~ Chapter 1 ~

~ Chapter 1 ~Tools of Algebra

Algebra I

Lesson 1-1 Using Variables

Lesson 1-2 Exponents & Order of Operations

Lesson 1-3 Exploring Real Numbers

Lesson 1-4 Adding Real Numbers

Lesson 1-5 Subtracting Real Numbers

Lesson 1-6 Multiplying & Dividing Real Numbers

Lesson 1-7 The Distributive Property

Lesson 1-8 Properties of Real Numbers

Lesson 1-9 Graphing Data on the Coordinate Plane

Chapter Review

Algebra I

Using Variables Chap 1 Diagnosing Readiness

AnswersLesson 1-1

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Using VariablesNotes

Lesson 1-1

Variable – a symbol that represents one or more numbers.

Examples – x, y, q, r, s, n …

Algebraic Expression – a mathematical phrase that can include numbers, variable,

and operation symbols. (no equal sign)

Examples ~ 2n , 4+8 , n , 27x – 4y … 9

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Using Variables

NotesLesson 1-1

Writing an Algebraic Expression

Add – Terms -> sum, altogether, more than,greater than… Subtract – terms -> difference, minus, less than… Multiply – terms -> product, times, multiplied by, twice, triple…

Divide – terms -> quotient, divided by,half, third…

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Writing an Algebraic Expression

Add – Terms -> sum, altogether, more than, greater than… Subtract – terms -> difference, minus, less than… Multiply – terms -> product, times, multiplied by, twice, triple…

Divide – terms -> quotient, divided by,half, third…

Using VariablesLesson 1-1

Notes


Lesson 1-1

Examples:Five more than a number n + 5

The difference of five and a number 5 - x

Five less than x x - 5

The product of five and a number 5n

The quotient of a number and five n ÷ 5 Main

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Lesson 1-1

More complex algebraic expressions:

(1)Two times a number plus five

2n + 5

(2)Seven less than five times a number

5x - 7

(3)Four more than the quotient of a number and six

(n ÷ 6) + 4Main Menu


Lesson 1-1

Equation – a mathematical sentence that uses an equal sign. (Ex: 2+3 = 5, 4x=8,…)

Open sentence – an equation that contains one or more variables. (Ex: 2x=8, 3x+2y = 10)

Writing an Equation

Track One Media sells all CD’s for $12 each. Write an equation for the total cost of a given number of CD’s.

Know: The total cost is 12 times the number of CD’s

Define: Let n = of CD’s Let c = total cost

Write: c = 12n or 12n = c

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Lesson 1-1

Number of Hours

Total Pay

4 $32

6 $48

8 $64

10 $80Know:Number of hours times 8 equals the total payDefine:Let n = number of hours Let t = total payWrite:8n = t or t = 8n Main

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Using VariablesNotes – Practice

Problems

Lesson 1-1

the quotient of 6.3 and b

6.3 ÷ b

s minus ten

s - 10

9 less than a number

n - 9

The sum of twice a number and thirty-one

2x + 31

The product of one half of a number and one fourth of the same number

½ n ( ¼ n)Main Menu

Using VariablesHomework

Lesson 1-1

Practice 1-1

~ all ~

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Exponents & Order of Operations

Daily Math Review

Lesson 1-2

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Using VariablesHomework -

Answers

Lesson 1-1

?????? Questions ???????

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NotesLesson 1-2

Simplify – replace an expression with its simplest name or form.

Exponents – A number that shows repeated multiplication. (In 24 ~ 4 is the exponent)

Base – The number that is multiplied repeatedly in a power. (In 24 ~ 2 is the base)

Power – has two parts, a base and an exponent, and has the form an.

Order of operations – GEMS – (1) grouping symbols; (2) Exponents; (3) Multiply & Divide (left to right) (4) Subtract & Add (left to right)

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NotesLesson 1-2

Simplifying a Numerical Expression

25 – 8 * 2 + 32

14 + 2 * 4 – 22

3 + 5 – 6 ÷ 2

6 – 10 ÷ 5

143 2

123

12 3

4

2 1

18

18

5

4

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NotesLesson 1-2

Evaluating an Algebraic Expression

3a – 23 ÷ b for a = 7 and b = 4

3 * 7 – 23 ÷ 4

Example: A shirt costs $22.85 plus sales tax. What is the total cost of the shirt?

Expression - p + p * r

( p = price; r = tax)

c = $22.85 + $22.85(0.07) = $22.85 + $1.5995 =

c = $24.45

12 3419

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PracticeLesson 1-2

3 * 6 – 42 ÷ 2

4 * 7 + 4 ÷ 22

53 + 90 ÷ 10

Evaluate the following for c = 2 and d = 5

4c – 2d ÷ c c4 – d * 2

3 6

d + 6c ÷ 4 40 – d2 + cd * 3

8 45Main Menu


NotesLesson 1-2

Expressions with parenthesis

15(13 – 7) ÷ (8 - 5) (5 + 3) ÷ 2 + (52 – 3)

15(6) ÷ (3) (8) ÷ 2 + (25 – 3)

90 ÷ 3 4 + 22

30 26

Expressions with Exponents

(cd)2 for c = 7 & d = 19

(7 * 19)2

(133)2 = 17,689 Main Menu


NotesLesson 1-2

m2n for m = 5 & n = 4

52 * 4 = 25 * 4 = 100

Evaluate the following for m=3, q=4, p=7

qp2 + q2p m(pq)2

4*72 + 42*7 3(7*4)2

4*49 + 16*7 3(28)2

196 + 112 = 308 3(784) = 2,352

Simplifying an expression

2[(13-7)2 ÷3]

24Main Menu


PracticeLesson 1-2

5[4 + 3(22+1)] 28 ÷ [(19 -7) ÷ 3]

5[4 + 3(4+1)] 28 ÷ [(12) ÷ 3]

5[4 + 3(5)] 28 ÷ [4] = 7

5[4 + 15]

5[19] = 95

9 + [4 – (10 – 9)2]3

9 + [4 – (1)2]3

9 + [4 – 1]3

9 + [3]3

9 + 27 = 36 Main Menu


HomeworkLesson 1-2

Practice 1-2

~ even ~

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Exploring Real Numbers

Daily Math Review

Lesson 1-3

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?????? Questions ???????


Homework - Answers

Lesson 1-2

?????? Questions ???????Main Menu

Exploring Real Numbers NotesLesson 1-3

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Natural Numbers – counting numbers ~ 1, 2, 3… (not 0)

Whole Numbers – non-negative integers ~ 0, 1, 2, 3, 4…

Integers – whole #’s & their opposites ~ …-2, -1,0,1,2…

Rational Numbers – numbers that can be written as a/b where b ≠ 0. Decimal form is a terminating or repeating decimal.

Irrational Numbers – numbers that cannot be expressed in the form a/b where a & b are integers. (Ex ~ π, √10, 0.101001000…)

Classify the following

-12 -4.67

integer, rational number rational number

5 5/12

natural number, whole number, rational number integer, rational number

Exploring Real Numbers NotesLesson 1-3

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Counterexample – Any example that proves a statement false…

(a)All Whole numbers are rational numbers T or F

(b) All integers are whole numbers. T or F

(c) The square of a number is always greater than the number. T or F

(d) All whole numbers are integers. T or F

(e) No fractions are whole numbers. T or F

Inequality

(>, <, ≥ , ≤ , ≠ ~ used to compare the value of two expressions)

Ordering fractions

(1)Write fractions as a decimal and then compare

(2) Find the common denominator, convert, and then compare

Absolute value – distance a number is from 0. l-19l = 19 l22l = 22

Adding Real Numbers NotesLesson 1-4

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Identity Property of Addition - n + 0 = n, for every real number n.

Inverse Property of Addition – n + (-n) = 0 (additive inverse is the opposite of a number)

Rules for Adding

•Numbers with the same signs – add and keep the sign.

•Numbers with different signs – subtract, answer takes the sign of the number with the greatest absolute value.

Examples

-7 + (-4) -26.3 + 8.9

-11 -17.4

-3/4 + (-1/2) 8/9 + (-5/6)

-1 ¼ 1/18

Adding & Subtracting Real Numbers NotesLesson 1-4 & 1-5

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Evaluating Expressions

-n + 8.9 for n = -2.3 t + (-4.3) for t = -7.1

11.2 -11.4

Matrix

29.3 3.1 -3 -3.9

14.6 1.2 + -4 2

12.1 3.3 2.7 -5

Subtracting Real Numbers

Leave, change, opposite… (then use the rules for addition)

3 – 5 = 3 + (-5) = -2 3-(-5) = 3 + 5 = 8

¾ - (-11/12) =

Subtracting Real Numbers NotesLesson 1-5

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Absolute Values

l 5-11 l = l 7 – 8 l =


-a – b for a = -3 & b = -5

-(-3) – (-5) = 3 + 5 = 8

Subtract with Matrices

-3 4 _ -5 -6

0 -1 -9 -4

Real Numbers

HomeworkLesson 1-3, 1-4, & 1-

5

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Practice 1-3 - every 3rd problem

Practice 1-4 – every 3rd problem

Practice 1-5 – every 3rd problem & #35

Adding Real Numbers

Daily Math Review

Lesson 1-4

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Real Numbers

Homework - Answers

Lesson 1-3 & 1-4

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Real Numbers

Homework - Answers

Lesson 1-5

?????? Questions ???????

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Multiplying & Dividing Real Numbers NotesLesson 1-6

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Identity Property of Multiplication ~ 1 * n = n

Multiplication Property of Zero ~ n * 0 = 0

Multiplication Property of -1 ~ -1 * n = -n

Rules for Multiplying & Dividing

(1)Like/same signs – answer is positive.

(2) Different signs – answer is negative.

Simplifying Expressions

-6 * -5 = -2( -15/3) = -2.7 * 4.1 =

30 10 -11.07

-43 (-2)4 -(3/4)2

-(4*4*4) (-2)(-2)(-2)(-2) -( ¾ * ¾)

-64 16 -9/16

Multiplying & Dividing Real Numbers NotesLesson 1-6

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-(cd) (-2)(-3)(cd) for c= -8 and d= -7

-(-8*(-7)) (-2)(-3)(-8*(-7))

-56 336

3x ÷ 2z + y ÷ 10 2z+x/2y for x = 8, y = -5, & z = -3

3(8) ÷ 2(-3) + (-5) ÷ 10 [2(-3) + (8)]/2(-5)

24 ÷ (-6) + (-5/10) [-6 + 8]/-10

-4 + (-1/2) 2/-10

-4 ½ - 1/5

Inverse Property of Multiplication ~ a ≠ 0, a (1/a) = 1

x/y x = -3/4 and y = -5/2

-3/4 ÷ (-5/2) (the reciprocal or multiplicative inverse is used)

-3/4 x (-2/5) = 6/20 = 3/10

Properties of Real Numbers

NotesLesson 1-8

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The Distributive Property NotesLesson 1-7

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Distributive Property ~ a(b + c) = ab + ac; (b + c)a = ba + ca

a(b – c) = ab – ac; (b – c)a = ba – ca

Simplifying Expressions

13(103) = 13(100 + 3) 24(98) =

= 13(100) + 13(3)

= 1300 + 39

= 1339

6(m + 5) 2(3-7t) (0.4 + 1.1c)(3)

6m + 30 6 – 14t 1.2 + 3.3c

Terms, constants, and coefficients…

6a2 – 5ab + 3b – 12 (a2, ab, and b are all unlike terms)

Like terms are combined to simplify an expression…

3x2 + 5x2 7y + 6y -9w3 - 3w3

Multiplying & Dividing Real Number & The Distributive Property

Notes & Homework

Lesson 1-6 & 1-7

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Writing an Expression…

-2 times the quantity t plus 7

-2(t + 7)

The product of 14 and the quantity 8 plus w

14(8 + w)


NotesLesson 1-8

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Commutative Property of Addition ~ a + b = b + a

Commutative Property of Multiplication ~ a * b = b * a

Associative Property of Addition ~ (a + b) + c = a + (b + c)

Associative Property of Multiplication ~ (a * b) * c = a * (b * c)

Identity Property of Addition ~ a + 0 = a

Identity Property of Multiplication ~ a * 1 = a

Inverse Property of Addition ~ a + (-a) = 0

Inverse Property of Multiplication ~ a (1/a) = 1

Distributive Property

Multiplication Property of Zero

Multiplication Property of -1

Identify the property… 9+7 = 7+9 1m = m np = pn 2+0=2


NotesLesson 1-8

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Using Deductive Reasoning – logically justifying the reason (the why) for each step in simplifying an expression using properties, definitions, or rules.

Simplify the expression… Justify each step

7z – 5(3 + z)

Step Reason

7z – 15 - 5z Distributive Property

7z +(-15) + (-5z) Rules for subtraction

7z + (-5z) + (-15) Commutative property of addition

2z + (-15) addition of like terms

2z – 15 rules for subtraction

Properties of Real Numbers & Graphing on the Coordinate Plane

NotesLesson 1-8 & 1-9

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2(3t – 1) + 2

Step Reason

6t – 2 + 2 Distributive property

6t +(-2) + 2 Rules/defn of subtraction

6t addition

HomeworkPractice 1-7 & 1-8 even;-)

Subtracting Real Numbers

Quiz Lesson 1-1 to 1-4

Lesson 1-5

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NotesLesson 1-8

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Homework Answers

Lesson 1-8 & 1-9

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Graphing on the Coordinate Plane

NotesLesson 1-9

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Graphing Data on the Coordinate Plane

A coordinate plane has an x-axis (horizontal)

and a y-axis (vertical)

An ordered pair (x, y) are the numbers that

identify the specific location of a point.

Quadrant I

Quadrant II

Quadrant III

Quadrant IV

origin(0,0)

(+,+)(-,+)

(-,-) (+,-)

x-axis

y-axis

origin


NotesLesson 1-8 & 1-9

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Identifying & Graphing Points

Use the (x, y) location to identify the location of a point.

Graph C(0,3); D (2,4); E(-1,-4); F(-3,0)

Quadrant? (-2,0) (4,-1) (-3,-5) (2.7,3.6)

Can we find the dimensions of a shape when we graph it?

Scatter Plot

Graph that relates data of two different sets. Scattered points do not form a line. (Usually graphed in Quadrant I)

A trend line can show trend of the data in a scatter plot.


PracticeLesson 1-8 & 1-9

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Homework

Practice 1-8 & 1-9 odd


Homework Answers

Lesson 1-8 & 1-9

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Daily Math Review

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~ Chapter 1 ~Chapter Review

Algebra I Algebra I

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~ Chapter 1 ~Chapter Review

Algebra I Algebra I

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