~ Chapter 1 ~
Tools of Algebra
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
Using Variables
Chap 1 Diagnosing Readiness Answers
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Using Variables
Notes
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
Using
Variables
Notes
Notes
Writing an Algebraic Expression
 Add – Terms -> sum, altogether, more than,
greater
than,
greater
than…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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Using Variables
Notes
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
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Using Variables
Notes
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) + 4
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Using Variables
Notes
 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 Variables
Notes
Number of
Hours
4
6
8
10
Total Pay
$32
$48
$64
$80
Know:
Number of hours times 8 equals the total pay
Define:
Let n = number of hours
Let t = total pay
Write:
8n = t or t = 8n
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Using Variables
Notes – Practice Problems
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)
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Using Variables
Homework
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Exponents & Order of
Operations
Daily Math Review
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Using Variables
Homework - Answers
?????? Questions ???????
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Exponents & Order of
Operations
Notes
 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
Notes
Simplifying a Numerical Expression
3
2
4 1
18
25 – 8 * 2 + 32
3
2
4
1
18
14 + 2 * 4 – 22
2
3
1
5
3+5–6÷2
2
1
6 – 10 ÷ 5
4
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Exponents & Order of
Operations
Notes
Evaluating an Algebraic Expression
3a – 23 ÷ b
2
for a = 7 and b = 4
4 1 3
3 * 7 – 23 ÷ 4
19
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
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Exponents & Order of
Operations
Practice
3 * 6 – 42 ÷ 2
4 * 7 + 4 ÷ 22
53 + 90 ÷ 10
Evaluate the following for c = 2 and d = 5
4c – 2d ÷ c
3
c4 – d * 2
6
d + 6c ÷ 4
8
40 – d2 + cd * 3
45
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Exponents & Order of
Operations
Notes
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
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Exponents & Order of
Operations
Notes
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]
24
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Exponents & Order of
Operations
Practice
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
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Exponents & Order of
Operations
Homework
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Exploring Real Numbers
Daily Math Review
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Exponents & Order of
Operations
Homework - Answers
?????? Questions ???????
?????? Questions ???????
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Exploring Real Numbers
Notes
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
5
natural number, whole number,
integer, rational number
rational number
5/12
rational number
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Exploring Real Numbers
Notes
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
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Adding Real Numbers
Notes
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
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Adding & Subtracting Real Numbers
Notes
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
14.6
1.2
12.1
3.3
+
-3
-3.9
-4
2
2.7
-5
Subtracting Real Numbers
Leave, change, opposite… (then use the rules for addition)
3 – 5 = 3 + (-5) = -2
¾ - (-11/12) =
3-(-5) = 3 + 5 = 8
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Subtracting Real Numbers
Notes
Absolute Values
l 5-11 l =
l7–8l=
Evaluating Expressions
-a – b for a = -3 & b = -5
-(-3) – (-5)
=3+5=8
Subtract with Matrices
-3
4
0
-1
_
-5
-6
-9
-4
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Real Numbers
Homework
Practice 1-3 - every 3rd problem
Practice 1-4 – every 3rd problem
Practice 1-5 – every 3rd problem & #35
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Adding Real Numbers
Daily Math Review
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Real Numbers
Homework - Answers
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Real Numbers
Homework - Answers
?????? Questions ???????
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Multiplying & Dividing Real Numbers
Notes
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) =
30
-43
-(4*4*4)
-64
-2.7 * 4.1 =
10
(-2)4
-11.07
-(3/4)2
(-2)(-2)(-2)(-2)
16
-( ¾ * ¾)
-9/16
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Multiplying & Dividing Real Numbers
Notes
Evaluating Expressions
-(cd)
(-2)(-3)(cd)
-(-8*(-7))
for c= -8 and d= -7
(-2)(-3)(-8*(-7))
-56
336
3x ÷ 2z + y ÷ 10
2z+x/2y
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
for x = 8, y = -5, & z = -3
- 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
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Properties of Real Numbers
Notes
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The Distributive Property
Notes
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
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Multiplying & Dividing Real Number & The
Distributive Property
Notes & Homework
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)
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Properties of Real Numbers
Notes
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 Main
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Properties of Real Numbers
Notes
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
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Properties of Real Numbers & Graphing on
the Coordinate Plane
Notes
2(3t – 1) + 2
Step
Reason
6t – 2 + 2
Distributive property
6t +(-2) + 2
Rules/defn of subtraction
6t
addition
Homework
Practice 1-7 & 1-8 even;-)
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Subtracting Real Numbers
Quiz Lesson 1-1 to 1-4
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Properties of Real Numbers
Notes
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Properties of Real Numbers & Graphing
on the Coordinate Plane
Homework Answers
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Graphing on the Coordinate Plane
Notes
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.
(0,0)
origin
(-,+)
Quadrant II
Quadrant III
(-,-)
y-axis
(+,+)
Quadrant I
x-axis
Quadrant IV
(+,-)
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Properties of Real Numbers & Graphing on
the Coordinate Plane
Identifying & Graphing Points
Notes
Use the (x, y) location to identify the location of a point.
Graph C(0,3);
Quadrant? (-2,0)
D (2,4);
(4,-1)
E(-1,-4);
(-3,-5)
F(-3,0)
(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.
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Properties of Real Numbers & Graphing
on the Coordinate Plane
Practice
Homework
Practice 1-8 & 1-9 odd
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Properties of Real Numbers & Graphing
on the Coordinate Plane
Homework Answers
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Daily Math Review
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~ Chapter 1 ~
Chapter Review
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~ Chapter 1 ~
Chapter Review
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