R.1 Operations with Integers

Math92 Syllabus
History:

  • There is  strong evidence of mathematics usage in pre-history
  • Zero was developed in both India and Mexico (Mayan civilization) independently in 600 C.E.
  • Negative numbers  first appeared in ancient China (200 B.C.E. to 220 C.E.)

Definitions:

  • Set: A set is a collection of objects.
    • A set with no elements is called the empty set or a null set written as {} or ϕ
  • Natural Numbers: The natural number consists of the set {1, 2, 3, ….}
  • Whole Numbers: The whole number consists of the set {0, 1, 2, 3…}
  • Integers:  The integers number consists of the set {...-3, -2, -1, 0, 1, 2, 3….}

Classify Natural Vs Whole Vs Integers:

  1. The number of students in our math class each day of the week.
    • Whole number; there will be no students on the weekend
  2. The number of students in our math class on Monday or Wednesday.
    • Natural number
  3. Average low temperature in Alaska
    • Integer; as it could be negative or zero or positive

Number Line:

Scale:  The scale of a number line is the distance between the evenly spaced tick marks on the number line.












Graphing Points:

Graph A=-2, B=4 & C=3 on a number line
















Inequalities:



























Examples:

 -5   >   -8
  3    >  -3
 -2    <   0

Absolute Values:

Absolute Value:  The absolute value of a number is the distance between the number and zero.
|number| is the notation of absolute value

Examples:
|-6| = 6
|-23| = 23
|17| = 17

Opposite Numbers:

Opposite Number:  Numbers that have same absolute value but different signs are called opposite numbers













Order of Operations:

PEMDAS - Left to Right

Parentheses
Exponents
Multiplication / Division
Addition / Subtraction 
Math92 Schedule

R.2 Operations with Fractions

Prime Factorization:

Prime Numbers:  Prime numbers are natural numbers greater than 1 that are divisible  only by themselves and 1.
{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, …..}

Factor: A factor of a number divides number evenly.  It is a multiple of a number.
6 / 3 = 2  or 3 * 2 = 6  Here 2 and 3 are factors of 6.
Is 1 factor of 6?


Example:






















Greatest Common Factor - GCF:

Write prime factorization of each number
The GCF is the product of the common prime fators

Example:  Find GCF for 8 and 20
Prime factorizaion
8 =2*2*2
20 = 5*2*2
GCF is 2*2 = 4

Numerator  and Denominator:

In a fraction a/b, a is numerator and b is denminator.

Fraction is derived from Latin word “Fractus” meaning broken

A Fraction is represented as a part of whole.
For example:







Reduce Fractions to Lowest Terms:

Find GCF for numerator and denominator
Factor the numetor and denominator, using GCF as one of the factors
Divide out the GCF from the numerator and denominator.  The remaining fraction will be reduced fraction o the lowest term

Example:
16/36 = 4*4/9*4 = 4/9
25/75 = 5*5/5*5*3 = 1/3

Equivalent Fraction:

When a/b = ?/d, find missing factor c such that c= b*c =d
Multiply numerator & denominator by missing factor c which is a*c/b*c.  This will give equivalent fraction  with denominator d.

Example:
3/8 = ?/32
(3*4) /(8*4) = 12/32

Types of Fractions:




















Addition of Fractions - with like denominators:

Add numerators
Place resulting sum over common denominator
Simplify fraction if applicable

















​​What happens when the denominators are unlike denominators?

Finding Least Common Multiplier - LCM:

Find LCM for 6 and 15
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, ….
Multiples of 15: 15, 30, 45, 60, 75, 90, ….
LCM = 30

Using prime numbers:
6 = 2 x 3
15 = 3 x 5
LCM = 2 x 3 x 5 = 30

Find LCM for 12 and 15
12 = 2 x 2 x 3
15 =  3 x5
LCM = 2 x 2 x 3 x 5 = 60

Find LCM for 5, 6, 10
5 = 5
6 = 2 x 3
10 = 2 x 5
LCM = 2 x 3 x 5 = 30

Addition of Fractions - with unlike denominator:













































R.3 Operations with Decimals & Percents

Place Value:

Notice  when the ending conatins a ‘ths’, it represents a place value to the right of the decimal point.

    3                  4              5                 .                   2                     8                        0                            1
Hundreds     Tens        Ones     Decimal        Tenths      Hundredth     Thousandth   Ten-thousandth
place              place       place      point             place            place                place                place
 

Relationship between fraction & decimal:

To write a finite decimal as a fraction, identify the place value of the last nonzero digit.
Example:
.25 = 25/100 = ¼
0.36 = 36/100 = 9/25
Try 1.6

Fraction in Decimal form:

To write fraction into decimal form, use long division.

dividend/divisor = quotient + remainder/divisor
                                            quotient
                             = divisor)dividend

Example:
4/5 =0.8
3/8 = 0.375

Graphing decimals on a number line:

Graph 1.5

Graph -0.7



Rounding decimals:

  1. Underline the given place value in the number
  2. Look at the digit directly to the right of the underlined place
  3. Rounding down - If the digit is 0, 1, 2, 3 or 4 then delete all digits to the right of the underlined place.
  4. Rounding up - If the digit is 5, 6, 7, 8, or 9, the add 1 to the underlined place and delete all the digits to the right.

Examples:
Round to tenths place

1.69  rounds to 1.7
53.25 rounds to 53.3
53.24 rounds to 53.2

Addition & Subtraction of Decimals:

Addition:
3.4725 + 1.7354 = 5.2079
3.472 + 1.7354 = 5.20

Subtraction:
4.283 - 1.624 = 2.659
3.472 - 1.7354 = 3.4720 - 1.7354 = 1.7366

Multiplying & Dividing Decimals:

Ignore the decimal points and multiply as if they were two integers.
Count the number of digits after the decimal point in both numbers.
Starting at the far right of the result, move the decimal point left the same number of places as number of digits in both decimals
Example:
3.75 * 2.4 = 9
22.94 / 3.1 = 229.4/31 = 7.4

Order of Operation with Decimals:

The order of operations agreement is used the same way with decimals.
Example:
18 / 3.6 -1.4 +6.8 = 10.4
3.4 + 2.8 * 4.6 - 8.1 = 8.18

Percent:

Percent: Percent means per 100.
1% = 1/100  which means p% = p/100

Examples:

65% = 65/100 = .65 = 13/20
40% = 40/100 = .4 = 2/5

Always reduce to lowest term

Coverting between Percents, Decimals and Fractions:
To convert a percent to a decimal:
Move the decimal point two places to the left and remove the % sign.  This is same as multiplying by 1% = 1/100 = 0.01

To convert a decimal to a percent:
Move the decimal point two places to the right, and add the % sign.  This is the same as multiplying by 100.


R.4 The Real Number System

Rational Numbers:

Rational Numbers:  The rational numbers consist of the set of numbers that are ratios of integers.  This category includes all of the integers, finite decimals, and infinite repeating decimals.

Below are examples of rational numbers:

  1.  0.25 = 25/100 = 1/4
  2. 1/3 = 0.33333
  3. 5 = 5/1
  4. The mothly rent for an apartment.  (Integer - rational)
  5. The balance in a checking account. (decimals -  rational)
  6. The average monthly charge for a cell phone (decimal - rational)


Irrational Numbers:

Irrational numbers can be expressed by infinite,non-repeating decimals.  These numbers cannot be written as a ratio of integers.
⫪ is irrational number
√2 is irrational number
√5 is irrational number

Real Number System:

Real Number System: The real number system consists of the set of all rational and irrational numbers.

Classify numbers:

-20 - Integer, Rational
0.58 - Decimal, Rational
3/7 - Rational
√7 - Irrational

Exact and Approximate Answers:


⫪ = 3.1415926535…  continues indefinitely.
~3.14 is an approximation
⅓ = 0.33333… repeats indefinitely.
~0.3 is an approximation

1.1 Exponents, Order of Operations, and Properties of Real Number

Evaluate Exponents:

Natural Number Exponent:  If a is a real number and n is the natural number, then
an = a * a * a * …  n times (a is multiplied n times)
a is called the base and n is called the exponent.

2 * 2 * 2 * 2 is expanded form
24 is in exponential form

Note: Exponents are sometimes referred as powers.

Zero as an Exponent:

Zero as an exponent a0 = 1 for all a ≠ 0

Example:
30 = 1
(-41)0  = 1
(23/12)0  = 1

0does not exist or is said to be undefined

Practice Problems:

Write in exponential form:
9 * 9 * 9 * 9 * 9 * 9 * 9    =  97
(-3) * (-3) * (-3) * (-3) * (-3)  = (-3)5
(4/7) *   (4/7) *  (4/7) *  (4/7)  = (4/7)4

Write in expanded form:
53  = 5 * 5 * 5
(-7)8 = (-7) * (-7) * (-7) * (-7) * (-7) * (-7) * (-7) * (-7)

Calculate value using calculator:
37 = 2187
(⅙)4  = (1/1296)
(1.42)3 = 2.863288

Scientific Notation:

Scientific Notation:  A number in the form a * 10n where 1 ≤ |a| < 10 and n is an integer is said to be in scientific notation.
When a number is so large that it cannot be displayed all at once on the calculator screen; th calculator displays it in scientific notation.

Try 813 = 5.437558139E11 or 5.43755813911  (exponent on 10)

This is same as 5.437558139 * 1011 = 543755813900
543755813900 is the standard form

Convert from Scientific Notation to Standard Form:

The exponent on 10 is positive n:  Multiply by 10n, which is equivalent to moving the decimal point n places to the right.

Example:  
4.53 * 105  = 4.53 * 100000 = 453000
-2.748 * 104 = -2.748 * 10000 = 27480


Order of Operations:

PEMDAS (L to R)
Evaluate everything inside the grouping symbols first, inside to outside working from left to right
Evaluate all exponents and divisions working from left to right
Do all multiplications and divisions in the order they occur, working from left to right
Do all additions and subtractions in the order they occur, working from left to right

Examples:
8/4 - 16 + 9 * 3 - 52
= 2 - 16 + 27 - 25
= -12

Commutative Property:

If a and b are real numbers, then the commutative properties are as follow:

Commutative property of addition is a + b = b + a  
eg: 8+5 = 5+8

Commutative property of multiplication is a * b = b * a  
eg: 3*5 = 5*3

Commutative word comes from french word means “to switch or substitute”

Associative Property:

If a, b & c are any real numbers, then the associative properties are as follows:

Associative property of addition is a + (b + c) = (a +b) + c
    eg: 2 + ( 5 + 8) = (2 + 5 ) + 8

Associative property of multiplication is a * (b * c) = (a * b) * c
    eg: -2 * ( 3 * 8) = (-2 * 3) * 8

Distributive Property:

If a, b, c are real numbers, then the distributive property states:

a(b + c) = a * b + a * c
a(b - c) = a * b - a * c

Example:
5 * (6 + 7) = 5 * 6 + 5 * 7
5 * ( 6 - 7 ) = 5* 6 - 5 * 7​

1.2 Algebra and Working with Variables

Constant, Variable, Coefficient:

Variable:  A variable is a symbol, usually a letrer, used to represent a quantity

Note: O is not a recommended letter to represent variable as it can be easily confused with number 0

Constant: A constant is a value that does not change

Coefficient:  The coefficient of a variable term is the numercial factor of the term

Example: 4x has 4 as coefficient

Problem:

Stephanie earns $9.25 per hour working as a tutor.  Her weekly pay can be computed as 9.25h, where h is the number of hours she worked in a week.
If Stephanie works 12 hours in a week, what is her weekly pay?
If stephanie works  26 hours in a week, what is her weekly pay?

Evaluating Expressions:

Substitute the given value of variable into the expression
Simplify using the order of operations agreement (PEMDAS - L to R)

Example:  
12r - 16 for r = 1/2
8(n + 3) - 15/ (n + 2) for n = 3


Unit Conversion:

Unity Fraction: A unity fraction is a fraction that has units in the numerator and deniminator and whose simplified values is equal to 1. Also known as conversion factors.

Example:
100cms  = 1 m,
100cm/1m is a unity fraction
Find the unity fraction. 

Steps to convert units:

Always place unit that are wanted in final answer in the numerator
Place the other unit (to be converted)  in the denominator

Multiply the given value by unity fraction.  Divide out the common units

Reduce to the lowest terms

Examples:
Convert 254 feet to meters (1m ≈ 3.28 feet)
254 ft * 1m/3.28ft = 77.44m

Convert 15 gallons to liters (1 gal = 3.785 l)
15 gal*3.785 l/1 gal = 56.775 l

Defining Variables:
Write a sentence that explains what the variable represents

Weekly pay check for an hourly paid employee
a = amount of pay in dollars
h = Number of hours worked in a week/month

Time you drive from San Diego to Los Angeles  depends on the speed
t = time to drive from SD to LA in hours
s = speed of the car in mph
d = distance (miles)  

Practice Problems:
The amount of money (in dollars) a family spends  a week on gas depends on the number of miles they drive that week.
a = amount of money in dollars
n = number of miles driven

The temperature outdoors (in degrees Fahrenheit) depends on the time of the day
F = temperature in Fahrenheit
t = time of the day

Translating Phrases into Algebraic Expressions:

Quotient of a number and 12  → n/12
The product of 7 and sum of a number and 12 → 7(n+12)
The sum of 7 divided by a number and 2 → 7/x + 3
The difference between -3 times a number and 8 → -3x-8
Five times a number divided by 12 → 5x/12


Generating Expressions from Tables:

Many times a input-output table is used to generate an expression

Input is independent variable - data to be entered in the table.  The independent variable is not changed or controlled by another variable.

Output is dependent variable - results based on input.  The dependent variable tends to change by the independent variable.

1.3 Simplifying Expressions

Like Terms:

Term:  A constant (a number) or a product of a constant and a variable (or variables) is called a term.

Example: -4a,  ⅔ km2,  x,  3

Like Terms:  Two or more terms that have the same variable part, where the variables are raised to the same exponents, are called like terms.

Example: 3x and 5x
-25a2bc3 and 10a2bc3

Practice Problems:

Identify the number of terms and the like terms in each expression

2x + 8 + 7x
3 terms; like terms: 2x & 7x

7xy + 8x - 2xy - 3y
4 terms; like terms: 7xy & 2xy

t5 + 4t3 + 3t5 - t3
4 terms; like terms: t5 & 3t5 , 4t3 & t3

Addition & Subtraction Properties:

Two or more terms can be added or subtracted only if they are like terms.  Add or subtract the coefficients of the like terms.  The variable part will remin the same.

5x +7x = 12x
-25a2bc3 + 10a2bc3 = -15a2bc3

Practice Problems:
7y + 20y = 27y
a - 4b - 6a + 5b = -5a + b
-3b + 11 - 7b -9 = -10b + 2
5y2 - 3y - 2 + 8y2 - 7y - 9 = 13y2 - 10y - 11

Multiplication & Distributive Properties:

When multiplying  a single term by a constant, multiply the coefficient by the constant, and the variable parts stay the same
5(4xy) = 20xy

When multiplying a multiple term expression by a constant, multiply the coefficient of each term by the constant, and the variable parts will stay the same
3(2x-7y) = 6x - 21y

Practice Problems:

6(9w3) = 54w3
7(3g5 + 5g3 + 2g - 8) = 21g5 + 35g3 + 14g - 56
-3(12st + 5t2 - 9s) = -36st -15t2 + 27s

Simplifying Expressions:

Simplify following expressions

4(2c + 5) - 3c + 11  = 5c + 31
5m + 7(3n - 4) - (6m + 2n -18)  = -m + 19n -10
⅓(6b - 15a) - ¼(16a - 8b)  = -9a + 4b
2.1(3b + 5) + 6.1b +10  = 12.4b + 20.5

1.4 Graphs and the Rectangular Coordinate System

Reading a Table:

The table gives the data on the number of days on which ozone levels exceeded the EPA’s health based standard



                                                                What is the maximum number of days on which the ozone levels exceeded the EPA’s health based                                                                                         standard?  When did this occur?
                                                                 
                                                                In what year were there 5 days on which ozone levels exceeded the EPA’s health based standard?















Bar Graphs:

Data are easier to examine if they are visually presented in a graph.
Horizontal axis is called x-axis.  This is where input values are plotted.
Vertical axis is called y-axis.  This is where output values are plotted.

Example of a bar graph:
Height of each bar
represents the output value
for corresponding input vlaue


Reading Bar Graphs:



















For which month did 6 students have their birthday?
For which months less than 2 students have their birthdays?
How many students had their birthdays in December?

Creating Bar Graph:

  1. The input values will be plotted on x-axis and output values on y-axis
  2. Evenly and consistently space the x-axis so that it includes the input values (data)
  3. Evenly and consistently space the y-axis so that it includes the output values (data)
  4. Draw the vertical bars of uniform width.  Center each bar over the appropriate input value.  Draw the bars of height corresponding to the output value.





2  Linear Equations & Inequalities with One Variable

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3  Linear Equations with Two Variables

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8  Modeling Data And Functions

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4  Systems of Linear Equations

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5  Exponents and Polynomials

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6  Factoring and Quadratic Equations

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7  Factoring and Quadratic Equations

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10  Square Roots & Radicals

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11 Quadratic Functions

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