Download Notes - Little Chute Area School District

Algebra
Notes: Getting Ready for Algebra
Math Basics
Natural Numbers
Whole Numbers
Integers
Rational Numbers
A mathematical operation is…
Mathematical Operations Part I
A binary operation is…
Four Main Binary Operations
1.
2.
3.
4.
Mathematical Operations Part II
A unary operation is…
Unit One Notes
Pg. 1
Algebra
Notes: Getting Ready for Algebra
Exponents
“Raising to an Exponent” is …
Evaluating Exponents
Long Way
Short Way
Roots
“Taking a Root” means to …
Evaluating Roots
Long Way
Short Way
Unit One Notes
Pg. 2
Algebra
Notes: Getting Ready for Algebra
Order of Operations: Introduction
Write a mathematical expression that could be used to model each of the following:
1. The total cost of buying a CD for $13.99, a book for $21.50, a candy bar for $.75, and a
shirt for $17.85.
2. The total cost of buying 7 CD’s for $13.99 each.
3. The total cost of buying a squash for $1.56 and 3 zucchinis for $2.10 each.
Which Price is correct?
Order of Operations
In a series of mathematical operations, there is a correct order in which to complete the
operations.
1.
2.
3.
4.
What Does Evaluate Mean?
To evaluate an expressions means…
Unit One Notes
Pg. 3
Algebra
Notes: Getting Ready for Algebra
Example
Evaluate:
(6 + 5)2 – 3(4) + 24÷6
When to work from left to right.
3(4)÷6(3)
7–3+9
Other Grouping Symbols
A fraction bar acts like a parentheses:
For example:
CPR
C
P
R
Unit One Notes
Pg. 4
Algebra
Notes: Getting Ready for Algebra
Practice Problems
Evaluate each of the following expressions using the correct order of operations. Show all of
your work.
6(2)  3 2
3[12 - 2(3)] + 44÷16
14  7  5
The Number Line
Concept of Opposites
The numbers 5 and -5 are called opposites because…
In fact, the – sign is often read as…
Combining Positives and Negatives
Positives and Negatives have the characteristic that …
Unit One Notes
Pg. 5
Algebra
Notes: Getting Ready for Algebra
Adding Signed Numbers
When adding two signed numbers, keep in mind two things:
1.
2.
Adding Example #1
Adding Example #2
Adding Example #3
Subtracting Signed Numbers
When subtracting two signed numbers, keep in mind three things:
1.
2.
3.
Unit One Notes
Pg. 6
Algebra
Notes: Getting Ready for Algebra
An Easy Example
Another Easy Example
A Little Bit Tougher
The Subtraction Rule
Subtraction means …
7–3
MEANS
-10 – (-6)
MEANS
-10 – (-13)
MEANS
a–b
MEANS
Unit One Notes
Pg. 7
Algebra
Notes: Getting Ready for Algebra
Multiple Operation Problems: Addition Only
15 + 7 + (-13) + 7 + (-20)
Multiple Operation Problems: Addition and Subtraction
20 - 9 + (-17) + 13 - (-19)
Multiplying Integers: The Pattern Approach
A New Pattern
Unit One Notes
Pg. 8
Algebra
Notes: Getting Ready for Algebra
Summary of Multiplication
Linking Division to Multiplication
24 ÷ 4 = x is the same as…
24 ÷ -4 = x is the same as…
Summary of Division
Guided Practice
7 + (-3) – 9 – (-8)
2(-3) - 6(-4) + 18 ÷ (-9)
3(9 – 12) – 4[-7 + 2(3)]
Unit One Notes
Pg. 9
Algebra
Notes: Getting Ready for Algebra
Let’s Make Flow Charts For Combining Signed Numbers
Variables
A variable is…
Why would we want to use a letter to represent a number?
1. Sometimes we …
2. Sometimes we…
When We Don’t Want To Specify the Value of a Number
Additive Inverse Property:
A number plus its opposite always equals zero.
Specific
General
Unit One Notes
Pg. 10
Algebra
Notes: Getting Ready for Algebra
When We Don’t Know the Value of a Number
Barbara weighs 60 kg and is on a diet of 1600 calories per day, of which 850 are used
automatically by basal metabolism. She spends about 15 cal/kg/day times her weight doing
exercise. If 1 kg of fat contains 10,000 calories and we assume that the storage of calories is
100% efficient, what will Barbara’s weight be in the long run? How long will it take for her
weight to level off?
Algebraic Expressions
An algebraic expression is…
Evaluating Algebraic Expressions
Evaluate the expression 2xy + y2÷3 - 6 if x = 3 and y = 6.
Real Life Applications
Evaluating algebraic expressions takes place all of the time in real life when we use formulas.
D
r2
 1.47Tr
30 F
D = the estimated total stopping distance of a car in feet
r = the car’s speed in miles per hour,
F = a driving surface factor based on road conditions
T = the time it takes for the driver to react and press brakes
Unit One Notes
Pg. 11
Algebra
Notes: Getting Ready for Algebra
Application Problem
A car is traveling on a wet asphalt highway when a deer jumps into the road. The driver takes
.85 seconds to react and then applies the brakes. If the car is traveling at a rate of 60 mph, what
is the total stopping distance?
Like Terms
Like terms are…
Another Example
5x + 7y + 2xy – 8x + 6y + 13xy
Simplifying Expressions
Expressions can be simplified by…
5 – 3x2 + 2x – 8 + 6x
Try This One
Simplify the following expression by combining like terms.
5x + 7y + 2xy – 8x + 6y + 13xy
Unit One Notes
Pg. 12
Algebra
Notes: Getting Ready for Algebra
The Distributive Property
A Specific Example
4(3 + 2)
Solution #1
Solution #2
Examples of The Distributive Property
5(x + 2) =
(2x - 3y)5 =
-7(3 – 7x) =
Be Careful of This One
12 – (3x – 7)
This is called…
Unit One Notes
Pg. 13
Algebra
Notes: Getting Ready for Algebra
The Distributive Property With Division
Example:
Practice Time
1. 3(x + 1)
2. (4x – 8)7
3. -3(-2x + y)
4. 6 (-2x - 4)
More Practice
Simplify each expression by combining like terms.
3x2 + 4x + 8 – 7x2
5y + 6x – 8 – 7y – 9x
2(3x + 4) + 7x
 7  (5x 3) 
3(4 – 3x) - 2(6x – 8)
Unit One Notes
Pg. 14
16x  24
4
Algebra
Notes: Getting Ready for Algebra
Words to Symbols
A major skill of algebra is to be able to translate a verbal description into an algebraic
expression, equation, or inequality.
Ten less than seven times a number is bigger than nine.
The length of a rectangle is two less than three times its width
Defining a Variable
Once we have an algebraic representation of a problem, we often are required to determine the
value of one or more of the variables in the equation. To do this, it is important that we are able
to identify what each variable in an expression or equation stands for.
The process of stating the meaning of each variable in an expression or equation is called …
To define a variable,…
Look for phrases like…
Formula Example
Law of Thermal Expansion
The length that a metal pipe will expand by when subjected to heat is directly proportional to the
product of the pipe’s original length and the difference between the pipe’s original temperature
and its new temperature after it has been heated.
E = kL(T – T0)
E=
T=
K=
T0=
L=
Unit One Notes
Pg. 15
Algebra
Notes: Getting Ready for Algebra
Some key words and phrases that mean add are…
Some key words and phrases that mean subtract are…
Some key words and phrases that mean multiply are…
Some key words and phrases that mean divide are…
Some key words and phrases that mean = are…
Some key words and phrases that mean > are…
Some key words and phrases that mean < are…
Some key words and phrases that mean ≥ are…
Some key words and phrases that mean ≤ are…
Words To Symbols: the Process
1.
2.
3.
Unit One Notes
Pg. 16
Algebra
Notes: Getting Ready for Algebra
Example #1
Seven times a number is the same as 12 more than 3 times the number. Find the number
Example #2
Twenty-eight less than the product of two numbers.
A Subtle but IMPORTANT Difference
Twenty-eight is less than the product of two numbers.
Guided Practice
Translate each of the following phrases and sentences into algebraic form. Make sure to define a
variable or variables for each situation.
1. Seven more than two times a number is the same as five.
2. Jerry’s age is two years less than two times Bob’s age.
Unit One Notes
Pg. 17
Algebra
Notes: Getting Ready for Algebra
Generalizing a Real Life Problem
Daisy’s Daycare provides the following list of babysitting rates.
Full Time, 0 years-2 years
Full Time, 2 years – 4 years
Half Time/ 4K or 5K
School age before and after school
$125 per week
$110 per week
$95 per week
$50 per week
Joe and Sue have a 4 year old daughter who will be attending 4K classes for half of a day, four
days per week, and who will be at the day care for a full day on the fifth day of the week. This
situation does not neatly fit into one of the situations described above, so Daisy would like a little
help in trying to decide exactly how much to charge Joe and Sue per week for daycare.
Specific Solution
General Solution
Unit One Notes
Pg. 18