Alg 1 Notes 1 1 thru 1 9 - Ms. Zaleski's Math Classes -...

Algebra I Name: of 32

CLASS NOTES: §1 – 1 thru §1 – 9 Foundations of Algebra §1 – 1: Variables and Expressions Vocabulary:

• Variable: a symbol (usually a letter) that represents a quantity (number).

• Algebraic expression: a mathematical phrase that includes one or more variables.

• Numerical expression: a mathematical phrase involving numbers and operations symbols, but no variables.

Phrase Translation Addition

The _________________ of a number and 6

A number ________________________ by 5

3 __________________________ a number

Subtraction The ________________ between a number and 7

A number _________________________ by 6

3 __________________________ a number

9 ___________________ a number

Multiplication The ____________________ of 6 and a number

Five _____________________ a number

One half _________ a number

Division The ____________________ of a number and 4

A number _______________________ 8


EX 1 Write expressions using addition and subtraction.

(a) 32 more than a number n

(b) 58 less a number n

(c) 18 more than a number n

(d) 24 less than a number n

(e) 15 subtracted from a number n

Use your reasoning skills…

(a) Do the phrases 6 less a number y and 6 less than a number y mean the same thing? Explain.


EX 2 Write expressions using multiplication and division.

(a) 8 times a number n

(b) the quotient of a number n and 5

(c) 6 times a number n

(d) twice a number n

(e) half of a number n

EX 3 Write expressions with two operations.

(a) 3 more than twice a number x

(b) 9 less than the quotient of 6 and a number x


(c) the product of 4 and the sum of a number x and 7

(d) 8 less than the product of a number x and 4

(e) twice the sum of a number x and 8

(f) the quotient of 5 and the sum of 12 and the number x

EX 4 State the algebraic expressions as a word phrase.

(a) 3x

(b) x + 8

(c) 10x + 9

(d) n3

(e) 5x − 1


EX 5 Write a rule to describe a pattern.

The table below shows how the height above the floor of a house of cards depends on the number of levels. What is the rule for the height?

Number of Levels

Height

2 3.5 × 2( ) + 24

3 3.5 × 3( ) + 24

4 3.5 × 4( ) + 24

n ?

(a) Another group of students built another house of cards with n levels. Each card was 5 inches tall, and the height from the floor to the top of the house of cards was 34 + 5n inches. How tall would the house of cards be if the group added 1 more level of cards?

(b) Suppose you draw a segment from any one vertex of a rectangular polygon to the other vertices. A sample for a regular hexagon is shown below. Use the table to find a pattern. What is the rule for the number of non-overlapping triangles formed?

Number of Sides of a Polygon

Number of Triangles

4 4 − 2

5 5 − 2

6 6 − 2

n ?


§1 – 2: Order of Operations and Evaluating Expressions Vocabulary:

• Powers: a way to shorten how you represent repeated multiplication such as 2 × 2 × 2 × 2 × 2 .

• Exponent: tells you how many times to multiply the base.

• Base: What you are multiplying repeatedly.

• Simplify a numerical expression: replace it with its single numerical value.

• Evaluate an algebraic expression: replace each variable with its given value and simplify.

EX 1 Simplify each power.

(a) 107 (b) 0.2( )5

(c) 34 (d) 2

3

⎛

⎝⎜⎞

⎠⎟

3


EX 2 Simplify each expression.

(a) 2 + 3 × 5 (b) 6 − 2( )3 ÷ 2

(c) 24 − 15

(d) 5 • 7 − 42 ÷ 2

(e) 12 − 25 ÷ 5 (f) 4 + 34

7 − 2

Essential Understanding: When simplifying an expression, you need to perform operations in the correct order.

Order of Operations

1. Grouping – perform operations inside parentheses ( ), brackets [ ]. The fraction bar is a grouping symbol – it separates the numerator from the denominator. If you have multiple grouping symbols, simplify the innermost first.

2. Exponents – simplify powers

3. Multiply and Divide – from left to right

4. Add and Subtract – from left to right


(g) 10 − 10 ÷ 8 − 3( )⎡

⎣⎤⎦ (h) 2 + 8 − 4( ) ÷ 2⎡

⎣⎤⎦

(i) 3 + 5 + 8 × 2 ÷ 4 (j) 3 9 − 2 5 − 1( )⎡

⎣⎤⎦

EX 3 Evaluate each expression for x = 5 and y = 2 .

(a) x 2 + x − 12 ÷ y 2 (b) xy( )2 ÷ xy( )

EX 4 Evaluate each expression for a = 3 and b = 4 .

(a) 3b − a2 (b) 2b2 − 7a


EX 5 Evaluate the real world expression.

(a) What is an expression for the spending money you have left after depositing 2

5

of your wages in savings? Evaluate the expression for weekly wages of $40, $50, $75 and $100.

Wages w −2

5w Total Spending

Money

40 40 −2

540( )

50 50 −2

550( )

75 75 −2

575( )

100 100 −2

5100( )

(b) The shipping cost for an order at an online store is 1

10 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, $79, $95 and $103?

Order Total Cost with Shipping

43

79

95

103


§1 – 3: Real Numbers and the Number Line

Vocabulary:

• Radical: a mathematical word for “square root”.

• Perfect square: the square of an integer. For example, 25 is a perfect square

52( ) . • Set: a well-defined collection of objects.

• Element of a set: what each object in a set is called.

• Subset of a set: a set made up of some of the elements of another set.

EX 1 Simplify each expression.

(a) 81 (b) 9

16

(c) 64 (d) 25

(e) 1

36 (f)

81

121

Square Root

Definition: if

Example: because


EX 2 Estimate each square root. (a) Lobster eyes are made of tiny square regions. Under a microscope, the

surface of the eye looks like graph paper. A scientist measures the area of one of the squares to be 386 square microns. What is the approximate side length of the square to the nearest micron?

• Estimate 386 by finding the two closest perfect squares.

• Estimate 386 using a calculator.

(b) What is the value of 34 to the nearest integer?

Number Classification Natural Numbers:

Whole Numbers:

Integers:

Rational Numbers: any number that you can write in the form where a

and b are integers and b is not zero. Rational numbers can be written as a decimal that terminates or repeats.

Irrational Numbers: any number that cannot be written as a fraction. In

decimal form, irrational numbers do not terminate or repeat.


Some square roots are rational and some are irrational. If a whole number is not a perfect square, its square root is irrational. EX 3 Classify each as rational or irrational.

(a) 4 (b) 3

(c) 25 (d) 10 EX 4 To which subsets of the real numbers does each number belong? (a) 15 (b) −1.4583

(c) 57 (d) 3

10

(e) 9 (f) −0.45 EX 5 Compare the two quantities using an inequality.

(a) 17 and 41

3

(b) 129 and 11.52


EX 6 What is the order from least to greatest?

(a) 4 ,0.4,−2

3, 2 ,− 1.5

(b) 3.5,− 2.1, 9 ,−7

2, 5


§1 – 4: Properties of Real Numbers Vocabulary:

• Equivalent expressions: expressions that have the same value

• Properties: relationships that are always true for real numbers

• Deductive reasoning: process of reasoning logically from given facts to a conclusion.

• Counterexample: an example showing a statement is not true.

Properties of Real Numbers

Commutative Property: changing the order does not change the sum or product.

• Addition

• Multiplication

Associative Property: changing the grouping does not change the sum or product.

• Addition

• Multiplication

Identity Property:

• Addition

• Multiplication

Zero Property of Multiplication:

Multiplication Property of -1:


EX 1 What property is illustrated by each statement? (a) 42 • 0 = 0 (b) y + 2.5( ) + 28 = y + 2.5 + 28( )

(c) 10x + 0 = 10x (d) 4x • 1 = 4x

(e) x + y + z( ) = x + z + y( ) EX 2 Use the properties to help with mental math. (a) A movie ticket costs $7.75. A drink costs $2.40. Popcorn costs $1.25.

What is the total cost for a ticket, a drink and popcorn? (b) A can holds 3 tennis balls. A box holds 4 cans. A case holds 6 boxes.

How many tennis balls are in 10 cases? EX 3 Use the properties and deductive reasoning to simplify each

expression. (a) 5 3n( ) (b) 4 + 7b( ) + 8


(c) 6xyy

(d) 6 + 4h + 3( ) EX 4 Is the statement true or false? If false, give a counterexample.

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

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

(c) For all real numbers j and k, j • k = k + 0( ) • j

(d) For all real numbers m and n, m n + 1( ) = mn + 1


§1 – 5: Adding and Subtracting Real Numbers Vocabulary:

• Absolute value of a number is its distance from 0 on the number line. Absolute value is always positive.

• Opposites: two numbers that are the same distance from 0 on the number line but lie on opposite sides of 0.

• Additive inverses: a number and its opposite.

EX 1 What is each sum? (a) −12 + 7 (b) −18 + −2( )

(c) −4.8 + 9.5 (d) 3

4+ −

5

6

⎛

⎝⎜⎞

⎠⎟

(e) −16 + −8( ) (f) −11 + 9

Adding Real Numbers Adding Numbers with the Same Sign – add the absolute values of the two numbers. The sum has the same sign as the numbers added. Example:

Adding Numbers with Different Signs – subtract the absolute values of the two numbers. The sum has the same sign as the number with the bigger absolute value. Example:


(g) 9 + −11( ) (h) −6 + −2( )

EX 2 What is each difference? (a) −8 − −13( ) (b) 3.5 − 12.4

(c) 9 − 9 (d) 4.8 − −8.7( ) The properties you learned in the last section apply only to addition, not subtraction. However, they do apply to both positive and negative numbers. You can use these properties to reorder and simplify expressions.

Inverse Property of Addition

Subtracting Real Numbers To subtract a real number, add its opposite:

Example:


EX 3 A reef explorer dives 25 ft to photograph brain coral and then rises 16 ft to travel

over a ridge before diving 47 ft to survey the base of the reef. Then the diver rises 29 ft to see an underwater cavern. What is the location of the cavern in relation to sea level?

EX 4 A robot submarine dives 803 ft to the ocean floor. It rises 215 ft as the water

gets shallower. Then the submarine dives 2619 ft into a deep crevice. Next, it rises 734 ft to photograph a crack in the wall of the crevice. What is the location of the crack in relation to sea level?


§1 – 6: Multiplying and Dividing Real Numbers Vocabulary:

• Reciprocal or multiplicative inverse: of a real number in the form ab

is ba

.

If you multiply… Your answer is…

€

+( ) • +( ) +

€

−( ) • −( ) +

€

+( ) • −( ) or

€

−( ) • +( ) -

EVEN number of negatives multiplied +

ODD number of negatives multiplied -

EX 1 What is each product? (a) 12 −8( ) (b) 24 0.5( )

(c) −3

4•1

2 (d) −3( )2

(e) 6 −15( ) (f) 12 0.2( )

(g) −7

10

3

5

⎛

⎝⎜⎞

⎠⎟ (h) −4( )2


Recall square roots. All numbers really have two square roots, one positive, one

negative. Consider this… 52 = 25 and also −5( )2 = 25 , so 25 = 5 or

25 = −5 . We say that 25 = ±5 . EX 2 What is the simplified form of each expression?

(a) − 25 (b) ±4

49

(c) ± 16 (d) − 121

Dividing Real Numbers The quotient of two real numbers with different signs is negative. Example:

The quotient of two real numbers with the same sign is positive. Example:

Division Involving 0 Example: is undefined

is undefined

Ways of Writing Division


EX 3 A sky diver’s elevation changes by −3,600 ft in 4 minutes after the parachute opens. What is the average change in the sky diver’s elevation each minute?

EX 4 You make five withdrawals of equal amounts from your bank account. The total

amount you withdraw is $360. What is the change in your account balance each time you make a withdrawal?

EX 5 Divide the following fractions.

(b) 3

4÷ −

5

2

⎛

⎝⎜⎞

⎠⎟

(a) find xy

if x = −3

4 and y = −

2

3

Inverse Property of Multiplication

For every nonzero real number a, there is a multiplicative inverse such that

.

Example:


Concept Byte: Operations with Rational and Irrational Numbers Common Core State Standard: N.RN.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a non-zero rational number and an irrational number is irrational. Rational + Rational = Rational Rational × Rational = Rational Rational + Irrational = Irrational Rational × Irrational = Irrational


§1 – 7: The Distributive Property

EX 1 Simplify each expression using the distributive property. (a) 3 x + 8( ) (b) 5b − 4( ) −7( )

(c) 5 x + 7( ) (d) 12 3 +1

6t

⎛

⎝⎜⎞

⎠⎟

(e) 0.4 + 1.1c( )3 (f) 2y − 1( ) −y( ) EX 2 What sum or difference is equivalent to each expression?

(a) 7x + 25

(b) 4x − 163

The Distributive Property

Algebra: Examples:


(c) 11 + 3x6

(d) 15 + 6x12

(e) 4 − 2x8

Recall that the Multiplication Property of -1 states that −1 • x = −x . To simplify an expression such as − x + 6( ) , you can rewrite the expression as −1 x + 6( ) . EX 3 What is the simplified form of each expression? (a) − 2y − 3x( ) (b) − a + 5( )

(c) − −x + 31( ) (d) − 4x − 12( ) (e) − 6m − 9n( )


Using the distributive property for mental math… EX 4 (a) Deli sandwiches cost $4.95 each. What is the total cost of 8 sandwiches? (b) Julia commutes to work on the train 4 times each week. A round-trip

ticket costs $7.25. What is her weekly cost for tickets? Use mental math. Vocabulary…

• Term: a number, a variable or the product of a number and one or more variables.

• Constant: a term that has no variable.

• Coefficient: the numerical factor of a term.

• Like terms: terms that have the same variable factors.

EX 5 6a2 − 5ab + 3b − 12

(a) What are the terms in this expression?

(b) What is the constant in this expression?

(c) What are the coefficients in this expression?


Terms: 7a and −3a 4x 2 and 12x 2 6ab and −2a xy 2 and x 2y

Like Terms?

EX 6 Simplify by combining like terms.

(a) 8x 2 + 2x 2 (b) 5x − 3 − 3x + 6y + 4

(c) 3y − y (d) 7y 3z − 6yz 3 + y 3z EX 7 Simplify by using the distributive property and combining like terms. (a) 2 − 3 x + 1( ) (b) 3 4x − 9( ) + 2x + 1

(c) 2x + 3 + x( )4 (d) 5x + 7 x − 3( )

(e) 4 2x + y( ) + 8 x − 3y( ) (f) 4a + 6b − a + 3b


(g) 7 9k − 3( ) − 5 6k − 5( ) − 3k (h) 8n − 3 3n − 1( ) §1 – 8: An Introduction to Equations Vocabulary…

• Equation: a mathematical sentence that uses an equals sign (=).

• Open sentence: an equation that contains one or more variables that may be true or false depending on the value of the variable.

• Solution of an equation: the value of the variable that makes an equation true.

EX 1 State whether each equation is true, false or open. (a) 24 + 18 = 20 + 22 (b) 7 • 8 = 54 (c) 2x − 14 = 54 (d) 3y + 6 = 5y − 8 (e) 16 − 7 = 4 + 5 (f) 32 ÷ 8 = 2 • 3 EX 2 (a) Is x = 6 a solution of the equation 32 = 2x + 12?


(b) Is m =1

2 a solution of the equation 6m − 8 = −5 ?

In real-world problems, the word “is” can indicate equality. You can represent some real-world situations using an equation. EX 3 An art student wants to make a model of the Mayan Great Ball Court in Chichén

Itzá, Mexico. The length of the court is 2.4 times its width. The length of the student’s model is 54 in. What should the width of the model be?

EX 4 The length of the ball court at La Venta is 14 times the height of its walls. Write

an equation that can be used to find the height of a model that has a length of 49 cm.

EX 5 Use mental math to find the solution of each equation.

(a) x + 8 = 12 (b) a8

= 9

(c) 12 − y = 3


§1 – 9: Patterns, Equations and Graphs

EX 1 Identify the following on the coordinate plane above.

(a) x-axis (b) y-axis (c) origin

(d) quadrants (e) ordered pair (f) coordinates


EX 2 (a) Is 3,10( ) a solution of the equation y = 4x ?

(b) Is 5,20( ) a solution of the equation y = 4x ?

Alg 1 Notes 1 1 thru 1 9 - Ms. Zaleski's Math Classes -...

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