~ chapter 1 ~ tools of algebra algebra i lesson 1-1 using variables lesson 1-2 exponents & order...
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~ 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 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
Main Menu
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
Using VariablesNotes
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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Using VariablesNotes
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
Using VariablesNotes
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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Using VariablesNotes
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
Exponents & Order of Operations
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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Exponents & Order of Operations
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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Exponents & Order of Operations
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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Exponents & Order of Operations
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
Exponents & Order of Operations
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
Exponents & Order of Operations
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
Exponents & Order of Operations
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
?????? Questions ???????
Exponents & Order of Operations
Homework - Answers
Lesson 1-2
?????? Questions ???????Main Menu
Exploring Real Numbers NotesLesson 1-3
Main Menu
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 =
Evaluating Expressions
-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
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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Evaluating Expressions
-(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
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)
Properties of Real Numbers
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
Properties of Real Numbers
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;-)
Properties of Real Numbers & Graphing on the Coordinate Plane
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
Properties of Real Numbers & Graphing on the Coordinate Plane
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.
Properties of Real Numbers & Graphing on the Coordinate Plane
PracticeLesson 1-8 & 1-9
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Homework
Practice 1-8 & 1-9 odd
Properties of Real Numbers & Graphing on the Coordinate Plane
Homework Answers
Lesson 1-8 & 1-9
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