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COLLEGE ALGEBRA MT 221

COLLEGE ALGEBRA MT 221. Jeaneth Balaba Lecturer/Instructor. ISHRM, 1ST SEMESTER, 2014. Good morning!. Self-introduction ISHRM Vision and Mission Classroom policy Grading system Other clarifications Trivia information Lesson introduction Lesson proper. CLASSROOM POLICY.

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COLLEGE ALGEBRA MT 221

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  1. COLLEGE ALGEBRAMT 221 Jeaneth Balaba Lecturer/Instructor ISHRM, 1ST SEMESTER, 2014

  2. Good morning! • Self-introduction • ISHRM Vision and Mission • Classroom policy • Grading system • Other clarifications • Trivia information • Lesson introduction • Lesson proper

  3. CLASSROOM POLICY • Attendance & Punctuality • Classroom Behavior & Language • Grading System • Attendance 10% • Quiz 15% • Activity 15% • Exams 60% [(100%/100%/30%/70%)] • Overall Rating: Prelim Rating 100%/ Mid-Term Rating 100%/ Pre-final Rating 30% & Final Period Rating 70% • Personal Profile (Index card) – Quiz #1

  4. INDEX CARD Name: 1x1 Photo Subject and Section: Course and Year: Age/Birthday: Home Location: E-mail address: Course expectation: (1-2 sentences) *Note: Leave the back portion of the index card blank*

  5. Making the connection… Brainstorm on what algebra and algebraic thinking is…

  6. DISCUSSION • Review: What is algebra? • How useful is algebra? What are its uses? • What are its uses in the culinary arts practice? Hotel and restaurant management? Business administration?

  7. Algebra – what is it? • Fundamental language of mathematics • Creates a mathematical model of a situation • Provides mathematical structure to use the model • Links numerical and graphical representation • Condenses large amounts of data into efficient statements

  8. Algebraic Habits of Mind • Analyze change • Understand functions • Variable (understand the idea and the variety of uses) • Interpret, create and move fluently between multiple representations for data

  9. Trivia: What is Algebra ? Mathematics dictionary: “The branch of mathematics that deals with the general properties of numbers and the generalizations arising therefrom. Make sense ?!? Eulers' Complete Introduction to Algebra (1767): “Algebra has been defined as the science which teaches how to determine unknown quantities by means of those that are known.”

  10. Trivia: The History of Algebra * Algebra has a long history even before Leonhard Euler (1707 – 1783) and Colin Maclaurin (1698-1746). * The use of algebra predates the use of symbols which we often use for manipulation. * Oldest text: by Muhammad ibn Musa al-Khwarizmi (c. 780- 850), ninth century Baghdad, The Condensed Book on the Calculation of al-Jabr and al-Muqabala - translation: “One of the branches of knowledge needed in that division of philosophy known as mathematics is the science of al-jabr and al-muqabala which aims at the determination of numerical and geometric unknowns.” “Algebra” is derived from “al-jabr” - about solving equations from first use of the word – but use is even older.... Stamp issued by Soviet Union, 1983 for 1200th birthday.

  11. Using Al-jabr Al-jabr: the transposition of a subtracted quantity from one side to the other side by adding it to both sides. 3x + 2 = 4 - 2x 5x + 2 = 4 Al-muqabala: subtraction of equal amounts from both sides to reduce positive term. “Do to both sides equally” 5x + 2 = 4 5x = 2

  12. How was Algebra Written ? Algebraic operations were first performed before the use of symbols. Al-Khwarizmi's text books were written as step-by-step instructions, in words! Describing how to solve each problem. This form of rhetorical algebra are known as algorithms (Algortmi dixit means “al-Khwarizmi says”) Suppose the temperature is 20oC.. What is the temperature in degrees Fahrenheit ? The solution is this: You take the 20. This you multiply by nine; the product is 180. Divide this by five; the result is 36. To this you add 32; the sum is 68. This is the temperature in degrees Fahrenheit that you sought.

  13. Tools for Equation Manipulation 1. You can add, multiply, divide, double, halve, subtract, or perform an other operation you like, provided that you do exactly the same to both sides of an equation. 2. You can always replace an expression by any other expression which is equal to it

  14. LESSON 1 OUTLINE • The Real Number System • Properties of Real Numbers

  15. I. The Real Number System

  16. Real Numbers • Real numbers consist of all the rational and irrational numbers. • The real number system has many subsets: • Natural Numbers • Whole Numbers • Integers

  17. Natural Numbers • Natural numbers are the set of counting numbers. {1, 2, 3,…}

  18. Whole Numbers • Whole numbers are the set of numbers that include 0 plus the set of natural numbers. {0, 1, 2, 3, 4, 5,…}

  19. Integers • Integers are the set of whole numbers and their opposites. {…,-3, -2, -1, 0, 1, 2, 3,…}

  20. Rational Numbers • Rational numbers are any numbers that can be expressed in the form of , where a and b are integers, and b ≠ 0. • They can always be expressed by using terminating decimals or repeating decimals.

  21. Terminating Decimals • Terminating decimals are decimals that contain a finite number of digits. • Examples: • 36.8 • 0.125 • 4.5

  22. Repeating Decimals • Repeating decimals are decimals that contain a infinite number of digits. • Examples: • 0.333… • 7.689689… FYI…The line above the decimals indicate that number repeats.

  23. Irrational Numbers • Irrational numbers are any numbers that cannot be expressed as . • They are expressed as non-terminating, non-repeating decimals; decimals that go on forever without repeating a pattern. • Examples of irrational numbers: • 0.34334333433334… • 45.86745893… • (pi)

  24. Other Vocabulary Associated with the Real Number System • …(ellipsis)—continues without end • { } (set)—a collection of objects or numbers. Sets are notated by using braces { }. • Finite—having bounds; limited • Infinite—having no boundaries or limits • Venn diagram—a diagram consisting of circles or squares to show relationships of a set of data.

  25. Venn Diagram of the Real Number System Rational Numbers Irrational Numbers

  26. Example • Classify all the following numbers as natural, whole, integer, rational, or irrational. List all that apply. • 117 • 0 • -12.64039… • -½ • 6.36 • -3

  27. To show how these number are classified, use the Venn diagram. Place the number where it belongs on the Venn diagram. Rational Numbers Irrational Numbers Integers 6.36 Whole Numbers -12.64039… Natural Numbers -3 0 117

  28. Solution • Now that all the numbers are placed where they belong in the Venn diagram, you can classify each number: • 117 is a natural number, a whole number, an integer, and a rational number. • is a rational number. • 0 is a whole number, an integer, and a rational number. • -12.64039… is an irrational number. • -3 is an integer and a rational number. • 6.36 is a rational number. • is an irrational number. • is a rational number.

  29. FYI…For Your Information • When taking the square root of any number that is not a perfect square, the resulting decimal will be non-terminating and non-repeating. Therefore, those numbers are always irrational.

  30. II. Properties of Real Numbers Commutative Associative Distributive Identity + × Inverse + ×

  31. Commutative Properties • Changing the order of the numbers in addition or multiplication will not change the result. • Commutative Property of Addition states: 2 + 3 = 3 + 2 or a + b = b + a. • Commutative Property of Multiplication states: 4 • 5 = 5 • 4 or ab = ba.

  32. Associative Properties • Changing the grouping of the numbers in addition or multiplication will not change the result. • Associative Property of Addition states: 3 + (4 + 5)= (3 + 4)+ 5 or a + (b + c)= (a + b)+ c • Associative Property of Multiplication states: (2 • 3) • 4 = 2 • (3 • 4) or (ab)c = a(bc)

  33. Distributive Property • Multiplication distributes over addition.

  34. Additive Identity Property • There exists a unique number 0 such that zero preserves identities under addition. a + 0 = a and 0 + a = a • In other words adding zero to a number does not change its value.

  35. Multiplicative Identity Property • There exists a unique number 1 such that the number 1 preserves identities under multiplication. a ∙ 1 = a and 1 ∙ a = a • In other words multiplying a number by 1 does not change the value of the number.

  36. Additive Inverse Property • For each real number a there exists a unique real number –a such that their sum is zero. a + (-a) = 0 • In other words opposites add to zero.

  37. Multiplicative Inverse Property • For each real number a there exists a unique real number such that their product is 1.

  38. Let’s play “Name that property!”

  39. State the property or properties that justify the following. 3 + 2 = 2 + 3 Commutative Property

  40. State the property or properties that justify the following. 10(1/10) = 1 Multiplicative Inverse Property

  41. State the property or properties that justify the following. 3(x – 10) = 3x – 30 Distributive Property

  42. State the property or properties that justify the following. 3 + (4 + 5) = (3 + 4) + 5 Associative Property

  43. State the property or properties that justify the following. (5 + 2) + 9 = (2 + 5) + 9 Commutative Property

  44. 2. Which Property? 3 + 7 = 7 + 3 Commutative Property of Addition

  45. 3. Which Property? 8 + 0 = 8 Identity Property of Addition

  46. 5. Which Property? 6 • 4 = 4 • 6 Commutative Property of Multiplication

  47. 11. Which Property? 5 • 1 = 5 Identity Property of Multiplication

  48. 25. Which Property? 51/7 + 0 = 51/7 Identity Property of Addition

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