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# Algebra 1 Chapter 1 Lesson

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Tweet## Lezah Acosta

on 10 January 2013#### Transcript of Algebra 1 Chapter 1 Lesson

1-1 Variables and Expressions Vocabulary quantity- is anything that can be measured or counted. variable- is a symbol, usually a letter, that represents the value(s) of a variable quantity. algebraic expression- is a mathematical phrase that includes one or more variables. numerical expression- is a mathematical phrase involving numbers and operation symbols, but no variables. Sit Back, Relax, and get your thinking caps on! In this chapter, we will learn about...

Variables and Expressions

Order of Operations and Evaluating Expressions

Real Numbers and the Number line

Properties of Real Numbers

Adding and Subtracting Real Numbers

Multiplying and Dividing Real Numbers

The Distributive Property

An Introduction to Equations

Patterns, Equations, and Graphs Question 5+3-7

What is an algebraic expression for the word phrase?

Word Phrase Expression Clues

25 more than a number h h + 25 ADD- more than, plus

25 less a number h 25 - h SUBTRACT- less, less than

25 less than a number h h – 25 SUBTRACT- less, less than

25 times a number h 25 x h, 25 h, 25n MULTIPLY- times, product

the product of 25 and a number h 25 x h, 25 h, 25n MULTIPLY- times, product

the quotient of a number h and 25 n ÷ 25, n/25 DIVIDE- quotient let's feed those brain of yours some math! YAY! MATH! MATH?!?!?...

AH AH AH AHHHHHH

AH AH AH AH AH MATH! MATH<3 MATH! Welcome to Ms. Acosta's Math Class ^_^^_^^_^ NOT OFF! Questionnaire!

Q: What is an algebraic expression for 1998 less than a number p?

A:

a) 1998 - p

b) p + 1998

c) p - 1998

d) p x 1998 ANSWER:

c) p - 1998 Writing Expressions With Addition, Subtraction, Multiplication, and Division. Questionnaire!

Q: What is an algebraic expression for each word phrase in parts (A) and (B) ?

A) 1998 times a number p

B) the quotient of 1998 and a number p Question 2+2-2

What is an algebraic expression for the word phrase? Word Phrase Expression

14 more than twice a number h 14 + 2h

14 less than the quotient of 9 and a number h 9/h - 14

the product of 14 and the sum of a number h and 9 14(h + 9) Questionnaire!

What is an algebraic expression for 9 less than the product of a number n and 7? Twice the sum of a number n and 7? The quotient of 9 and the sum of 7 and a number x? Answers!

1. 7n - 9

2. 2(n + 7)

3. 9/x +7 Answers!

1. 7n - 9

2. 2(n + 7)

3. 9/x +7 Questionnaire!

What is an algebraic expression for 9 less than the product of a number n and 7? Twice the sum of a number n and 7? The quotient of 9 and the sum of 7 and a number x? Now it's time to translate algebraic expressions into d h e ! ! What word phrase can you use to represent the algebraic expression 8n? Expression 8n

8 x n r w o p r a s s ! A number and a variable side by side indicate a product. Words 8 times a number n the product of 8 and a number n Question 3 + 3 - 3 Questionnaire!

What word phrase can you use to represent the algebraic expression 9n + 1? It's........the sum of nine times a number x and 1 Writing a Rule to Describe a pattern! 1-2 Order of Operations and Evaluating Expressions Vocabulary! power- has two parts, a base and an exponent.

exponent- tells you how many times to use the base as a factor.

base-a number that is multiplied repeatedly

simplify- to replace an expression with its simplest

evaluate- to substitute a given number for each variable, and then simplify. Question 5+3-7 What is the simplified form of the expression?

9^5 = 9 x 9 x 9 x 9 x 9 = 59049

(.5)^5= .5 x .5 x .5 x .5 x .5 = .03125 Questionnaire!

What is the simplified form of the expression (1/4)^4 ? (1/4)^4 = 1/4 x 1/4 x 1/4 x 1/4 Now take note that when simplifying an expression, you need to perform operations in the correct order. The clue is... PEMDAS P-arenthesis

E-xponents

M-ultiplication

D-ivision

A-ddition

s-ubtraction left to right left to right Question 2 + 2- 2 What is the simplified form of each expression?

(8 - 4)^3 / 8 = ?

3^2 + 3 / 4 = ? (8 - 4)^3 / 8 = 4^3 / 8 Subtract inside the parenthesis

= 64 / 8 Simplify the power

Answer = 32 Divide (3^2 + 3) / 4 = (9 + 3) / 4 Simplify the power

= 12 / 4 Add

Answer = 3 Divide Question 3 + 3 - 3 What is the value of the expression for x = 2 and y = 5?

x^2 + y -x + 15 1-3 Real Numbers and the Number Line square root- a number a such that a^2 = the square root of b. The square root of √b is the principal square root. -(square root sign) b is the negative square root.

radicand- the expression under the radical symbol.

radical- is both the radical symbol and radicand form. rational number is any number that you can write in the form a/b, where a and b are integers and b is not equal to 0. A rational number is also a terminating decimal or a repeating decimal

Ex: 6.75 or .51515151

Natural numbers- (1,2,3,...)

Whole numbers- (0,1,2,3,...)

Integers- (...-2, -1, 0, 1, 2, 3,...)

irrational number- cannot be represented as the quotient of two integers.

real numbers- rational and irrational numbers. set- is a well-defined collection of objects.

element of the set- members of a set

subset- consist of elements from the given set.

*You can list the elements of a set within braces . x^2 + y -x + 15 = 2^2 + 5 - 2 + 15 Substitute 2 for x and 5 for y.

= 4 + 5 - 2 +15 Simplify power.

= 22 Add and subtract from left to right. Question 4 + 4 - 4 What is an expression for the spending money you have left after depositing 3/5 of your wages in savings? Evaluate the expression for weekly wages of $50, $60, $85, and 100. So we know that our savings equals 3/4 of wages.

Now, we need an expression for spending money and the amount of spending money for various weekly wages. The expression:

spending money = wages minus 3/5 of wages

w - 3/5(w)

w = your wages

*This expression determines how much money you have left for spending money after depositing 3/5 of your wages in savings. What is an expression for the spending money you have left after depositing 3/5 of your wages in savings? Evaluate the expression for weekly wages of $50, $60, $85, and 100. Wages (w) w - 3/5(w) Spending Money

50 50 - 3/5(50) 20

60 60 - 3/5(60) 24

85 85 - 3/5(85) 34

100 100 - 3/5(100) 40 The shipping cost for an order at an online store is 1/4 the cost of the items you order. What is an expression for the total cost of a given order? What are the total costs for orders of $43. The expression:

c + 1/4(c)

43 + 1/4(43)

43 + 10.75

$53.75 Answer The shipping cost for an order at an online store is 1/4 the cost of the items you order. What is an expression for the total cost of a given order? What are the total costs for orders of $43. QUESTIONNAIRE!! perfect square- the square of an integer. ( 16 is a perfect square because 4^2 = 16). inequality- is a mathematical sentence that compares the values of two expressions using an inequality symbol. Question 1 + 1 - 1 What is the simplified form of each expression?

49 = 7 7^2 = 49, so 7 is a square root of 49

25/36 = 5/6 (5/6)^2 = 25/36, so 5/6 is a square root of 25/36 Questionnaire!

What is the simplified form of each expression?

144 225 1/9 16/49 * 12 * * 15 * * 1/3 * * 4/7 * Question 2 + 2 - 2 Mr. Francis measures the area of one of the squares to be 600 square microns. What is the approximate side lengths of the square to the nearest micron? METHOD 1 Estimate 600 by finding the two closest perfect squares.

The perfect squares closest to 600 are 576 and 625

24^2 = 576

600

25^2 = 625

Since 600 is closer to 576, 600 is approximately 24,

and the side length is about 24 microns. METHOD 2 Estimate 600 using a calculator.

600 24.9 Use the square root function on you calculator.

The side length of the square is about 24 microns. Questionnaire!

What is the value of 78 to the nearest integer? * 9 * Numbers can be classified by their characteristics. Some types of numbers can be represented on the number line. To which subsets of the real numbers does each number belong?

9 natural numbers, whole numbers, integers, rational numbers

-5.987 rational numbers (terminating decimal)

600 irrational numbers (it's not a perfect square) Writing Expressions With Addition, Subtraction, Multiplication, and Division. Questionnaire! To which subsets of the real numbers does each number belong?

1/2 0 15 rational numbers * whole numbers, integers, and rational numbers * irrational number Question 4 + 4 - 4 What is an inequality that compares the numbers 28 and 5 1/6? 28 = 5.2915... Write the square root as a decimal.

5 1/6 = 5.1666666667 Write the fraction as a decimal.

28 5 1/6 Compare using an inequality symbol. Questionnaire!

What is an inequality that compares the numbers 81 and 9.5? 81 9.5 Graphing and Ordering Real Numbers Question 5 + 5 - 5 What is the order of 9 , 1.1, 5 , 8/9, and -1.1 from least to greatest? First step is to write the numbers that are not in decimal form as decimals. 9 = 3.0

5 = 2.236...

8/9 = .8888888889 Second step is to graph all five numbers on the number line to order the numbers, and read the graph from left to right. -2 -1.5 -1 -.5 0 .5 1 1.5 2 2.5 3 9 5 -1.1 8/9 1.1 THE NUMBER LINE From least to greatest, the numbers are -1.1, 8/9, 1.1, 5 , and 9 Questionnaire!

What is the order of -3, 4 , -7/5, -1, 3 from least to greatest. -3, -7/5, -1, 4 , 3 * * * * * * * * * * * * 1-4 Properties of Real Numbers NOW LET US MOVE ON TO.... Equivalent expressions- are two algebraic expressions if they have the same value for all values of the variable(s). Let a and b be any real numbers

Changing the order of the addends does not change the sum. Changing the order of the factors does not change the product. j Ob e c t i v e To write

algebraic expressions j Ob e c t i v e * To simplify expressions involving exponents.

* To use the order of operations to evaluate expressions. j Ob e c t i v e To classify, graph, and compare real numbers

To find and estimate square roots * * j Ob e c t i v e * To identify and use properties of real numbers Properties of Real Numbers Algebra Example

Addition a + b = b + a 2 + 4 = 4 + 2

Multiplication a x b = b x a 3 x 4 = 4 x 3 Commutative Properties of Addition and Multiplication Associative Properties of Addition and Multiplication Let a be any real number. Identity Properties of Addition and Multiplication The sum of any real number and 0 is the original number. The product of any real number and 1 is the original number. Algebra Example

Addition a + 0 = a 12345 + 0 = 12345

Multiplication a x 1 = a 6789 x 1 = 6789 Let a be any real number Zero Property of Multiplication The product of a and 0 is 0 Algebra Example

a x 0 = 0 25 x 0 = 0 Let a be any real number Multiplication Property of -1 The product of -1 and a is -1 Algebra Example

-1 x a = -a -1 x 20 = -20 Identifying Properties * * * * * * * * Question 1 + 1 - 1 What property is illustrated by each statement?

(x + 10) + 2 = x + (10 +2) Associative Property of Addition

56 x 0 = 0 Zero Property of Multiplication

10y + 0 = 10y Identity Property of Addition Questionnaire!

What property is illustrated in z + (e + l) = (z + e) + l ? Associative Property

of Addition :) :) :) :) * * * * Question 2 + 2 - 2 A movie ticket costs $7.00. A drink costs $1.50. Nachos costs $3.50. What is the total cost for a ticket, a drink, and nachos? Use mental math. (7.00 + 1.50) + 3.50 = 7.00 + (1.50 + 3.50) Associative Property of Addition.

= 7.00 + 5.00 Simplify inside parenthesis.

= 12.00 Add.

The total cost is $12.00 Spider man has to save 26 people from the fire, 18 from Ms. Lynch class, and kill 14 evil MoRgAns. How many people Spider man has to save and kill? Use MENTAL math. Q u e s i o n n a i r e t ? ? 58! WAY TO GO

HAKEEM! Simplify each expression. Problem 3 + 3 - 3 4(3n)

4(3n) = (4 x 3)n Associative Property of Multiplication.

= 12n Simplify. (5 + 3b) + 10

(5 + 3b) + 10 = (3b + 5) + 10 Commutative Property of Addition.

= 3b + (5 + 10) Associative Property of Addition.

= 3b + 15 Simplify. 4xy

y 4xy 4x y

y 1 y

4x y

1 y

4x 1 x / 1 = x and y / y = 1

4x Identity Property of Multiplication = x x x x = = Rewrite denominator using Identity Property of Multiplication Use rule for multiplying fractions: a/b x c/d =ac/bd = It's your turn !

Simplify each expression

5.9(.5n) 9 + (6h + 2) 2.95n * * * * * * * * 6h + 11 * * * * * * Deductive reasoning- is the process of reasoning logically from given facts to a conclusion.

Counterexample- an example showing that a statement is false. That's exactly what we just did. We showed that two expressions were equivalent by using deductive reasoning. :D Question 4 + 4 - 4 Is the statement true or false? If it is false, give a counterexample. For all real numbers a and b, a b = b + a

False. 2 3 = 3 + 2 is a counterexample.

For all real numbers a, b, and c, (a + b) + c = b + (a + c).

True.

(a + b) + c = (b + a) + c Commutative Property of Addition

= b + (a + c) Associative Property of Addition x x Questionnaire!

For all real numbers c and h, (c h)+ 1 = c (h + 1) ? ? ? ? ? (2 x 3) + 1 = 2 x (3 + 1)

6 + 1 = 2 x 3

7 = 6 False!! x x 1-5 Adding and Subtracting Real Numbers j Ob e c t i v e To find sums and differences of real numbers. Know : YOU CAN ADD OR SUBTRACT ANY REAL NUMBERS USING A NUMBER LINE MODEL. Question 1 + 1 - 1 What is each sum? Use a number line 4 + 5 4 + 5 = 9 2 3 4 5 6 7 8 9 10 11 12 Start at 4 Move 5 units to the right. -3 -2 -1 0 1 2 3 4 5 6 7 move 5 units left -4 + 5 -4 + 5 = 1 -5 -4 -3 -2 -1 0 1 2 3 4 5 move 5 units right 4 + (-5) 4 + (-5) = -1 -4 + (-5) -4 + (-5) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 move 5 units to the left What is -10 + 5? -5 *

-14 + 8

-14 + 8 = -6 The difference of the absolute value is 6. The negative addend has the greater absolute value. The sum is negative.

-9 + (-9)

-9 + (-9) = -18 The addends have the same sign (negative), so add their absolute values. The sum is negative. Adding Real Numbers Adding Numbers With the Same Sign To add numbers with the same sign, add their absolute values. The sum has the same sign as the addends. Examples: 2 + 3 = 5 -2 + -3 = -5 * * * * Adding Numbers With Different Signs To add two numbers with different signs, subtract their absolute values. The sum has the same sign as the addend with the greater absolute value. Example: -2 + 3 = 1 2 + (-3) = -1 For example, the absolute value of 5 is 5 and the absolute value of -5 is 5. -5 -4 -3 -2 -1 0 1 2 3 4 5 * * * * * * * * * Question 2 + 2 - 2 What is each sum? The absolute value of a number is its distance from 0 on a number line.

3/6 + (-3/4) =

3/6 + (-3/4) = 6/12 + (-9/12) Find the least common denominator.

= -3/12

= -1/4 The difference of the absolute value is -3/12 (which is simplified to -1/4). The negative addend has the greater absolute value. The sum is negative. -5 + 11

-5 + 11 = 6 The difference of the absolute value is 6. The positive addend has the greater absolute value. The sum is positive. What is the sum of

-19 + (-10) -29 * * Opposites - two numbers that are the same distance from 0 on a number line but lie in opposite directions. -4 -3 -2 1 0 1 2 3 4 5 6 3 units 3 units -3 and 3 are the same distance from 0. So -3 and 3 are opposites. Additive inverses - a number and its opposite. Inverse Property of Addition For every real number a, there is an additive inverse -a such that a + (-a) = -a + a = 0

Examples: 15 + (-15) = 0 -15 + 15 = 0 method to find the sum of a number and its opposite Subtracting Real Numbers To subtract a real number, add its opposite: a - b = a + (-b).

Example: 9 - 14 = 9 + (-14) = -5 9 - (-14) = 9 + 14 = 23 Question 3 + 3 - 3 What is each difference? -4 - (-14) = -4 + 14 The opposite of -14 is 14. So add 14.

= 10 5 - 12 = 5 + (-12) The opposite of 12 is -12. So add -12.

= -7 14 - 14 = 14 + (-14) The opposite of 14 is -14. So add -14.

= 0 Inverse Property of Addition. Questionnaire!

* * * * * * * * * * * * * * * * * * *

What is 9.5 - (-10) ? 19.5 * * * * Question 4 + 4 - 4 A hungry child dove 14ft to look for fish and then rises 15ft to catch her breath. Then she dove 20ft again. When she finally caught a mahi mahi, she dove up 22ft to put her fish in her boat. How high is Jenee above sea level? 0 - 14 + 15 - 20 + 22 Write an expression.

= 0 + (-14) + 15 + (-20) + 22 Use rule for subtracting real numbers.

= 0 + 15 + 22 + (-14) + (-20) Commutative Property of Addition

= 0 + (15 + 22) + [(-14) + (-20)] Group addends with the same sign.

= 0 + 37 + (-34) Add inside grouping symbols

= 37 + (-34) Identity Property of Addition

= 3 Use rule for adding numbers with different signs.

She is 3ft above sea level. Questionnaire!

A robot submarine dives 500 ft to the ocean floor. It rises 215 ft as the water gets shallower. Then the submarine dives 2916 ft into a deep crevice. Next, it rises 700 ft to photograph a crack in the wall of the crevice. What is the location of the crack in relation to sea level? -2501 1-6 Multiplying and Dividing Real Numbers j Ob e c t i v e * To find products and quotients of real numbers. Multiplying Real Numbers The product of two real numbers with different signs is negative.

Example: 4( -3 ) = -12 2( -4 ) = -8 The product of two real numbers with the same sign is positive.

Example: 4( 3 ) = 12 2( 4 ) = 8 Question 1 + 1 - 1 What is each product? 5(-12) = -60 The product of two numbers with different signs is negative. 12.5(3) =37.5 The product of two numbers with the same sign is positive. * * * * * * * * * * * * ** ** ** * * * * * * * * * * * * * * Questionnaire!

* ? *

What is the product of 15(-.5)? -7.5 A negative square root is represented by -

Every positive real number has a positive and a negative square root.

The symbol in front of the radical indicates both square roots. - + Question 2 + 2 - 2 What is the simplified form of each expression? - 9 = -3 (-3)^2 = 9

4/49 = 4/49 (2/7)^2 = 4/49 (-2/7)^2 = 4/49 + _ + - What is the simplified form of - 144 -12 **** **** **** Dividing Real Numbers The quotient of two real numbers with different signs is negative.

Example: -10 / 2 = -5 10 / (-2) = -5 The quotient of two real numbers with the same sign is positive.

Example: 10 / 2 = 5 -10 / (-2) = 5 Division Involving 0 The quotient of 0 and any nonzero real number is 0. The quotient of any real number and 0 is modified.

Examples: 0 / 4 = 0 4 / 0 = undefined Question 3 + 3 - 3 A sky diver's elevation changes by -3000 ft in 3 min after the parachute opens. What is the average change in the sky diver's elevation each minute?

-3000 / 3 = -1000 The numbers have different signs, so the quotient is negative.

The sky diver's average change in elevation is -1000 ft per minute. You make five withdrawals of equal amounts from your bank account. The total amount you withdraw is $465. What is the change in your account balance each time you make a withdrawal? Questionnaire! -93 Inverse Property of Multiplication For every nonzero real number a, there is a multiplicative inverse 1/a such that a(1/a) = 1

Examples: The multiplicative inverse of -4 is -1/4 because -4(-1/4) = 1 The reciprocal of a nonzero real number of the form a/b is b/a. The product of a number and its reciprocal is 1, so the reciprocal of a number is its multiplicative inverse. This applies for dividing fractions. a/b c/d = a/b d/c . . - . Question 4 + 4 - 4 What is the value of x/y when x = -3/4 and y = 1/4 = x y

= -3/4 1/4

= -3/4 4/1

= -3 - . . - . . . x/y What is the value of -8/9 1/2 ? - . . -16/9 Questionnaire! * * * * * * * * ***** ***** *

* *

* 1-7 The Distributive Property To use the Distributive Property to simplify expressions j Ob e c t i v e How to do the Distributive Property? For every real number a, b, and c:

a(b - c) = ab - ac 4(5 - 4) = 4(5) - 4(4)

(b - c)a = ba - ca (25 - 5)2 = 25(2) - 5(2) Also... SSSSsssshhhhh!!! What is the simplified form of each expression?

4(x + 5) = 4(x) + 4(5) Distributive Property

= 4x + 20 Simplify Question 1 + 1 - 1 What is the simplified form of expression 9(x + 11) ? Questionnaire! 9x + 99 A fraction bar indicates division and may act as a grouping symbol.

Any fraction a/b can also be written as a 1/b.

a 1 . b b = a What sum or difference is equivalent to 5x + 3

5 ? Question 2 + 2 - 2 5x + 3 1

5 5 = (5x + 3) Write division as multiplication. 1 1

5 5 (5x) + (3) Distributive property. 1x + 3

5 Simplify. = = Questionnaire!

What sum of difference is equivalent to 4 - 2x

8 ? 1/2 - 1/4x Question 3 + 3 - 3 What is the simplified form of -(3x - 5y) ?

-(3x - 5y) = -1(3x - 5y) Multiplication Property of -1.

= (-1)(3x) + (-1)(-5y) Distributive Property.

= -3x + 5y Simplify. Multiplication Property of -1 states that -1 x = -x

You can write -(x + 2) as -1(x + 2) . What is the simplified form of each expressions?

-(c - 9) -c + 9 Questionniare! ************** * * Question 4 + 4 - 4 A gatorade cost cost $2.95. What is the total cost for 9 gatorades? Use mental math.

9(2.95) = 9(3 - 0.05) Think of 2.95 as 3 - 0.05

= 9(3) - 9(0.05) Distributive Property

= 27 - .45 Multiply Mentally

= 26.55 Subtract Mentally

The total cost for 9 gatorades is $26.55 Bria picks up trash from Ms. Lynch's classroom for $1.25. She does this 5 times a week. How much is her salary each week? Use mental math. $6.25 Questionnaire! Term- is a number, a variable, or the product of a number and one or more variables.

Constant- is a term that has no variable.

Coefficient- is a numerical factor of a term. Like terms have the same variable factors. Terms 6c and -9c 4h^2 and 7h^2 2s^2 and 2s

Variable Factors c and c h^2 and h^2 s^2 and s

Like terms? Yes Yes No An algebraic expression in simplest form has no like terms or parenthesis.

NOT SIMPLIFIED SIMPLIFIED

4(3x - 9 + 4x) 15x + 9 Question 5 + 5 - 5 What is the simplified form of the expression 5x^2 - 5 + 4x - 6 + 2x?

5x^2 - 5 + 4x - 6 + 2x = 5x^2 + (-5) + 4x + (-6) + 2x Rewrite as a sum.

= 5x^2 + 4x + 2x + (-5) + (-6) Commutative Property.

= 5x^2 + (4 + 2)x + (-5) + (-6) Distributive Property.

= 5x^2 + 6x - 11 Simplify. Questionnaire!

What is the simplified form of 7y^2x - 6y^2 + y^2x? 8y^2x - 6y^2 1-8 An Introduction to Equations To solve equations using tables and mental math. j Ob e c t i v e An equation is a mathematical sentence that uses an equal sign (=). An equation is true if the expressions on either side of the equal sign are equal (2 + 2 = 4). An equation is false if the expressions on either side of the equal sign are not equal (2 + 2 = 5). An equation is an open sentence if it contains one or more variables and mat be true of false depending on the values of its variables. Question 1 + 1 - 1 Is the equation true, false, or open? Explain. 34 + 23 = 18 + 39 True, because both expressions equal to 57.

6 7 = 41 False, because 6 7 = 42 and 42 is not equal to 41.

7x + 5 = 19 Open, because there is a variable. . . Is the equation true, false, or open?

3y + 6 = 9y - 2 OPEN! cuz it has a variable ;) Questionnaire! A solution of an equation containing a variable is a value of the variable that makes the equation Question 2 + 2 - 2 Is x = 9 a solution of the equation 6x + 14 = 69?

6x + 14 = 69

6(9) + 14 = 69 Substitute 9 for x

68 = 69 Simplify

No, x = 9 is not a solution of the equation. Is h = 4 a solution of the equation 7h - 8 = 20? YES!!!!! Question 3 + 3 - 3 An art student wants to make a model of a baseball court. The length of the court is 1.5 times its width. The length of the student is 60 in. What should the width of the model be?

Relate: The length is 1.5 times width

Define: Let w = the width of the model.

Write: 60 = 1.5 w

1.5 60 = 1.5 w 1.5

40 = w

The width should be 40 inches. . . - . . - . . The length of Hazel's hair is 2.75 times its width. Write an equation that can be used to find the width if the length of her hair is 33 inches. 2.75w = 33 Q u e s t i o n n a i r e ! 60 in w Question 4 + 4 - 4 What is the solution of each equation? Use mental math.

Think Solution Check

x + 6 = 15 What number plus 6 equals 12? 9 9 + 6 = 15 (Correct)

a/15 = 3 What number divided by 15 equals 3? 45 45/15 = 3 (Correct) * Questionnaire!

What is the solution of 17 - x = -3 x = 20 Question 5 + 5 - 5 What is the solution of 4n + 9 = 33

Make a table of values. Choose a starting value using mental math. 4(1) + 9 = 13 and 4(10) + 9 = 49, so 1 is too low and 10 is too high. n 4n + 9 Value of 4n + 9

4 4(4) + 9 25

5 4(5) + 9 29

6 4(6) + 9 33 The value of 4n + 9 increases as n increases, so try greater values of n. When n = 6, 4n + 9 = 33. SO the solution is 6 What is the solution of 19 - 6b = -35 ? b = 9 Questionnaire! *********************************************** Question 3 + 3 - 3 + 3 What is an estimate of the solution of -9x - 5 = 28? To estimate the solution, find the integer values of x between which the solution must lie. -9(0) - 5 = -5 and -9(1) - 5 = -14. If you try greater values of x, the value of -9x - 5 farther from 28. x -9x - 5 Value of -9x - 5

-1 -9(-1) - 5 4

-2 -9(-2) - 5 13

-3 -9(-3) - 5 22

-4 -9(-4) - 5 31 Now the values of -9x -5 are getting closer to 28 28 is between 22 and 31, so the solution is between -3 and -4. Questionnaire!

What is the solution of 3x + 3 = -22 ? Between -8 and -9 WOOOOOOHOOOOOOO!!!

LAST LESSON!

UP NEXT! 1-9 Patterns, Equations, and Graphs To use tables, equations, and graphs to describe relationships. O B E J E C T I V Know that you can use an equation with two variables to represent the relationship between two varying quantities. A solution of an equation with two variables x and y is any ordered pair (x, y) that makes the equation true. Question 1 + 1 - 1 Is (3, 12) a solution of the equation y = 4x?

y= 4x

12 = 4 3 Substitute 3 for x and 12 for y.

12 = 12 So (3, 12) is a solution of y = 4x. . ? Nope! Questionnaire!

Is (5, -18) a solution of the equation y = -3x ? Question Both Nathan and her sister Isabelle were born on January 1, but Nathan was born 2 years before Isabelle. How can you represent the relationship between Nathan's age and Isabelle's age in different ways? Step 1: MAKE A TABLE Nathan's and Isabelle's Ages(Years)

Isabelle's Age 1 2 3 4 5 6 7 8 9 10

Nathan's Age 3 4 5 6 7 8 9 10 11 12 2 + 2 - 2 Step 2: Write an Equation

Let x = Isabelle's age. Let y = Nathan's age. From the table, you can see that y is always 2 greater than x.

So y = x + 2 Step 3: Draw a graph. . . . . . . . . . Inductive Reasoning- is the process of reaching a conclusion based on an observed pattern. You can use inductive reasoning to predict values. Question 3 + 3 - 3 The table shows the relationship between the number of light blue tiles and the total number of tiles in each figure. Extend the pattern. What is the total number of tiles in a figure with 9 light blue tiles? Number of Light Blue tiles, x

1

2

3

4

5 Total Number of Tiles, y

2

4

6

8

10 Method 1:

Draw a graph. Method 2:

Write an equation.

y = 2x The total number of tiles is 2 times the number of light blue tiles.

=2(9) Substitute 9 for x.

=18 Simplify.

The total number of tiles is 18. Question 4 + 4 - 4 What is a rule for the height? Give the rule in words and as an algebraic expression. Number of levels

2

3

4

n Height (in.)

(2.5 x 2) + 20

(2.5 x 3) + 20

(2.5 x 4) + 20

? Rule in Words - Multiply the number of levels by 2.5 and add 20.

Rule as an Algebraic Expression - The variable n represents the number of levels in the house of cards. Level = 2.5 in.

Table height = 20 in 2.5n + 20 (This expression lets you find the height for n levels.) 1 2 3 4 5 6 7 8 9 10 20 18 16 14 2 4 6 8 10 12 . . . . . . Blue Tiles T

o

t

a

l

T

i

l

e

s RING!!!!!!!!!!!!!!!!!!!!!!!! SIT DOWN!.........and um... NO HOMEWORK!

Class your dismiss except for Hakeem for constantly standing up and Bria you have work detail till final exams are done. .... -_- -_- -_- -_- -_- * * * * * By: Hazel Acosta

Special thanks to Wi-Fi

Full transcriptVariables and Expressions

Order of Operations and Evaluating Expressions

Real Numbers and the Number line

Properties of Real Numbers

Adding and Subtracting Real Numbers

Multiplying and Dividing Real Numbers

The Distributive Property

An Introduction to Equations

Patterns, Equations, and Graphs Question 5+3-7

What is an algebraic expression for the word phrase?

Word Phrase Expression Clues

25 more than a number h h + 25 ADD- more than, plus

25 less a number h 25 - h SUBTRACT- less, less than

25 less than a number h h – 25 SUBTRACT- less, less than

25 times a number h 25 x h, 25 h, 25n MULTIPLY- times, product

the product of 25 and a number h 25 x h, 25 h, 25n MULTIPLY- times, product

the quotient of a number h and 25 n ÷ 25, n/25 DIVIDE- quotient let's feed those brain of yours some math! YAY! MATH! MATH?!?!?...

AH AH AH AHHHHHH

AH AH AH AH AH MATH! MATH<3 MATH! Welcome to Ms. Acosta's Math Class ^_^^_^^_^ NOT OFF! Questionnaire!

Q: What is an algebraic expression for 1998 less than a number p?

A:

a) 1998 - p

b) p + 1998

c) p - 1998

d) p x 1998 ANSWER:

c) p - 1998 Writing Expressions With Addition, Subtraction, Multiplication, and Division. Questionnaire!

Q: What is an algebraic expression for each word phrase in parts (A) and (B) ?

A) 1998 times a number p

B) the quotient of 1998 and a number p Question 2+2-2

What is an algebraic expression for the word phrase? Word Phrase Expression

14 more than twice a number h 14 + 2h

14 less than the quotient of 9 and a number h 9/h - 14

the product of 14 and the sum of a number h and 9 14(h + 9) Questionnaire!

What is an algebraic expression for 9 less than the product of a number n and 7? Twice the sum of a number n and 7? The quotient of 9 and the sum of 7 and a number x? Answers!

1. 7n - 9

2. 2(n + 7)

3. 9/x +7 Answers!

1. 7n - 9

2. 2(n + 7)

3. 9/x +7 Questionnaire!

What is an algebraic expression for 9 less than the product of a number n and 7? Twice the sum of a number n and 7? The quotient of 9 and the sum of 7 and a number x? Now it's time to translate algebraic expressions into d h e ! ! What word phrase can you use to represent the algebraic expression 8n? Expression 8n

8 x n r w o p r a s s ! A number and a variable side by side indicate a product. Words 8 times a number n the product of 8 and a number n Question 3 + 3 - 3 Questionnaire!

What word phrase can you use to represent the algebraic expression 9n + 1? It's........the sum of nine times a number x and 1 Writing a Rule to Describe a pattern! 1-2 Order of Operations and Evaluating Expressions Vocabulary! power- has two parts, a base and an exponent.

exponent- tells you how many times to use the base as a factor.

base-a number that is multiplied repeatedly

simplify- to replace an expression with its simplest

evaluate- to substitute a given number for each variable, and then simplify. Question 5+3-7 What is the simplified form of the expression?

9^5 = 9 x 9 x 9 x 9 x 9 = 59049

(.5)^5= .5 x .5 x .5 x .5 x .5 = .03125 Questionnaire!

What is the simplified form of the expression (1/4)^4 ? (1/4)^4 = 1/4 x 1/4 x 1/4 x 1/4 Now take note that when simplifying an expression, you need to perform operations in the correct order. The clue is... PEMDAS P-arenthesis

E-xponents

M-ultiplication

D-ivision

A-ddition

s-ubtraction left to right left to right Question 2 + 2- 2 What is the simplified form of each expression?

(8 - 4)^3 / 8 = ?

3^2 + 3 / 4 = ? (8 - 4)^3 / 8 = 4^3 / 8 Subtract inside the parenthesis

= 64 / 8 Simplify the power

Answer = 32 Divide (3^2 + 3) / 4 = (9 + 3) / 4 Simplify the power

= 12 / 4 Add

Answer = 3 Divide Question 3 + 3 - 3 What is the value of the expression for x = 2 and y = 5?

x^2 + y -x + 15 1-3 Real Numbers and the Number Line square root- a number a such that a^2 = the square root of b. The square root of √b is the principal square root. -(square root sign) b is the negative square root.

radicand- the expression under the radical symbol.

radical- is both the radical symbol and radicand form. rational number is any number that you can write in the form a/b, where a and b are integers and b is not equal to 0. A rational number is also a terminating decimal or a repeating decimal

Ex: 6.75 or .51515151

Natural numbers- (1,2,3,...)

Whole numbers- (0,1,2,3,...)

Integers- (...-2, -1, 0, 1, 2, 3,...)

irrational number- cannot be represented as the quotient of two integers.

real numbers- rational and irrational numbers. set- is a well-defined collection of objects.

element of the set- members of a set

subset- consist of elements from the given set.

*You can list the elements of a set within braces . x^2 + y -x + 15 = 2^2 + 5 - 2 + 15 Substitute 2 for x and 5 for y.

= 4 + 5 - 2 +15 Simplify power.

= 22 Add and subtract from left to right. Question 4 + 4 - 4 What is an expression for the spending money you have left after depositing 3/5 of your wages in savings? Evaluate the expression for weekly wages of $50, $60, $85, and 100. So we know that our savings equals 3/4 of wages.

Now, we need an expression for spending money and the amount of spending money for various weekly wages. The expression:

spending money = wages minus 3/5 of wages

w - 3/5(w)

w = your wages

*This expression determines how much money you have left for spending money after depositing 3/5 of your wages in savings. What is an expression for the spending money you have left after depositing 3/5 of your wages in savings? Evaluate the expression for weekly wages of $50, $60, $85, and 100. Wages (w) w - 3/5(w) Spending Money

50 50 - 3/5(50) 20

60 60 - 3/5(60) 24

85 85 - 3/5(85) 34

100 100 - 3/5(100) 40 The shipping cost for an order at an online store is 1/4 the cost of the items you order. What is an expression for the total cost of a given order? What are the total costs for orders of $43. The expression:

c + 1/4(c)

43 + 1/4(43)

43 + 10.75

$53.75 Answer The shipping cost for an order at an online store is 1/4 the cost of the items you order. What is an expression for the total cost of a given order? What are the total costs for orders of $43. QUESTIONNAIRE!! perfect square- the square of an integer. ( 16 is a perfect square because 4^2 = 16). inequality- is a mathematical sentence that compares the values of two expressions using an inequality symbol. Question 1 + 1 - 1 What is the simplified form of each expression?

49 = 7 7^2 = 49, so 7 is a square root of 49

25/36 = 5/6 (5/6)^2 = 25/36, so 5/6 is a square root of 25/36 Questionnaire!

What is the simplified form of each expression?

144 225 1/9 16/49 * 12 * * 15 * * 1/3 * * 4/7 * Question 2 + 2 - 2 Mr. Francis measures the area of one of the squares to be 600 square microns. What is the approximate side lengths of the square to the nearest micron? METHOD 1 Estimate 600 by finding the two closest perfect squares.

The perfect squares closest to 600 are 576 and 625

24^2 = 576

600

25^2 = 625

Since 600 is closer to 576, 600 is approximately 24,

and the side length is about 24 microns. METHOD 2 Estimate 600 using a calculator.

600 24.9 Use the square root function on you calculator.

The side length of the square is about 24 microns. Questionnaire!

What is the value of 78 to the nearest integer? * 9 * Numbers can be classified by their characteristics. Some types of numbers can be represented on the number line. To which subsets of the real numbers does each number belong?

9 natural numbers, whole numbers, integers, rational numbers

-5.987 rational numbers (terminating decimal)

600 irrational numbers (it's not a perfect square) Writing Expressions With Addition, Subtraction, Multiplication, and Division. Questionnaire! To which subsets of the real numbers does each number belong?

1/2 0 15 rational numbers * whole numbers, integers, and rational numbers * irrational number Question 4 + 4 - 4 What is an inequality that compares the numbers 28 and 5 1/6? 28 = 5.2915... Write the square root as a decimal.

5 1/6 = 5.1666666667 Write the fraction as a decimal.

28 5 1/6 Compare using an inequality symbol. Questionnaire!

What is an inequality that compares the numbers 81 and 9.5? 81 9.5 Graphing and Ordering Real Numbers Question 5 + 5 - 5 What is the order of 9 , 1.1, 5 , 8/9, and -1.1 from least to greatest? First step is to write the numbers that are not in decimal form as decimals. 9 = 3.0

5 = 2.236...

8/9 = .8888888889 Second step is to graph all five numbers on the number line to order the numbers, and read the graph from left to right. -2 -1.5 -1 -.5 0 .5 1 1.5 2 2.5 3 9 5 -1.1 8/9 1.1 THE NUMBER LINE From least to greatest, the numbers are -1.1, 8/9, 1.1, 5 , and 9 Questionnaire!

What is the order of -3, 4 , -7/5, -1, 3 from least to greatest. -3, -7/5, -1, 4 , 3 * * * * * * * * * * * * 1-4 Properties of Real Numbers NOW LET US MOVE ON TO.... Equivalent expressions- are two algebraic expressions if they have the same value for all values of the variable(s). Let a and b be any real numbers

Changing the order of the addends does not change the sum. Changing the order of the factors does not change the product. j Ob e c t i v e To write

algebraic expressions j Ob e c t i v e * To simplify expressions involving exponents.

* To use the order of operations to evaluate expressions. j Ob e c t i v e To classify, graph, and compare real numbers

To find and estimate square roots * * j Ob e c t i v e * To identify and use properties of real numbers Properties of Real Numbers Algebra Example

Addition a + b = b + a 2 + 4 = 4 + 2

Multiplication a x b = b x a 3 x 4 = 4 x 3 Commutative Properties of Addition and Multiplication Associative Properties of Addition and Multiplication Let a be any real number. Identity Properties of Addition and Multiplication The sum of any real number and 0 is the original number. The product of any real number and 1 is the original number. Algebra Example

Addition a + 0 = a 12345 + 0 = 12345

Multiplication a x 1 = a 6789 x 1 = 6789 Let a be any real number Zero Property of Multiplication The product of a and 0 is 0 Algebra Example

a x 0 = 0 25 x 0 = 0 Let a be any real number Multiplication Property of -1 The product of -1 and a is -1 Algebra Example

-1 x a = -a -1 x 20 = -20 Identifying Properties * * * * * * * * Question 1 + 1 - 1 What property is illustrated by each statement?

(x + 10) + 2 = x + (10 +2) Associative Property of Addition

56 x 0 = 0 Zero Property of Multiplication

10y + 0 = 10y Identity Property of Addition Questionnaire!

What property is illustrated in z + (e + l) = (z + e) + l ? Associative Property

of Addition :) :) :) :) * * * * Question 2 + 2 - 2 A movie ticket costs $7.00. A drink costs $1.50. Nachos costs $3.50. What is the total cost for a ticket, a drink, and nachos? Use mental math. (7.00 + 1.50) + 3.50 = 7.00 + (1.50 + 3.50) Associative Property of Addition.

= 7.00 + 5.00 Simplify inside parenthesis.

= 12.00 Add.

The total cost is $12.00 Spider man has to save 26 people from the fire, 18 from Ms. Lynch class, and kill 14 evil MoRgAns. How many people Spider man has to save and kill? Use MENTAL math. Q u e s i o n n a i r e t ? ? 58! WAY TO GO

HAKEEM! Simplify each expression. Problem 3 + 3 - 3 4(3n)

4(3n) = (4 x 3)n Associative Property of Multiplication.

= 12n Simplify. (5 + 3b) + 10

(5 + 3b) + 10 = (3b + 5) + 10 Commutative Property of Addition.

= 3b + (5 + 10) Associative Property of Addition.

= 3b + 15 Simplify. 4xy

y 4xy 4x y

y 1 y

4x y

1 y

4x 1 x / 1 = x and y / y = 1

4x Identity Property of Multiplication = x x x x = = Rewrite denominator using Identity Property of Multiplication Use rule for multiplying fractions: a/b x c/d =ac/bd = It's your turn !

Simplify each expression

5.9(.5n) 9 + (6h + 2) 2.95n * * * * * * * * 6h + 11 * * * * * * Deductive reasoning- is the process of reasoning logically from given facts to a conclusion.

Counterexample- an example showing that a statement is false. That's exactly what we just did. We showed that two expressions were equivalent by using deductive reasoning. :D Question 4 + 4 - 4 Is the statement true or false? If it is false, give a counterexample. For all real numbers a and b, a b = b + a

False. 2 3 = 3 + 2 is a counterexample.

For all real numbers a, b, and c, (a + b) + c = b + (a + c).

True.

(a + b) + c = (b + a) + c Commutative Property of Addition

= b + (a + c) Associative Property of Addition x x Questionnaire!

For all real numbers c and h, (c h)+ 1 = c (h + 1) ? ? ? ? ? (2 x 3) + 1 = 2 x (3 + 1)

6 + 1 = 2 x 3

7 = 6 False!! x x 1-5 Adding and Subtracting Real Numbers j Ob e c t i v e To find sums and differences of real numbers. Know : YOU CAN ADD OR SUBTRACT ANY REAL NUMBERS USING A NUMBER LINE MODEL. Question 1 + 1 - 1 What is each sum? Use a number line 4 + 5 4 + 5 = 9 2 3 4 5 6 7 8 9 10 11 12 Start at 4 Move 5 units to the right. -3 -2 -1 0 1 2 3 4 5 6 7 move 5 units left -4 + 5 -4 + 5 = 1 -5 -4 -3 -2 -1 0 1 2 3 4 5 move 5 units right 4 + (-5) 4 + (-5) = -1 -4 + (-5) -4 + (-5) -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 move 5 units to the left What is -10 + 5? -5 *

-14 + 8

-14 + 8 = -6 The difference of the absolute value is 6. The negative addend has the greater absolute value. The sum is negative.

-9 + (-9)

-9 + (-9) = -18 The addends have the same sign (negative), so add their absolute values. The sum is negative. Adding Real Numbers Adding Numbers With the Same Sign To add numbers with the same sign, add their absolute values. The sum has the same sign as the addends. Examples: 2 + 3 = 5 -2 + -3 = -5 * * * * Adding Numbers With Different Signs To add two numbers with different signs, subtract their absolute values. The sum has the same sign as the addend with the greater absolute value. Example: -2 + 3 = 1 2 + (-3) = -1 For example, the absolute value of 5 is 5 and the absolute value of -5 is 5. -5 -4 -3 -2 -1 0 1 2 3 4 5 * * * * * * * * * Question 2 + 2 - 2 What is each sum? The absolute value of a number is its distance from 0 on a number line.

3/6 + (-3/4) =

3/6 + (-3/4) = 6/12 + (-9/12) Find the least common denominator.

= -3/12

= -1/4 The difference of the absolute value is -3/12 (which is simplified to -1/4). The negative addend has the greater absolute value. The sum is negative. -5 + 11

-5 + 11 = 6 The difference of the absolute value is 6. The positive addend has the greater absolute value. The sum is positive. What is the sum of

-19 + (-10) -29 * * Opposites - two numbers that are the same distance from 0 on a number line but lie in opposite directions. -4 -3 -2 1 0 1 2 3 4 5 6 3 units 3 units -3 and 3 are the same distance from 0. So -3 and 3 are opposites. Additive inverses - a number and its opposite. Inverse Property of Addition For every real number a, there is an additive inverse -a such that a + (-a) = -a + a = 0

Examples: 15 + (-15) = 0 -15 + 15 = 0 method to find the sum of a number and its opposite Subtracting Real Numbers To subtract a real number, add its opposite: a - b = a + (-b).

Example: 9 - 14 = 9 + (-14) = -5 9 - (-14) = 9 + 14 = 23 Question 3 + 3 - 3 What is each difference? -4 - (-14) = -4 + 14 The opposite of -14 is 14. So add 14.

= 10 5 - 12 = 5 + (-12) The opposite of 12 is -12. So add -12.

= -7 14 - 14 = 14 + (-14) The opposite of 14 is -14. So add -14.

= 0 Inverse Property of Addition. Questionnaire!

* * * * * * * * * * * * * * * * * * *

What is 9.5 - (-10) ? 19.5 * * * * Question 4 + 4 - 4 A hungry child dove 14ft to look for fish and then rises 15ft to catch her breath. Then she dove 20ft again. When she finally caught a mahi mahi, she dove up 22ft to put her fish in her boat. How high is Jenee above sea level? 0 - 14 + 15 - 20 + 22 Write an expression.

= 0 + (-14) + 15 + (-20) + 22 Use rule for subtracting real numbers.

= 0 + 15 + 22 + (-14) + (-20) Commutative Property of Addition

= 0 + (15 + 22) + [(-14) + (-20)] Group addends with the same sign.

= 0 + 37 + (-34) Add inside grouping symbols

= 37 + (-34) Identity Property of Addition

= 3 Use rule for adding numbers with different signs.

She is 3ft above sea level. Questionnaire!

A robot submarine dives 500 ft to the ocean floor. It rises 215 ft as the water gets shallower. Then the submarine dives 2916 ft into a deep crevice. Next, it rises 700 ft to photograph a crack in the wall of the crevice. What is the location of the crack in relation to sea level? -2501 1-6 Multiplying and Dividing Real Numbers j Ob e c t i v e * To find products and quotients of real numbers. Multiplying Real Numbers The product of two real numbers with different signs is negative.

Example: 4( -3 ) = -12 2( -4 ) = -8 The product of two real numbers with the same sign is positive.

Example: 4( 3 ) = 12 2( 4 ) = 8 Question 1 + 1 - 1 What is each product? 5(-12) = -60 The product of two numbers with different signs is negative. 12.5(3) =37.5 The product of two numbers with the same sign is positive. * * * * * * * * * * * * ** ** ** * * * * * * * * * * * * * * Questionnaire!

* ? *

What is the product of 15(-.5)? -7.5 A negative square root is represented by -

Every positive real number has a positive and a negative square root.

The symbol in front of the radical indicates both square roots. - + Question 2 + 2 - 2 What is the simplified form of each expression? - 9 = -3 (-3)^2 = 9

4/49 = 4/49 (2/7)^2 = 4/49 (-2/7)^2 = 4/49 + _ + - What is the simplified form of - 144 -12 **** **** **** Dividing Real Numbers The quotient of two real numbers with different signs is negative.

Example: -10 / 2 = -5 10 / (-2) = -5 The quotient of two real numbers with the same sign is positive.

Example: 10 / 2 = 5 -10 / (-2) = 5 Division Involving 0 The quotient of 0 and any nonzero real number is 0. The quotient of any real number and 0 is modified.

Examples: 0 / 4 = 0 4 / 0 = undefined Question 3 + 3 - 3 A sky diver's elevation changes by -3000 ft in 3 min after the parachute opens. What is the average change in the sky diver's elevation each minute?

-3000 / 3 = -1000 The numbers have different signs, so the quotient is negative.

The sky diver's average change in elevation is -1000 ft per minute. You make five withdrawals of equal amounts from your bank account. The total amount you withdraw is $465. What is the change in your account balance each time you make a withdrawal? Questionnaire! -93 Inverse Property of Multiplication For every nonzero real number a, there is a multiplicative inverse 1/a such that a(1/a) = 1

Examples: The multiplicative inverse of -4 is -1/4 because -4(-1/4) = 1 The reciprocal of a nonzero real number of the form a/b is b/a. The product of a number and its reciprocal is 1, so the reciprocal of a number is its multiplicative inverse. This applies for dividing fractions. a/b c/d = a/b d/c . . - . Question 4 + 4 - 4 What is the value of x/y when x = -3/4 and y = 1/4 = x y

= -3/4 1/4

= -3/4 4/1

= -3 - . . - . . . x/y What is the value of -8/9 1/2 ? - . . -16/9 Questionnaire! * * * * * * * * ***** ***** *

* *

* 1-7 The Distributive Property To use the Distributive Property to simplify expressions j Ob e c t i v e How to do the Distributive Property? For every real number a, b, and c:

a(b - c) = ab - ac 4(5 - 4) = 4(5) - 4(4)

(b - c)a = ba - ca (25 - 5)2 = 25(2) - 5(2) Also... SSSSsssshhhhh!!! What is the simplified form of each expression?

4(x + 5) = 4(x) + 4(5) Distributive Property

= 4x + 20 Simplify Question 1 + 1 - 1 What is the simplified form of expression 9(x + 11) ? Questionnaire! 9x + 99 A fraction bar indicates division and may act as a grouping symbol.

Any fraction a/b can also be written as a 1/b.

a 1 . b b = a What sum or difference is equivalent to 5x + 3

5 ? Question 2 + 2 - 2 5x + 3 1

5 5 = (5x + 3) Write division as multiplication. 1 1

5 5 (5x) + (3) Distributive property. 1x + 3

5 Simplify. = = Questionnaire!

What sum of difference is equivalent to 4 - 2x

8 ? 1/2 - 1/4x Question 3 + 3 - 3 What is the simplified form of -(3x - 5y) ?

-(3x - 5y) = -1(3x - 5y) Multiplication Property of -1.

= (-1)(3x) + (-1)(-5y) Distributive Property.

= -3x + 5y Simplify. Multiplication Property of -1 states that -1 x = -x

You can write -(x + 2) as -1(x + 2) . What is the simplified form of each expressions?

-(c - 9) -c + 9 Questionniare! ************** * * Question 4 + 4 - 4 A gatorade cost cost $2.95. What is the total cost for 9 gatorades? Use mental math.

9(2.95) = 9(3 - 0.05) Think of 2.95 as 3 - 0.05

= 9(3) - 9(0.05) Distributive Property

= 27 - .45 Multiply Mentally

= 26.55 Subtract Mentally

The total cost for 9 gatorades is $26.55 Bria picks up trash from Ms. Lynch's classroom for $1.25. She does this 5 times a week. How much is her salary each week? Use mental math. $6.25 Questionnaire! Term- is a number, a variable, or the product of a number and one or more variables.

Constant- is a term that has no variable.

Coefficient- is a numerical factor of a term. Like terms have the same variable factors. Terms 6c and -9c 4h^2 and 7h^2 2s^2 and 2s

Variable Factors c and c h^2 and h^2 s^2 and s

Like terms? Yes Yes No An algebraic expression in simplest form has no like terms or parenthesis.

NOT SIMPLIFIED SIMPLIFIED

4(3x - 9 + 4x) 15x + 9 Question 5 + 5 - 5 What is the simplified form of the expression 5x^2 - 5 + 4x - 6 + 2x?

5x^2 - 5 + 4x - 6 + 2x = 5x^2 + (-5) + 4x + (-6) + 2x Rewrite as a sum.

= 5x^2 + 4x + 2x + (-5) + (-6) Commutative Property.

= 5x^2 + (4 + 2)x + (-5) + (-6) Distributive Property.

= 5x^2 + 6x - 11 Simplify. Questionnaire!

What is the simplified form of 7y^2x - 6y^2 + y^2x? 8y^2x - 6y^2 1-8 An Introduction to Equations To solve equations using tables and mental math. j Ob e c t i v e An equation is a mathematical sentence that uses an equal sign (=). An equation is true if the expressions on either side of the equal sign are equal (2 + 2 = 4). An equation is false if the expressions on either side of the equal sign are not equal (2 + 2 = 5). An equation is an open sentence if it contains one or more variables and mat be true of false depending on the values of its variables. Question 1 + 1 - 1 Is the equation true, false, or open? Explain. 34 + 23 = 18 + 39 True, because both expressions equal to 57.

6 7 = 41 False, because 6 7 = 42 and 42 is not equal to 41.

7x + 5 = 19 Open, because there is a variable. . . Is the equation true, false, or open?

3y + 6 = 9y - 2 OPEN! cuz it has a variable ;) Questionnaire! A solution of an equation containing a variable is a value of the variable that makes the equation Question 2 + 2 - 2 Is x = 9 a solution of the equation 6x + 14 = 69?

6x + 14 = 69

6(9) + 14 = 69 Substitute 9 for x

68 = 69 Simplify

No, x = 9 is not a solution of the equation. Is h = 4 a solution of the equation 7h - 8 = 20? YES!!!!! Question 3 + 3 - 3 An art student wants to make a model of a baseball court. The length of the court is 1.5 times its width. The length of the student is 60 in. What should the width of the model be?

Relate: The length is 1.5 times width

Define: Let w = the width of the model.

Write: 60 = 1.5 w

1.5 60 = 1.5 w 1.5

40 = w

The width should be 40 inches. . . - . . - . . The length of Hazel's hair is 2.75 times its width. Write an equation that can be used to find the width if the length of her hair is 33 inches. 2.75w = 33 Q u e s t i o n n a i r e ! 60 in w Question 4 + 4 - 4 What is the solution of each equation? Use mental math.

Think Solution Check

x + 6 = 15 What number plus 6 equals 12? 9 9 + 6 = 15 (Correct)

a/15 = 3 What number divided by 15 equals 3? 45 45/15 = 3 (Correct) * Questionnaire!

What is the solution of 17 - x = -3 x = 20 Question 5 + 5 - 5 What is the solution of 4n + 9 = 33

Make a table of values. Choose a starting value using mental math. 4(1) + 9 = 13 and 4(10) + 9 = 49, so 1 is too low and 10 is too high. n 4n + 9 Value of 4n + 9

4 4(4) + 9 25

5 4(5) + 9 29

6 4(6) + 9 33 The value of 4n + 9 increases as n increases, so try greater values of n. When n = 6, 4n + 9 = 33. SO the solution is 6 What is the solution of 19 - 6b = -35 ? b = 9 Questionnaire! *********************************************** Question 3 + 3 - 3 + 3 What is an estimate of the solution of -9x - 5 = 28? To estimate the solution, find the integer values of x between which the solution must lie. -9(0) - 5 = -5 and -9(1) - 5 = -14. If you try greater values of x, the value of -9x - 5 farther from 28. x -9x - 5 Value of -9x - 5

-1 -9(-1) - 5 4

-2 -9(-2) - 5 13

-3 -9(-3) - 5 22

-4 -9(-4) - 5 31 Now the values of -9x -5 are getting closer to 28 28 is between 22 and 31, so the solution is between -3 and -4. Questionnaire!

What is the solution of 3x + 3 = -22 ? Between -8 and -9 WOOOOOOHOOOOOOO!!!

LAST LESSON!

UP NEXT! 1-9 Patterns, Equations, and Graphs To use tables, equations, and graphs to describe relationships. O B E J E C T I V Know that you can use an equation with two variables to represent the relationship between two varying quantities. A solution of an equation with two variables x and y is any ordered pair (x, y) that makes the equation true. Question 1 + 1 - 1 Is (3, 12) a solution of the equation y = 4x?

y= 4x

12 = 4 3 Substitute 3 for x and 12 for y.

12 = 12 So (3, 12) is a solution of y = 4x. . ? Nope! Questionnaire!

Is (5, -18) a solution of the equation y = -3x ? Question Both Nathan and her sister Isabelle were born on January 1, but Nathan was born 2 years before Isabelle. How can you represent the relationship between Nathan's age and Isabelle's age in different ways? Step 1: MAKE A TABLE Nathan's and Isabelle's Ages(Years)

Isabelle's Age 1 2 3 4 5 6 7 8 9 10

Nathan's Age 3 4 5 6 7 8 9 10 11 12 2 + 2 - 2 Step 2: Write an Equation

Let x = Isabelle's age. Let y = Nathan's age. From the table, you can see that y is always 2 greater than x.

So y = x + 2 Step 3: Draw a graph. . . . . . . . . . Inductive Reasoning- is the process of reaching a conclusion based on an observed pattern. You can use inductive reasoning to predict values. Question 3 + 3 - 3 The table shows the relationship between the number of light blue tiles and the total number of tiles in each figure. Extend the pattern. What is the total number of tiles in a figure with 9 light blue tiles? Number of Light Blue tiles, x

1

2

3

4

5 Total Number of Tiles, y

2

4

6

8

10 Method 1:

Draw a graph. Method 2:

Write an equation.

y = 2x The total number of tiles is 2 times the number of light blue tiles.

=2(9) Substitute 9 for x.

=18 Simplify.

The total number of tiles is 18. Question 4 + 4 - 4 What is a rule for the height? Give the rule in words and as an algebraic expression. Number of levels

2

3

4

n Height (in.)

(2.5 x 2) + 20

(2.5 x 3) + 20

(2.5 x 4) + 20

? Rule in Words - Multiply the number of levels by 2.5 and add 20.

Rule as an Algebraic Expression - The variable n represents the number of levels in the house of cards. Level = 2.5 in.

Table height = 20 in 2.5n + 20 (This expression lets you find the height for n levels.) 1 2 3 4 5 6 7 8 9 10 20 18 16 14 2 4 6 8 10 12 . . . . . . Blue Tiles T

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t

a

l

T

i

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e

s RING!!!!!!!!!!!!!!!!!!!!!!!! SIT DOWN!.........and um... NO HOMEWORK!

Class your dismiss except for Hakeem for constantly standing up and Bria you have work detail till final exams are done. .... -_- -_- -_- -_- -_- * * * * * By: Hazel Acosta

Special thanks to Wi-Fi