95 per pair.

Ex:3 Simplify each expression.

a) [pic] b) [pic] c) [pic]

Ex:4 Simplify each expression.

a) [pic] b) [pic] c) [pic]

Ex:5 Simplify each expression.

a) [pic] b) [pic] c) [pic]

Ex:6 Write an expression for each phrase.

a) -2 times the quantity t plus 7.

b) The product of 14 and the quantity 8 plus w.

2-2 Solving Two-Step Equations

Steps to Solving two-step equations:

1. Use the addition or subtraction property of equality to get a variable alone on one side of the equation.

2. Use the multiplication or division property of equality to write an equivalent equation in which the variable has a coefficient of 1.

Ex:1 Solve each equation. Check your answer.

a) [pic] b) [pic] c) [pic]

Ex:2 Suppose tulips are on sale for $0.60 per bulb. What number of bulbs can you order if you have $14 to spend and shipping costs $3?

Ex:3 Solve each equation.

a) [pic] b) [pic] c) [pic]

Ex:4 Solve [pic] Justify each step.

Ex:5 Solve [pic] Justify each step.

2-3 Solving Multi-Step Equations

Ex:1 Solve each equation. Check your answer.

a) [pic] b) [pic]

c) [pic] d) [pic]

Ex:2 A Carpenter is building a rectangular fence for a playground. One side of the playground is the wall of a building 70 ft. wide. He plans to use 340 ft. of fencing material. What is the length of the playground if the width is 70 ft.?

Ex:3 Solve each equation.

a) [pic] b) [pic]

Ex:4 Solve each equation.

a) [pic] b) [pic]

Ex:5 Solve each equation.

a) [pic] b) [pic]

Steps for solving multi-step equations:

1. Clear the equation of fractions and decimals.

2. Use the distributive property to remove parentheses on each side.

3. Combine like terms on each side.

4. Undo addition or subtraction.

5. Undo multiplication or division.

2-4 Equations With Variables on Both Sides

Identity -

****The only way to move a variable from one side of an equation to the other is to add or subtract.

Ex:1 Solve each equation.

a) [pic] b) [pic]

c) [pic] d) [pic]

Ex:2 A hairdresser is considering ordering a certain shampoo. Company A charges $4 per 8-oz bottle plus a $10 handling fee per order. Company B charges $3 per 8-oz bottle plus a $25 handling fee per order. How many bottles must the hairdresser buy to justify using Company B?

Ex:3 Determine whether each equation is an identity or whether it has no solution. (Please remember that these are special cases… the earlier problems had solutions.)

a) [pic] b) [pic]

3-1 Inequalities and Their Graphs

Solution of an inequality –

Ex:1 Is each number a solution of [pic]?

a) -5 b) -4.1 c) 8 d) 0

Ex:2 Is each number a solution of [pic]?

a) 1 b) 2 c) 3 d) 4

Ex:3 Graph each inequality.

a) [pic] b) [pic] c) [pic]

Ex: 4 Write an inequality for each graph. (You will have to copy from Mrs. Cafin)

a) b)

|< |> |[pic] |[pic] |

|Less than |Greater than |Less than or equal to |Greater than or equal to |

|Fewer than |More than |No more than |No less than |

| |Exceeds |At most |At least |

3-2 Solving Inequalities Using Addition and Subtraction

Equivalent inequalities –

Addition Property of Inequality:

For every real number a, b, and c,

If [pic], then [pic]

If [pic], then [pic]

(Also true for [pic] and [pic] )

Ex:1 Solve [pic]. Graph your solution.

Ex:2 Solve [pic]. Graph and check your solution.

Subtraction Property of Inequality:

For every real number a, b, and c,

If [pic], then [pic]

If [pic], then [pic]

(Also true for [pic] and [pic] )

Ex:3 Solve [pic]. Graph and check your solution.

3-3 Solving Inequalities Using Multiplication and Division

Multiplication Property of Inequality for [pic]:

For every real number a, b, and for [pic],

If [pic], then [pic]

If [pic], then [pic]

(Also true for [pic] and [pic] )

Ex:1 Solve each inequality. Graph and check your solution.

a) [pic] b) [pic] c) [pic]

Multiplication Property of Inequality for [pic]:

For every real number a, b, and for [pic],

If [pic], then [pic]

If [pic], then [pic]

(Also true for [pic] and [pic] )

Ex:2 Solve each inequality. Graph and check the solution.

a) [pic] b) [pic] c) [pic]

Division Property of Inequality for [pic]:

For every real number a and b, and [pic],

If [pic], then [pic]

If [pic], then [pic]

(Also true for [pic] and [pic] )

Division Property of Inequality for [pic]:

For every real number a and b, and [pic],

If [pic], then [pic]

If [pic], then [pic]

(Also true for [pic] and [pic] )

Ex:3 Solve the inequality. Graph and check your solution.

a) [pic] b) [pic] c) [pic]

Ex:4 Students in the school band are selling calendars. They earn $0.40 on each calendar they sell. Their goal is to earn more than $327. Write and solve an inequality to find the fewest number of calendars they can sell and still reach their goal.

3-4 Solving Multi-Step Inequalities

Ex:1 Solve each inequality.

a) [pic] b) [pic]

c) [pic] d) [pic]

Ex:2 The band is making a rectangular banner that is 20 ft. long with trim around the edges. What are the possible widths the banner can be if there is no more than 48 ft. of trim?

Ex:3 Solve each inequality.

a) [pic] b) [pic]

c) [pic]

Ex:4 Solve [pic]

Ex:5 Solve [pic]

3-5 Compound Inequalities

Compound inequality –

Ex: 1 Write a compound inequality that represents each situation. Graph the solutions.

a) All real numbers greater than -2 but less than 9.

b) The books were priced between $3.50 and $6.00, inclusive.

**inclusive means between two numbers including the two numbers**

Ex:2 Solve each inequality. Graph your solution.

a) [pic]

b) [pic]

c) [pic]

Ex: 3 Your test grades in science so far are 83 and 87. What possible grades can you make on your next test to have an average between 85 and 90 inclusive?

*Compound inequalities joined by “or” means both inequalities are true.

Ex:4 Write an inequality that represents all real numbers that are at most -5 or at least 3. Graph your solution.

Ex:5 Solve the compound inequality [pic] or [pic]. Graph your solution.

4-1 Ratio and Proportion

Ratio –

Rate –

Unit rate –

Unit analysis –

Proportion –

Extremes of a proportion –

Means of a proportion –

Cross products –

Ex:1 Find the unit rate for apple juice that costs $1.20 for 32 oz.

Ex:2 A sloth travels 0.15 miles per hour. Convert this speed to feet per minute.

Cross Products of a Proportion:

If [pic] then [pic]

Ex:3 Solve each proportion.

a) [pic] b) [pic] c) [pic]

Ex:4 Suppose you walk 2 miles in 35 minutes.

a) Write a proportion to find how far you would walk in an hour if you were to continue at the same rate.

b) Solve the proportion.

Ex:5 Solve each proportion.

a) [pic] b) [pic]

c) [pic] d) [pic]

4-2 Proportions and Similar Figures

Similar figures –

Similar Triangles: (Copy Mrs. Cafin’s triangles below.)

[pic] and [pic]

(Corresponding angles are equal) (Sides are proportional)

Scale drawing –

Scale –

Ex:1 In the figure below, [pic]. Find LM. (Copy the triangles from Mrs. Cafin below.)

Ex:2

a) A tree casts a 26 ft. shadow. A boy standing nearby casts a 12 ft. shadow. His height is 4.5 ft. How tall is the tree?

b) A house casts a 56 ft. shadow. A girl standing nearby casts a 7.2 ft. shadow. Her height is 5.4 ft. What is the height of the house?

Ex:3 The scale of a map is 1 inch:10 miles. The map distance from Valkaria to Gifford is 2.25 inches. Approximately how far is the actual distance?

4-3 Proportions and Percent Equations

[pic] Total will always be the number after “of”.

Ex:1 What is 20% of 70?

Ex:2 What percent of 40 is 20?

Ex:3 15 is 40% of what number?

Ex4 Use fractions to estimate each answer.

a) 49% of 280 is what number?

b) What is 65% of 334?

c) What is 74.2% of 44?

d) 11% of 521 is what number?

Ex:5 A store advertises sneakers on sale for 33% off. The original price of sneakers was $56.

a) Estimate the amount the sneakers have been marked down.

b) Estimate the sale price of the sneakers.

4-4 Percent of Change

Percent of Change –

Percent of Increase –

Percent of Decrease -

Ex:1

a) find the percent of change if the price of a CD increases from $12.99 to $13.99. Round to the nearest percent.

b) Find the percent of change if the CD is on sale, and its price decreases from $13.99 to $12.99. Round to the nearest percent.

Ex: 2 The number of alpaca owners increased from 146 in 1991 to 2,919 in 2000. Find the percent of increase. Round to the nearest percent.

Greatest possible error –

Ex:3 You measure a picture for the yearbook and record its height as 9 cm. What is your greatest possible error?

Ex:4 You measure a wall of your room as 8 feet high and 12 feet wide. Find the minimum and maximum possible areas of the wall.

Percent error:

Ex: 5

a) You measure the length of a table as 168 inches. Find the percent error in this measurement.

b) You measure the length of the table as 168.0 inches. Find the percent error in this measurement.

Ex:6 Suppose you measure your math book and recorded its dimensions as 1 in x 9 in x 10 in. Find the percent error in calculating its volume.

End Algebra Notes

8-1 Relations and Functions

Relation –

Domain –

Range –

Function –

Mapping –

Ex:1 Is each relation a function?

a) {(0,5),(1,6),(2,4),(3,7)}

b) {(0,5),(1,5),(2,6),(3,7)}

c) {(0,5),(0,6),(1,6),(2,7)}

Vertical-line Test –

Ex:2 Graph the relation shown in the table. Use the vertical line test to see if the relation is a function.

|Domain Value |Range Value |

|-3 |5 |

|-5 |3 |

|3 |5 |

|5 |3 |

[pic]

8-2 Equations With Two Variables

Solution –

Ex:1 Find the solution of each equation for [pic]

a) [pic]

b) [pic]

c) [pic]

Linear equation –

Ex:2 Graph each linear equation.

a) [pic]

[pic]

b) [pic]

[pic]

c) [pic]

[pic]

8-3 Slope and y-intercept

Slope –

Positive Negative

Slope Slope

Slope = m = vertical change = rise

Horizontal change run

Given two points [pic] and [pic] [pic]

Ex:1 Find the slope for the following:

a) (2, 3) (4, 5) b) (-1, 8) (10, -9) c) (0, 5) (6, -8)

Special Case Slopes

Slope = 0 Slope is undefined.

Ex:2 Find the slope of the following:

a) (1, 2) (1, 8) b) (-2, 3) (5, 3)

y-intercept –

Slope-intercept form-

Ex:3 Write the following in slope-intercept form then graph.

a) slope = [pic]

y-intercept = 5

[pic]

b) Slope = 3

y-intercept = 0

[pic]

c) Slope = [pic]

y-intercept = -8

[pic]

8-3 Continued Transforming an Equation into Slope-Intercept Form

Change each equation into slope-intercept form by solving for y.

Ex:1 [pic] Ex2: [pic]

Ex:3 [pic] Ex:4 [pic]

8-4 Writing Rules for Linear Functions

function notation –

function rule –

Ex:1 A long-distance phone company charges its customers a monthly fee of $4.95 plus 9¢ for each minute of a long-distance call.

a) Write a function rule that relates the total monthly bill to the number of minutes a customer spent on long-distance calls.

b) Find the total monthly bill if the customer made 90 minutes of long-distance calls.

When writing a function rule from a table, you can use slope intercept form.

Ex:2 Write a rule for each linear function.

a)

|x |f(x) |

|-1 |-2 |

|0 |0 |

|1 |2 |

|2 |4 |

|x |f(x) |

|2 |3 |

|0 |-5 |

|-2 |-13 |

|-4 |-21 |

b)

Ex:3 Write a function rule for the linear function graphed below.

(Copy the line from Mrs. Cafin.)

[pic]

8-5 Scatter Plots

scatter plot –

Ex:1 Refer to the graph used in example 1 on page 423 in your Pre-Algebra book.

a) Describe the person represented by point B.

b) How many people have exactly 12 years of education?

Ex:2 Use the table to make a scatter plot of the elevation and precipitation data.

|City |Elevation Above Sea Level (ft) |Mean Annual Precipitation (in) |

|Atlanta |1050 |51 |

|Boston |20 |42 |

|Chicago |596 |36 |

|Honolulu |18 |22 |

|Miami |11 |56 |

|Phoenix |1072 |8 |

|Portland |75 |44 |

|San Diego |40 |10 |

|Wichita |1305 |29 |

[pic]

Positive Correlation –

Negative Correlation –

No Correlation –

Ex:3 Use the scatter plot below. Is there a positive correlation, a negative correlation, or no correlation between temperatures and amounts of precipitation? Explain.

(Copy points from Mrs. Cafin.)

[pic]

8-6 Solve by Graphing

Trend line –

Ex:1 Use the data in the table below. Suppose this year there are 16 wolves on the island. Predict how many moose are on the island.

|Year |Wolf |Moose |

|1982 |14 |700 |

|1983 |23 |900 |

|1984 |24 |811 |

|1985 |22 |1062 |

|1986 |20 |1025 |

|1987 |16 |1380 |

|1988 |12 |1653 |

|1989 |11 |1397 |

|1990 |15 |1216 |

|1991 |12 |1313 |

|1992 |12 |1600 |

|1993 |13 |1880 |

|1994 |15 |1800 |

|1995 |16 |2400 |

|1996 |22 |1200 |

|1997 |24 |500 |

|1998 |14 |700 |

|1999 |25 |750 |

[pic]

[pic]

8-7 Solving Systems of Linear Equations

System of linear equations –

Solution of the system of linear equations –

(Copy the graphs from Mrs. Cafin.)

No Solution One Solution Infinitely many solutions

Ex:1 Solve each system by graphing. Check each solution.

a) [pic] b) [pic]

[pic] [pic]

Ex:2 Solve each system by graphing.

a) [pic] b) [pic]

[pic][pic][pic]

Ex:3 Find two numbers with a difference of 2 and a sum of -8.

[pic]

8-8 Graphing Linear Inequalities

Linear inequality –

Graphing linear inequalities:

Step 1- replace the >, or < them make the line dotted.

Step 5- pick an ordered pair (a point) far away from the line.

Step 6- substitute that point into the original problem.

Step 7- work out the problem. If it’s true, shade the side of the line with that point. If it’s false, shade the other side.

Ex:1 Graph the following:

a) [pic] b) [pic]

[pic] [pic]

System of linear inequalities –

Ex:2 Solve each system by graphing.

a) [pic] b) [pic]

[pic] [pic]

9-1 Introduction to Geometry: Points, Lines, and Planes

Point-

Line-

Plane-

Segment-

Ray-

Ex:1 Use the figure to name each of the following.

[pic]

a) Four points.

b) Four different segments.

c) Five other names for [pic].

d) Five different rays.

Parallel-

Skew-

Ex:2 Name each of the following:

[pic]

a) Four segments that intersect [pic].

b) Three segments that are parallel to [pic].

c) Four segments skew to [pic].

Ex:3 Draw 2 intersecting lines. Then draw a segment that is parallel to one of the intersecting lines.

9-2 Angle Relationships and Parallel Lines

Acute angle-

Right angle-

Obtuse angle-

Straight angle-

Adjacent angles-

Vertical angles-

Congruent angles-

Supplementary angles-

Complementary angles-

Ex:1 Find the measures of [pic] if [pic].

[pic]

Transversal-

Corresponding angles-

Alternate interior angles-

**When 2 lines are parallel and have a transversal, corresponding angles are congruent and alternate interior angles are congruent.

Ex:2 In the diagram, p||q. Identify each of the following.

[pic]

a) Congruent corresponding angles.

b) Congruent alternate interior angles.

9-3 Classifying Polygons

Polygon-

Triangle-

Acute triangle-

Right triangle-

Obtuse triangle-

Equilateral triangle-

Isosceles triangle-

Scalene triangle-

Ex:1 Classify the triangle by its sides and angles.

[pic]

Quadrilateral-

Trapezoid-

Parallelogram-

Rectangle-

Rhombus-

Square-

Ex:2 Name the types of quadrilaterals that have at least one pair of parallel sides.

Regular polygon-

**You can use algebra to write a formula for the perimeter of a regular polygon.

Ex:3

a) Write a formula to find the perimeter of a regular hexagon.

b) Use the formula to find the perimeter if one side is 16 cm.

9-4 Draw a Diagram

**Ask Mrs. Cafin for geoboards and rubber bands**

Ex:1 How many diagonals does an octagon have?

Step 1- What’s an octagon?

Step 2- What’s a diagonal?

Step 3- Use the geoboard and rubber band to make an octagon.

Use diagonals to connect diagonals AC, AD, AE, AF, and AG. How many diagonals can you make from point A?

Step 4- How many new diagonals can you create from point B?

Step 5- How many new diagonals can you create from point C?

Step 6- Continue and make a table to keep track of the information.

|Vertex |# of Diagonals |

|A | |

|B | |

|C | |

|D | |

|E | |

|F | |

|G | |

|H | |

|Total | |

9-5 Congruence

Congruent figures-

Ex:1 In the figure, [pic]

[pic]

a) Name the corresponding congruent angles.

b) Name the corresponding congruent sides.

c) Find the length of [pic].

Identifying Congruent Triangles:

S.S.S.- Triangles with 3 congruent sides make congruent triangles.

S.A.S.- Triangles with 2 congruent sides and a congruent angle formed by those sides makes congruent triangles.

A.S.A.- Triangles with 2 congruent angles and a congruent side between those angles makes congruent triangles.

Ex:2 List the congruent corresponding parts of each pair of triangles. Write a congruency statement for the triangles.

(Copy the triangles from Mrs. Cafin.)

a) b)

9-6 Circles

[pic]

Circumference-

Formula for Circumference:

[pic] or [pic]

Ex:1 Find the circumference of the circles.

a) [pic] b) [pic]

***** Circles have 360 degrees inside them!**********

Ex:2 Make a circle graph for Jackie’s weekly budget.

|Jackie’s Weekly |Budget |

|Entertainment |20% |

|Food |20% |

|Transportation |10% |

|Savings |50% |

Steps to making a circle graph:

1. Draw a circle large enough to write labels inside.

2. Draw a radius.

3. Measure from the radius the amount of degrees the first category is. (Use a protractor).

4. Mark the circle at the proper degrees and create another radius at that mark. Label.

5. Using the new radius, measure the amount of degrees for the second category.

6. Mark the circle at the proper degrees and create another radius. Label.

7. Repeat steps 5 and 6 until the last category.

8. For the last category, use a protractor to check the measure of the empty section of the circle graph. Some error is ok because of graphing. If the measurement is more than 5( off, check the other categories and your work to find the degrees.

9. Label the last category if measurement is close to what the amount should be.

9-7 Constructions

perpendicular lines-

segment bisector-

perpendicular bisector-

angle bisector-

***Constructions are made with a compass and straightedge (unmarked ruler). An arc is a part of a circle you can draw with a compass.

Ex:1 Draw a line segment. Construct a segment twice the length of the segment you drew.

Ex:2 Draw an obtuse angle. Construct a congruent angle.

Ex:3 Draw a segment. Construct its perpendicular bisector.

Ex:4 Draw an obtuse angle. Construct its angle bisector.

9-8 Translations

Transformation-

Translation-

Image-

Ex:1 Draw the triangle translated by the following ways on the same graph.

a) 4 units to the left

b) 5 units down

c) 4 units to the left and 5 down

[pic]

When a translation moves like example 1 c, then you can write it with arrow notation. So, [pic] moved to [pic] can be written as [pic].

Ex:2 Use arrow notation to describe a translation of B(-1,5) to

B’(3,1).

Ex:3 Write a rule to describe the translation of quadrilateral ABCD to quadrilateral A’B’C’D’.

[pic]

9-9 Symmetry and Reflections

Reflectional symmetry-

Line of symmetry-

Ex:1 Draw all the lines of symmetry.

a) b)

Reflection-

Line of reflection-

Ex:2 Graph [pic] after a reflection over the x-axis.

[pic]

Ex:3 Graph the [pic] after each reflection over each line.

[pic]

a) [pic] b) [pic]

9-10 Rotations

Rotation-

Center of rotation-

Angle of rotation-

Rotational Symmetry-

Ex:1 Rotate the [pic] 90( about the origin.

[pic]

Ex:2 Tell if there is rotational symmetry. If so, what is the angle of rotation?

10-1 Area: Parallelograms

Area-

Altitude-

****Formulas for area on page 780 in Pre-Algebra book****

Ex:1 Find the area of each rectangle.

a)

[pic]

b)

[pic]

Ex:2 Find the area of each parallelogram.

a)

[pic]

b)

[pic]

10-2 Area: Triangles and Trapezoids

Altitude of a triangle-

Ex:1 Find the area of each triangle.

a)

[pic]

b)

[pic]

Ex:2 Find the area of the figure.

[pic]

Area of trapezoid = [pic] [pic]

Ex:3 Find the area of each trapezoid.

a)

[pic]

b)

[pic]

10-3 Area: Circle

Area of a circle:

[pic] ( is about 3.14

r is radius of the circle

A is area of the circle.

Ex:1 Find the exact area of a circle with radius 50 inches. (Exact means to use the ( symbol instead of 3.14.)

Ex:2 Find the approximate area of a circle with radius 6 miles. (Approximate means to use 3.14 for (.)

Ex:3 Find the shaded area of each figure to the nearest tenth.

a) b)

[pic] [pic]

10-4 Space Figures

Space figures-

Prism-

Pyramid-

Cylinder-

Cone-

Sphere-

Ex:1 Name each space figure by its base(s) and shape.

a) [pic] b) [pic]

c) [pic] d) [pic]

Net-

Ex:2 Name the figures you can create from each net.

a) [pic] b) [pic] c) [pic]

10-5 Surface Area: Prism & Cylinders

Surface Area-

Ex:1 Find the surface area of the rectangular prism.

[pic]

Lateral Area-

Lateral Area (LA) = perimeter of a base * height

[pic]

Surface Area (SA) = lateral area + 2 * area of base

[pic]

Ex:2 Find the surface area of the rectangular prism.

[pic]

Surface Area of a Cylinder:

[pic]

[pic]

Ex:3 Find the surface area of the cylindrical water tank.

[pic]

10-6 Surface Area: Pyramids, Cones, and Spheres

Slant height-

Surface Area of a Pyramid:

[pic] p = perimeter of base

[pic] B = area of base

Ex:1 Find the surface area of the square pyramid.

[pic]

Surface Area of a Cone:

[pic]

Ex:2 Find the surface area of this cone.

[pic]

Surface Area of a Sphere:

[pic]

Ex:3 Earth has an average radius of 3,963 mi. What’s the Earth’s approximate surface area to the nearest 1,000 square miles? Assume that Earth is a sphere.

10-7 Volume: Prisms & Cylinders

Volume-

Cubic unit-

Volume of a Prism: Page 780 has list

[pic]

V is volume

B is Base Area

h is height

Ex:1 Find the volume of the triangular prism.

[pic]

Volume of a Cylinder:

V = Bh

V is volume

B is Base Area

h is height

Ex:2

a) Find the volume of the cylinder to the nearest cubic foot.

[pic]

b) How does the volume of this cylinder compare to one having twice its dimensions?

10-8 Make a Model

Ex:1 A can company rolls rectangular pieces of metal that measure 8 in by 10 in to make the sides of cans. Which height, 8 in or 10 in, will make the can with the greater volume?

10-9 Volume: Pyramids, Cones, and Spheres

Volume for cones and pyramids:

[pic]

V is volume, B is base area, h is height

Ex:1 Find the volume of the cone.

[pic]

Ex:2 Find the volume of the square pyramid.

[pic]

Volume of a Sphere:

[pic]

V is volume and r is radius

Ex:3 Earth has an average radius of 3,963 mi. What is Earth’s approximate volume to the nearest 1,000,000 cubic miles? Assume that Earth is a sphere.

11-1 Square Roots and Irrational Numbers

Perfect square-

Square root-

Irrational number-

Ex:1 Simplify each square root.

a) [pic] b) [pic] c) [pic] d) [pic]

Ex:2 Estimate to the nearest integer.

a) [pic] b) [pic] c) [pic] d) [pic]

Ex:3 Identify each number as rational or irrational. Explain.

a) [pic] b) [pic] c) 0.53 d) [pic]

11-2 The Pythagorean Theorem

Legs-

Hypotenuse-

Pythagorean Theorem:

[pic] [pic]

Ex:1 The lengths of two sides of a right triangle are given. Find the length of the third side.

a) legs: 3 feet and 4 feet b) leg: 12 m, hypotenuse: 15 m

Ex:2 In a right triangle, the length of the hypotenuse is 15 m and the length of a leg is 8 m. What is the length of the other leg, to the nearest tenth of a meter?

Ex:3 What is the rise of a roof if the span is 22 feet and the rafter length is 14 feet? Round to the nearest tenth of a foot.

Converse of the Pythagorean Theorem:

If you substitute the sides of a triangle into [pic] and it works out to be true, then the triangle is a right triangle. Remember that c has to be the biggest side of the triangle.

Ex:4 Can you form a right triangle with the three lengths given? Explain.

a) 7 in., 8 in., [pic] in. b) 5 mm, 6 mm, 10 mm

11-3 Distance and Midpoint Formulas

Midpoint –

Distance Formula between two points [pic] and [pic] is

[pic]

Midpoint Formula between two points [pic] and [pic] is

Midpoint = [pic]

Ex:1 Find the distance between the two points. If necessary, round to the nearest tenth.

a) (3,8) and (2,4) b) (10,-3) and (1,0)

Ex:2 Find the perimeter of the triangle DEF. Round to the nearest tenth.

[pic]

Ex:3 Find the midpoint of each segment.

a) [pic] b) [pic]

13-1 Patterns and Sequences

Sequence-

Term-

Arithmetic Sequence-

Common Difference-

Geometric Sequence-

Common Ratio-

Ex:1 What’s the common difference in each sequence?

a) 8, 13, 18, 23, … b) 12, 9, 6, 3, …

Ex:2 Find the next three terms and describe the pattern.

a) 23, 19, 15, 11, … b) [pic]

Ex:3 Find the common ratio and the next three terms of each sequence. Then write a rule to describe the sequence.

a) 4, 12, 36, 108, … b) 4, 2, 1, 0.5, …

Ex:4 Tell if each sequence is arithmetic, geometric, or neither. Then find the next three terms.

a) 3, 9, 27, 81, …

b) 10, 13, 18, 25, …

c) -12, 12, -12, 12, …

d) 50, 200, 350, 500, …

13-2 Graphing Nonlinear Functions

Quadratic functions-

Parabola-

Absolute value functions-

Ex:1 Graph each function.

a) [pic] [pic]

b) [pic]

[pic]

Ex:2 Graph each function.

a) [pic]

[pic]

b) [pic]

[pic]

Ex:3 Graph each function.

a) [pic]

[pic]

b) [pic]

[pic]

13-3 Exponential Growth and Decay

Growth: Decay:

[pic] [pic]

Ex:1 For the function [pic] , make a table with integer values of x from 1 to 4. Then graph the function.

[pic]

Ex:2 For the function [pic] , make a table with integer values of x from 0 to 5. then graph the function.

[pic]

Ex:3 For the function [pic] , make a table with integer values of x from 0 to 5. Then graph the function.

[pic]

13-4 Polynomials

Monomial-

Ex:1 Is each expression a monomial? Explain.

a) [pic] b) [pic] c) [pic] d) [pic]

Polynomial-

Binomial-

Trinomial-

Ex:2 State whether the polynomial is a monomial, binomial, or trinomial.

a) [pic] b) [pic] c) [pic] d) [pic]

Ex:3 Evaluate each polynomial for [pic] and [pic]

a) 2mp b) 3m-2p

Ex:4 The polynomial [pic] gives the height, in feet, reached by fireworks in t seconds. If the fireworks explode 4 seconds after launch, at what height do they explode?

13-5 Adding and Subtracting Polynomials

Ex:1 Simplify.

a) [pic]

b) [pic]

Ex:2 Find the sum of [pic] and [pic]

Ex:3 Simplify [pic]

-----------------------

27.3 cm

16.8 cm

17.4 in.

8.6 in.

G

F

R

S

T

K

R

T

A

B

C

D

A

B

C

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