Download Chapter 1

Introduction and Chapter 1
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Introduction – part 1
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Course Syllabus (Reminder: prepare a folder for homework)
Prepare a 3-ring binder with dividers for class notes, homework,
quizzes, tests, or related information
Homework
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Check the answers for odd numbers on the back of the book, show your
work. No credit will be awarded if no work is shown.
Use 3 holes line papers with your name on the right upper corner with
your ID; write down the homework # and assignments.
Place your homework on the homework folder and is due Monday,
unless the test in on Wednesday (then it is due Wednesday).
A Scientific calculator is required.
Homework log – for your own record
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Write the date next to the homework number and homework log should be in
your homework folder with your homework
Failure to comply with instructions, will result point deduction
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Introduction – part 2
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Lecture notes will be on line, please print it out before the class
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Lecture notes may only contain the summary of lectures, examples will be
shown on the whiteboard, thus taking notes is essential part of the learning.
Beginning each class , there may be a warm-up
At the end of each class, there may be a quiz or an exit slip which contains
in-class work
Quizzes (counted as your grade) may be given randomly unannounced,
may contains vocabularies, homework problems.
Familiar with syllabus, and keep track your total score on your own.
Exit slip today: write down the following on the index card and turn it in
before you leave the class
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Your name, email address, phone number
What was your last math subject? When did you take it?
What is your career goal?
How may I as instructor help you to master this subject?
How are you as student going to help yourself to master this subject?
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Chapter 1 The Real Numbers and Geometry – Review (not to
cover in class, except section 1.7)
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Chapter 1 Objectives:
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Review of fractions, exponents and order of operations, geometry review, sets of numbers
and absolute value, addition and subtraction of real numbers, multiplication and division
of real numbers, algebraic expressions and properties of real numbers.
Definitions:
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Set – a collection of objects; notation: { }, e.g. A = {dog, cat, dolphin, panda} is a set.
Elements – the objects of a set, e.g. dog is an element of the set A
Natural numbers – the set of counting numbers, sometimes we call it counting numbers.
Natural numbers = {1, 2, 3, 4, ….}
Whole numbers – a set of natural numbers and zero, thus whole number s = {0, 1, 2,
……}.
Prime number - A natural number that is greater than 1 that is divisible (evenly) only by
itself and 1, e.g. 2, 3, 5, 7, 11 and 13 are the first six prime numbers.
Composite number – A natural number that is greater than 1 and is not a prime number is
a composite number, e.g. the numbers 4, 6, 8, 9 are the first four composite numbers. Any
composite number can be expressed as a product of its prime factor. The process of
finding its prime factors is called prime factorization of the number. Use factor tree to
find factorization.
Note that 1 is neither a prime nor a composite number.
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Review of Fractions (1.1 – 2)
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Integers – a set of positive integers and negative integers, i.e. integers =
{…, 5, 4, 3, 2, 1, 0, 1, 2, 3, 4, 5, ….}. An integer is either positive
or negative and zero is neither a positive nor a negative integer.
Rational numbers – is a set of numbers that can be expressed as a fraction
a
(or ratio of two integers), i.e. rational numbers = { b , where a and b are
integers and b  0; a is called the numerator, and b is called
denominator}. E.g. 2/3, 9/2, 5/1. Note that all integers n are rational
numbers, since n =n/1.
Note: a fraction describes a part of a whole quantity, and ba means a 
b
A fraction is in lowest terms when the numerator and denominator have
no common factors except 1.
Always reduce the fraction to the lowest terms.
An improper if its numerator is greater than or equal to its denominator,
and which can be expressed as a mixed number.
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Review of Fractions(1.1 – 3): operations
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a c
ac
Multiplication: b  d  b  d and b  0, d  0
Division: a  c  a  d and b  0, c  0, d  0
b d b c
Addition: with the same (or common) denominator ac  bc  a c b and c  0
a b a b
Subtraction: with the same (or common) denominator c  c  c and c  0
Addition or subtraction of mixed numbers with the same (or common)
denominator
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Add or subtract the integers, then add the fractions, or
 Convert to improper fractions, then add or subtract
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LCD: the least common denominator
Addition or subtraction of unlike denominator: rewrite the fractions with
same LCD, then do the operations (addition or subtraction), express answe
in lowest term
Hint: always reduce the fraction to the lowest term before apply any
operations.
Hw01: Pp13 – 16, #1, 5, 9, 11, 13, 17, 21, 25(a, c, g, i), 31, 35
Exponents and Order of Operations(1.2 – 1)
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Exponential expressions – is used to represent repeated multiplication of same
factor; the exponent indicate how many times the factor (base) is multiplied.
Example of the exponential expressions is 2 6 which means 2 is the base, and 6 is the
exponent and multiply 2 six times.
nth power of a is denoted as a n = aaaa, with n times where a is a real number
and n is a positive integer. E.g. evaluate (3)4 and 34
Operations of exponents – (1) rewrite the product to exponential form; (2) evaluate
the expressions
The order of operations agreement: Remember this “Please Excuse My Dear Aunt
Sally”
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Perform operations inside grouping symbols: ( ), [ ], the absolute symbol | |. And fraction bar.
Simplify exponential expressions
Multiplication and division as they occur from left to right
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Do addition and subtraction as they occur from left to right
2
2
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Examples: (a)
8
12  2 2
2
4 1
(b) 5[2 (3 + 7)4]
HW02: P21, #1, 5, 11, 15, 19, 23, 27, 31
3 9
3
5
(c) 18  2
(d)
9 6
3
8
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Review of Geometry(1.3 – 1)
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Identify angles A (with measurement of angle A, denoted by mA)
Acute angle – angle is greater than 0 and less than 90, 0 < mA <90
 Right angle – angle is equal to 90, mA =90
 Obtuse angle – angle is greater than 90 and less than 180, 90 < mA <180
 Straight angle – angle is equal to 180, mA =180
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A and B are complement Angles, if mA+ mB = 90
 A and B are supplement Angles, if mA+ mB = 180
 Vertical angles A and B are the pair of opposite angles when two lines intersect and
mA= mB
 Parallel lines are lines in the same plane that do not intersect.
 Perpendicular lines are lines that intersect at right angles.
 Identify triangles (by their angles or by their sides)
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Acute triangle – a triangle with all three angles are acute
Obtuse triangle – a triangle contains one obtuse angle
Right triangle – a triangle contains one right angle
Equilateral triangle – a triangle has three sides of equal length
Isosceles triangle – a triangle has two sides of equal length
Scalene triangle – a triangle has no sides of equal length
The sum of the measures of the angles of any triangle is 180
Review of Geometry(1.3 – 2) Formulas
Figure
Perimeter
Area
Rectangle (length l, width w)
P= 2 l + 2 w
A= lw
Square (side s)
P = 4s
A = s2
Triangle (height h, side a, b, c)
P= a + b + c
A = (1/2) b h
Parallelogram (height h, side a, b)
P = 2 a + 2b
A= b h
Trapezoid (height h, side a, c, b1, b2)
P = a + c + b1 + b2
A = (1/2) (b1 + b2) h
Circle (radius r )
p = 2r
A=  r2
Figure
Volume
Rectangle Solid (length l, width w, height h)
V= lwh
Cube (side s)
V = s2
Right Circular cylinder (height h, radius r)
V =  r2 h
Sphere (radius r)
V = (4/3)  r3
Right Circular cone (height h, radius r)
V = (1/3)  r 2 h
HW 03: Pp 30 – 34, #1 – 6, 7 – 51 every other odd, 55 – 61 odd.
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Set of numbers and absolute value (1.4 – 1)
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The set of integers includes the set of natural numbers, their negatives, and
zero and is denoted by { …. –3, –2, –1, 0, 1, 2, 3, …}
To graph a number on a number line is to locate the number on a number
line.
Identify the whole numbers, natural numbers and integers
p
A rational number is any number of the form q , where p and q are integers
and q  0
Note that the set of rational numbers includes: integers, whole numbers,
natural numbers, repeating decimals, terminating decimals, fractions and
mixed numbers. The set of numbers that cannot be written as the quotient
of two integers is called the set of irrational numbers. A decimal number
with non-repeating, non-terminating decimal is an irrational number.
The set of real numbers consists of the rational and irrational numbers.
Set of numbers and absolute value (1.4 – 2)
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Inequality symbols review
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During the recession, the number of employees at manufacturing company decreased by 850.
From October 2008 to March 2009, the number of Facebook users increased by over 23,000,000
After getting off the highway, Huda decreased his car’s speed by 25mph.
Additive inverse: Two numbers are additive inverses if they are the same distance
from 0 on the number line but on the opposite side of 0. Therefore, if a is any real
number, then a is its additive inverse. Examples: Find (5)
If a is any real number, then the absolute value of a, denoted by | a |, is
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 less than or equal to (compound symbol of “<“ and “=“)
 greater than or equal to (compound symbol of “>” and “=“)
 approximately equal to
Remember on the number line: as we move to the left, the numbers get smaller; as
we move to the right, the number s get larger
Use signed symbol to represent the change:
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< less than
> greater than
 not equal to
a if a  0
a if a < 0
In other words, | a | is the distance from 0, | a | is never negative (or less than 0)
HW 04, Pp41 – 42, #3, 5 – 10, 11 – 35 EOO, 45 – 49 odd
Addition and subtraction of real numbers (1.5)
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Adding integers using a number line
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Adding real numbers with the same sign – find the absolute value of each number and
add them. The sum will have the same sign.
Adding real numbers with different sign – subtract the smaller absolute value from the
larger, the sum will have the sign with larger number with larger absolute value.
a + (a) = 0 for any real number. (0 is additive identity)
a – b = a + (–b)
Applying the order of operations to real numbers – same as integers
Examples:
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15 – (21 + 5);
to
sum, more than, increased by
Difference between, less than, decreased by
math expression:
addition
subtraction
Examples:
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Simplify | 7 – 14|  | 4 – 2|;
English expression
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Adding positive number a, move a units to the right
Adding negative number b, move b units to the left (remember, subtraction is the same as adding negative
numbers)
9 more than 2 ;
27 less then 15 ;
14 increased by 6 ;
the sum of 27 and 7 decreased by 5
HW 05, Pp48 – 49, #5 –49EOO, 51 – 63 EOO
8 less than the sum of 11 and 3
Multiplication and division of real numbers (1.6)
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Multiplication of a number x is the repeated addition of the number x
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Simplify 24| 12 – 22;
5(–3) 4(2 – 3);
Times, product of
Divided by, quotient of
to
math expression:
multiplication
diivision
Examples:
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The quotient of two positive numbers is a positive number
The quotient of two negative numbers is a positive number
The quotient of a positive and a negative number is a negative number
English expression
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of two positive numbers is positive
of two negative numbers is positive
of a positive number and a negative number is negative
of any real number and zero is zero
Applying the order of operations to real numbers – same as integers
Examples:
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product
product
product
product
Evaluate exponential expression (note a negative base number raised to an odd power is
a negative number, and with an even power is a positive number)
Divide real numbers (Note: “sign” rule is the same as multiplication)
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The
The
The
The
The quotient of 84 and 7 ; The product of 5 and the sum of 15 and 6 ; half of the sum of 12 and 5
Twice the sum of 11 and 20 ;
the sum of 23 and 7, divided by the square of 3
HW 06, Pp56 – 57, #5 –53EOO, 55 – 69 odd
Algebraic Expressions and Properties of Real Numbers (1.7 – 1)
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Properties of the real numbers
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Commutative property of
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Associative property of
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Addition a + b = b + a
Multiplication a  b = b  a
Addition (a + b) + c = a + (b + c)
Multiplication (a  b)  c = a  (b  c)
Addition property of zero
a +0 = 0 + a
Multiplication property of one
a  1= 1  a
Inverse property of addition a + (a) = (a) + a = 0
Inverse property of multiplication a  (1/a) = (1/a) a = 1, a  0
Distributive property a (b + c) = a b + a c
An algebraic expression is a collection of numbers, variables, and grouping symbols connected by
operation symbols such as +, , , .
A coefficient a is a real number that multiply the variable, e.g. 3 is a coefficient for 3x2y. (Note that if
you substitute x and y with values, then 3 is a multiplication factor.)
Evaluate variable expressions – a variable expression is a mathematical expression with a variable x,
y, z, or others in it. E.g. 3x2y + 2z – x + 4 is a variable expression.
To evaluate an expression is to replace the variables with values, e.g. evaluate 3 – 2|3x -2y2| , when x
= 1 and y = 2
Example 1: The radius of the base of a cylinder is 3 in. and the height is 6 in. Find the volume of the
cylinder (round to the nearest hundredth) (A. 169.95 in3)
Example 2: Find the surface area of a right circular cone with a radius of 5cm and a slant height of
12 cm. Give both the exact area and an approximation to the nearest hundredth. (A. 85 cm2, or
267.04 cm2)
Variable Expressions (1.7 – 2)
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Simplify variable expressions – use combine the like terms.
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Examples
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Like terms – the terms with the same variable part. E.g. 2x and ax are like terms, 3xy and 5xy are
like terms.
Combine like terms is to add the coefficient of the like terms by using the Distributive Property.
Simplify
Simplify
Simplify
Simplify
2(x + y) +3(y – 3x)
(2x +xy – y) – (5x – 7xy + y)
4y – 2[ x – 3(x + y) – 5y
2x – 3[y – 3(x – 2y +4)]
Homework 1: Pp 69 – 70, #1 – 89 EOO.
Extra Examples: Variable Expressions and variable expressions
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Translate a verbal expression into a variable expression – use the following
keywords
Operation
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Key word
Example
Expression
Addition
More than, added to, the sum of ,
the total of, increased by
8 more than w, x is
added to 9, the sum of z
and 9, the total of r and
s, x increased by 7
w + 8, 9 + x, z + 9, r + x,
x+7
Subtraction
Less than, the difference between,
minus, decreased by
12 less than b, the
difference between x
and 1, z minus 7, 17
decreased by a
b – 12, x – 1, z – 7,
12 – a
Multiplication
Times, the product of, multiplied
by, of, twice
Negative 2 times c, the
product of x and y, 3
multiplied by n, threefourth s of m, twice d
– 2c, xy, 3n, (3/4)m, 2d
Division
Divided by, the quotient of, ratio
v divided by 15, the
quotient of y and 3, the
ratio of x to 7
v/15, y/3, x/7
Power
The square of or the second power
of, the cube of or the third power
of, the fifth power of
The square of x, the
cube of r, the fifth
power of a
x2 , r3 , a5
Variable Expressions and variable expressions
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Example 1: Translate and simplify “the total of five times a number and twice the difference
between the number and three”
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Example 2: Translate and simplify “a number decreased by the difference between eight and
twice the number”
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Example 3: Translate and simplify “fifteen minus one-half the sum of a number and ten”
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Example 4: Translate and simplify “the sum of three-eights of a number and five-twelfths of
the number”
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Example 5: A cyclist is riding at a rate that is twice the speed of a runner: Express the speed of
the cyclist in terms of the speed of the runner
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Example 6: A mixture of candy contains 3 lb more of milk chocolate than of caramel. Express
the amount of milk chocolate in the mixture in terms of the amount of caramel in the mixture.
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Example 7: The length of a rectangle is 2 ft more than 3 times the width. Express the length of
the rectangle in terms of the width.