Week 1_ GEC MATH_BSN (1).pdf

MATHEMATICS IN THE MODERN WORLD Instructor: Prince Steven Arvie C. Vitug, RME  Classroom Rules 1. Don’t be late. If you are late after 30 minutes. You will be considered ABSENT. Late and Absent Students will not have a recorded Pre-test.  Classroom Rules 2. Everyone deserves re...

MATHEMATICS IN THE MODERN WORLD Instructor: Prince Steven Arvie C. Vitug, RME  Classroom Rules 1. Don’t be late. If you are late after 30 minutes. You will be considered ABSENT. Late and Absent Students will not have a recorded Pre-test.  Classroom Rules 2. Everyone deserves respect, both students and teachers.  Classroom Rules 3. When someone else is talking, listen.  Classroom Rules 4. Raise your hand to talk.  Classroom Rules 5. Make sure you have everything you need before you enter the class. Requirements: a. Calculator (any type,except phone) b. ¼ and 1 whole Yellow Paper for Seatwork c. Short Bond Paper for Weekly Assessment d. Notebook specified for GEC MATH e. Black and Red Ballpen  Classroom Rules 6. Mobile phone use is strictly prohibited during class. Failure to comply will lead to confiscation. (Unless otherwise I allowed you to use your phone)  Classroom Rules 7. You can leave the classroom without asking the professor for permission if it is toilet break to avoid disrupting the class. But if other matters, make sure to ask the professor first. When leaving the classroom, one at a time only.  Classroom Rules 8. You are not allowed to eat and drink (except water) inside the classroom.  Classroom Rules 9. Don’t litter inside the classroom. Maintain the cleanliness of our classroom at all times.  Classroom Rules 10. Participate in all activities. Work hard and do your best!  Classroom Rules 11. When leaving after class, make sure to bring all your items and pick up any trash. YOU ARE AIMING TO BE PROFESSIONAL, THEN ACT LIKE ONE  Submission Rules 1. Seatwork will be submitted during class. No seatwork will be submitted late. Unless you have approved excuse slip presented. 2. Weekly Assessment will be submitted before the deadline. The deadline will depend upon the professor decision. After the deadline, you are not allowed to pass the weekly assessment. Unless, you have excuse slip presented.  Submission Rules 3. Quiz will be submitted during class. No Quizzes will be submitted late. Special Quiz is allowed only when you have approved excuse slip presented. Special Quiz will be conducted during CHAT (Consultation Hours and Tutorial) Course Description This course deals with nature of mathematics, appreciation of its practical, intellectual, and aesthetic dimensions, and application of mathematical tools in daily life. Course Description The course begins with an introduction to the nature of mathematics as an exploration of patterns (in nature and in the environment) and as an application of inductive and deductive reasoning. By exploring these topics, students are encouraged to go beyond the typical understanding of mathematics as merely a set of formulas but a source of aesthetic in patterns of nature, for example, and a rich language in itself (and of science) governed by logic and reasoning. Course Description The course then proceeds to survey ways in which mathematics provides a tool for understanding and dealing with various aspects of present-day living, such as managing personal finances, making social choices, appreciating geometric designs, understanding codes used in data transmission and security, and dividing limited sources fairly. These aspects will provide opportunities for actually doing mathematics in a broad range of exercises that bring out the various dimensions of mathematics as a way of knowing, and test the students’ understanding and capacity. Course Outline Nature of Mathematics Language of Mathematics Problem Solving and Reasoning Data Management Geometric Designs, Arts and Culture Codes Mathematics of Finance Logic Graph Theory Course Requirement Pretest Seatwork Weekly Assessment Assignment Quiz Term Exam Term output Course Requirement TERM OUTPUT PRELIM - COMPILATION OF MEASUREMENTS, FORMULAS AND CONVERSIONS MIDTERM - COMPILATION OF MEASUREMENTS, FORMULAS AND CONVERSIONS FINALS - COMPILATION OF MEASUREMENTS, FORMULAS AND CONVERSIONS Grading System  Week 1: Mathematics in our World Objectives: 1. To understand the mathematics of the modern world. 2. To gain awareness of the role of mathematics as well as our role in mathematics. 3. To develop one's understanding about patterns. 4. to explain the presence of Fibonacci numbers in nature. INTRODUCTION  STRATEGY R - epetition E - asy P - leasure O - ften H - abit  STRATEGY Success comes by REPEATING what you do daily.  STRATEGY Success comes by REPEATING what you do daily. Eventually, it will become EASY for you.  STRATEGY Success comes by REPEATING what you do daily. Eventually, it will become EASY for you. When these things became easy, we will have PLEASURE in doing them.  STRATEGY Success comes by REPEATING what you do daily. Eventually, it will become EASY for you. When these things became easy, we will have PLEASURE in doing them. We experience pleasure by doing this things OFTENly.  STRATEGY Success comes by REPEATING what you do daily. Eventually, it will become EASY for you. When these things became easy, we will have PLEASURE in doing them. We experience pleasure by doing this things OFTENly. And this will be a HABIT for us.  STRATEGY Through REPETITION it becomes EASY. If it’s EASY it is a PLEASURE. Then you’ll do it OFTEN, and that’s how it becomes a HABIT.  WHAT IS MATHEMATICS?  What is Mathematics? The word mathematics comes from the Greek word "mathema", which in the Ancient Greek language, means "that which is learnt", or "lesson" in modern Greek. Mathema is derived from "manthano" while the modern Greek equivalent is "mathaino"which means "to learn".  What is Mathematics? Voluminous studies have been conducted about the nature of mathematics since time immemorial. The rapid growth of mathematics and its applications over the past several years have led to several discussions, studies, essays, and arguments that examine its nature and importance. Mathematics is defined as the science of patterns and relationships. However, people still ask: What is it exactly that mathematicians do when they are doing mathematics? What really is mathematics?  What is Mathematics? Numerous definitions from different sources are given to “mathematics.” For example, the Encyclopedia Britannica defines mathematics as “the science of structure, order, and relations that has evolved from elemental practices of counting, measuring, and describing the shapes and characteristics of objects.” This definition is the one closest to the mathematics that is evident in the modern world today. CHARACTERISTICS OF MATHEMATICS  Characteristics of Mathematics Classification Logical Sequence Structure Characteristics of Precision and Generalization Mathematics Accuracy Mathematical Applicability Language and Abstractness Symbolism  Characteristics of Mathematics Classification Classification generates a series of mental relations through which objects are grouped according to similarities and differences depending on specific criteria such as shape, color, size, etc.  Characteristics of Mathematics Logical Sequence Ideas in mathematics need to flow in an order that makes sense. The sequence can naturally match what occurs in a text (main ideas) and what the reader needs to understand. It means that each step can be derived logically from the preceding steps.  Characteristics of Mathematics Structure A structure on a set is an additional mathematical object that is related to that given set in some particular characteristic or manner. The structure on a particular mathematical set will allow mathematicians to study the set further and find its relationship with other objects.  Characteristics of Mathematics Precision and Accuracy Accuracy is how close a measured value is to the actual (true) value. It is the degree to which a given quantity is correct and free from error. Precision, on the other hand, is how close the measured values are to each other.  Characteristics of Mathematics Abstractness Abstraction is the process of extracting the underlying essence of a mathematical concept by taking away any dependence on real-world objects.  Characteristics of Mathematics Symbolism The language of mathematics is the system used by mathematicians to communicate mathematical ideas using symbols instead of words. This language is uniquely constructed in such a way that all mathematicians understand symbolic notations and mathematical formulas.  Characteristics of Mathematics Applicability The applicability of mathematics can lie anywhere on a spectrum from the completely simple (trivial) to the utterly complex (mysterious). Mathematics can be used in all fields of human endeavor.  Characteristics of Mathematics Generalizations Making generalizations is fundamental to mathematics. It is a skill that must be developed among students. It is of vital importance in a functioning society. Introductory Activity Instructions: Identify what comes next in the following sequence of letters or numbers. 1. R, O, Y, G, B, I, ___ 2. A, E, I, M, Q, ___ 3. 1, 4, 9, 16, 25, 36, ___ 4. 1, 3, 6, 10, 15, 21, ___ 5. 0, 1, 1, 2, 3, 5, ____ Introductory Activity Instructions: Identify what comes next in the following sequence of letters or numbers. 1) R, O, Y, G, B, I, V This sequence represents the first letters of the colors in the rainbow: Red, Orange, Yellow, Green, Blue, Indigo, Violet. Introductory Activity Instructions: Identify what comes next in the following sequence of letters or numbers. 2) A, E, I, M, Q, U This sequence consists of every fourth letter in the alphabet: A (1), E (5), I (9), M (13), Q (17), U (21). Introductory Activity Instructions: Identify what comes next in the following sequence of letters or numbers. 3) 1, 4, 9, 16, 25, 36, 49 This sequence lists the squares of consecutive integers: 12, 22, 32, 42, 52, 62, 72 Introductory Activity Square numbers is a type of a figurate numbers, they are numbers that can form a perfect square. Introductory Activity Instructions: Identify what comes next in the following sequence of letters or numbers. 4) 1, 3, 6, 10, 15, 21, 28 This sequence represents the triangular numbers, which are the sums of the natural numbers: 1, 1+2, 1+2+3, 1+2+3+4, 1+2+3+4+5, 1+2+3+4+5+6, 1+2+3+4+5+6+7. Introductory Activity Triangular numbers is a type of a figurate numbers, they are numbers that can form an equilateral triangle. Introductory Activity Instructions: Identify what comes next in the following sequence of letters or numbers. 5) 0, 1, 1, 2, 3, 5, 8 This sequence is the Fibonacci sequence, where each number is the sum of the two preceding ones: 0, 1, 1, 2, 3, 5, 8. SEQUENCE  Sequence It refers to an ordered list of numbers or letters called terms, which are ascending, descending or determined by a pattern.  Sequence 1, 3, 5, 7, …. 1st 2nd 3rd 4th Term Term Term Term As shown above, the elements in the sequence are called terms. It is called sequence because the list is ordered and it follows. It is called sequence because the list is ordered and it follows a certain kind of pattern that must be recognized in order to see the expanse. DIFFERENT TYPE OF SEQUENCE Arithmetic Sequence It consists of terms with a common difference Arithmetic Sequence 2, 4, 6, 8, 10, 12, 14, 16... 2 2 2 2 2 2 2 Notice in the given example above, the common difference between two consecutive terms in the sequence is two. The common difference is the clue that must be figure out in a pattern in order to recognize it as an arithmetic sequence. Arithmetic Sequence Arithmetic Sequence Example 1: Arithmetic Sequence 1. Given the arithmetic sequence: 5, 11, 17, 23,... Find the 15th term. Example 2: Arithmetic Sequence 2. For the arithmetic sequence: 3,7,11,15… Find the sum of the first 20 terms. Geometric Sequence It composed of terms with a common ratio Geometric Sequence 2, 8, 32, 128,... 4 4 4 This is a geometric sequence where each term is obtained by multiplying the previous term by a constant factor called the common ratio. To determine the next term, you multiply the last given term by the common ratio, which you can find by dividing any term by the previous term. Geometric Sequence Geometric Sequence Example 3: Geometric Sequence 3. Given the geometric sequence: 2,6,18,54… Find the 10th term. Example 4: Geometric Sequence 4. For the geometric sequence: 2,8,32,128,... Find the sum of the first 6 terms. Week 1: Seatwork 1 Instructions: Choose the letter of the correct answer. 1. What are the next three terms of the sequence 4, 16, 36, 64, 100, …? a. 121, 144, 169 b. 121,169, 225 c. 144, 196, 256 d. 44, 169, 196 Week 1: Seatwork 1 2. In the arithmetic sequence 2, 5, 8, 11,..., what is the common difference? a. 2 b. 3 c. 4 d. 5 Week 1: Seatwork 1 3. If the 6th term of an arithmetic sequence is 20 and the common difference is 3, what is the first term? a. 5 b. 8 c. 11 d. 14 Week 1: Seatwork 1 4. What is the sum of the first 10 terms of the arithmetic sequence 5, 8, 11,...? a. 155 b. 180 c. 200 d. 225 Week 1: Seatwork 1 5. The 3rd term of an arithmetic sequence is 12, and the common difference is 4. What is the 6th term? a. 20 b. 24 c. 28 d. 32 Week 1: Seatwork 1 6. What is the 4th term of the geometric sequence 2, 6, 18,...? a. 54 b. 72 c. 81 d. 108 Week 1: Seatwork 1 7. In the geometric sequence 5, 15, 45, 135,..., what is the common ratio? a. 2 b. 3 c. 5 d. 7 Week 1: Seatwork 1 8. The 4th term of a geometric sequence is 12 and the 6th term is 48. What is the common ratio? a. 2 b. 3 c. 4 d. 6 Week 1: Seatwork 1 9. What is the sum of the first 5 terms of the geometric sequence 2, 6, 18, 54,...? a. 150 b. 180 c. 216 d. 242 Week 1: Seatwork 1 10. The sum of the first 7 terms of an arithmetic sequence is 154. If the first term is 10, what is the common difference? a. 4 b. 6 c. 7 d. 8  EXCHANGE PAPERS “An honest person earns respect, while a cheater buys temporary success with a permanent loss” Week 1: Seatwork 1 Instructions: Choose the letter of the correct answer. 1. What are the next three terms of the sequence 4, 16, 36, 64, 100, …? a. 121, 144, 169 b. 121,169, 225 c. 144, 196, 256 d. 44, 169, 196 Week 1: Seatwork 1 2. In the arithmetic sequence 2, 5, 8, 11,..., what is the common difference? a. 2 b. 3 c. 4 d. 5 Week 1: Seatwork 1 3. If the 6th term of an arithmetic sequence is 20 and the common difference is 3, what is the first term? a. 5 b. 8 c. 11 d. 14 Week 1: Seatwork 1 4. What is the sum of the first 10 terms of the arithmetic sequence 5, 8, 11,...? a. 155 b. 185 c. 200 d. 225 Week 1: Seatwork 1 5. The 3rd term of an arithmetic sequence is 12, and the common difference is 4. What is the 6th term? a. 20 b. 24 c. 28 d. 32 Week 1: Seatwork 1 6. What is the 4th term of the geometric sequence 2, 6, 18,...? a. 54 b. 72 c. 81 d. 108 Week 1: Seatwork 1 7. In the geometric sequence 5, 15, 45, 135,..., what is the common ratio? a. 2 b. 3 c. 5 d. 7 Week 1: Seatwork 1 8. The 4th term of a geometric sequence is 12 and the 6th term is 48. What is the common ratio? a. 2 b. 3 c. 4 d. 6 Week 1: Seatwork 1 9. What is the sum of the first 5 terms of the geometric sequence 2, 6, 18, 54,...? a. 150 b. 180 c. 216 d. 242 Week 1: Seatwork 1 10. The sum of the first 7 terms of an arithmetic sequence is 46. If the first term is 10, what is the common difference? a. 4 b. 6 c. 7 d. 8 Fibonacci Sequence This specific sequence was named after an Italian mathematician Leonardo Pisano Bigollo (1170-1250). He discovered the sequence while he was studying rabbits. The Fibonacci sequence is a series of numbers governed by some unusual arithmetic rule. The sequence is organized in a way a number can be obtained by adding the two previous numbers. Fibonacci Sequence Fibonacci Sequence 1, 1, 2, 3, 5, 8, 13, 21,... Fibonacci Sequence To explore a little bit more about the Fibonacci sequence, the location of the term was conventionally tagged as Fib(n). This means that Fib(1) = 1, Fib(2) = 1, Fib(3) = 2, Fib(4) = 3. In this method, the Fib(n) is actually referring to the nth term of the sequence. Fibonacci Sequence The amazing grandeur of Fibonacci sequence was also discovered in the structure of Golden rectangle. The golden rectangle is made up of squares whose sizes, surprisingly is also behaving similar to the Fibonacci sequence. Fibonacci Sequence A golden rectangle is a rectangle whose length to width ratio equal to the golden ratio, φ, which has a value of approximately 1.618, assuming the length is the larger value. Example 5: Fibonacci Sequence 5. Find the 12th term of the Fibonacci sequence.  Fibonacci Sequence The Fibonacci sequence is constantly encountered among flowering plants such as sunflower. Looking at the sunflower, you could see the seeds in the center spiraling in either clockwise or counterclockwise directions. When you count the number of spirals going to a particular direction, you would discover that the number of spirals corresponds to the numbers in the Fibonacci sequence. Take a closer look  Fibonacci Sequence From, the previous slide, the first set of line show 21 spiral seeds, the second set 34 and the third set 55. The numbers 21, 34 and 55 are terms of the Fibonacci Sequence. This fascinating sequence of numbers that abound in nature is also being depicted in the images that follow. Week 1: Seatwork 2 Instructions: Choose the letter of the correct answer. 1. What is the 10th term in the Fibonacci sequence? a. 34 b. 55 c. 89 d. 144 Week 1: Seatwork 2 Instructions: Choose the letter of the correct answer. 2. In the Fibonacci sequence, what is the sum of the 4th and 7th term? a. 16 b. 20 c. 24 d. 28 Week 1: Seatwork 2 Instructions: Choose the letter of the correct answer. 3. What is the ratio of the 10th term to the 9th term in the Fibonacci sequence? a. 1.5 b. 1.618 c. 1.732 d. 2.0 Week 1: Seatwork 2 Instructions: Choose the letter of the correct answer. 4. What is one way to decide if two numbers follow a Fibonacci sequence? a. If each number is prime b. If their sum is the same as their difference c. If their product is approximately the golden ratio d. If their ratio is approximately the golden ratio Week 1: Seatwork 2 Instructions: Choose the letter of the correct answer. 5. What is the 15th term in the Fibonacci sequence? a. 377 b. 610 c. 987 d. 1597 Week 1: Seatwork 2 6. Which is next in the sequence? Week 1: Seatwork 2 7. Which is next in the sequence? Week 1: Seatwork 2 8. Which is next in the sequence? Week 1: Seatwork 2 9. Which is next in the sequence? Week 1: Seatwork 2 10. Which of the figures below best fits the missing space in the Question Figure?  EXCHANGE PAPERS "Honesty may not always be easy, but it is always right; cheating may seem easy, but it is always wrong." Week 1: Seatwork 2 Instructions: Choose the letter of the correct answer. 1. What is the 10th term in the Fibonacci sequence? a. 34 b. 55 c. 89 d. 144 Week 1: Seatwork 2 Instructions: Choose the letter of the correct answer. 2. In the Fibonacci sequence, what is the sum of the 4th and 7th term? a. 16 b. 20 c. 24 d. 28 Week 1: Seatwork 2 Instructions: Choose the letter of the correct answer. 3. What is the ratio of the 10th term to the 9th term in the Fibonacci sequence? a. 1.5 b. 1.618 c. 1.732 d. 2.0 Week 1: Seatwork 2 Instructions: Choose the letter of the correct answer. 4. What is one way to decide if two numbers follow a Fibonacci sequence? a. If each number is prime b. If their sum is the same as their difference c. If their product is approximately the golden ratio d. If their ratio is approximately the golden ratio Week 1: Seatwork 2 Instructions: Choose the letter of the correct answer. 5. What is the 15th term in the Fibonacci sequence? a. 377 b. 610 c. 987 d. 1597 Week 1: Seatwork 2 6. Which is next in the sequence? Week 1: Seatwork 2 7. Which is next in the sequence? Week 1: Seatwork 2 8. Which is next in the sequence? Week 1: Seatwork 2 9. Which is next in the sequence? Week 1: Seatwork 2 10. Which of the figures below best fits the missing space in the Question Figure? Week 1: Weekly Assessment Instructions: Answer the following questions and show your solutions. Use short bond paper. 1. The Arithmetic Sequence has a first term of 10 and a fifth term of 34. What is the common difference? 2. If the last term of an arithmetic sequence is 73, the first term is 3, and the common difference is 5, how many terms are in the sequence? 3. In the geometric sequence: 5, 25,125,625, … Which nth term is equal to 15625? 4. If the last term of a geometric sequence is 1024, the first term is 2, and the common ratio is 2, how many terms are in the sequence? 5. In a modified Fibonacci sequence where the first two terms are F(1) = 3 and F(2) = 5. Find the 12th term. Week 1: Assignment Instructions: Review the following for the Pre-Test next meeting. 1. Mathematics as a Language 2. Common Signs and Symbols 3. Expressions and Sentences in Mathematics