GECC103 Mathematics in the Modern World SCP PDF

ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached SIMPLIFIED COURSE PACK (SCP) FOR SELF-DIRECTED LEARNING GECC103 – Mathematics in the...

ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached SIMPLIFIED COURSE PACK (SCP) FOR SELF-DIRECTED LEARNING GECC103 – Mathematics in the Modern World This Simplified Course Pack (SCP) is a draft version only and may not be used, published or redistributed without the prior written consent of the Academic Council of SJPIICD. Contents of this SCP are only intended for the consumption of the students who are officially enrolled in the course/subject. Revision and modification process of this SCP are expected. SCP-GECC103 | 1  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached By 2023, a recognized professional institution providing quality, Vision economically accessible, and transformative education grounded on the teachings of St. John Paul II. Serve the nation by providing competent JPCean graduates through quality teaching and learning, transparent governance, holistic Mission student services, and meaningful community-oriented researches, guided by the ideals of St. John Paul II. Respect Hard Work Perseverance Core Values Self-Sacrifice Compassion Family Attachment Inquisitive Ingenious Graduate Attributes Innovative Inspiring Course Code/Title GECC103 – Mathematics in the Modern World This course deals with nature of mathematics, appreciation of its Course Description practical, intellectual and aesthetic dimensions, and application of mathematical tools in daily life. Course Requirement Time Frame 54 Hours “Based 40” Cumulative Averaging Grading System Grading System Periodical Grading = Attendance (5%) + Participation (10%) + Quiz (25%) + Exam (60%) Final-Final Grade = Prelim Grade (30%) + Midterm Grade (30%) + Final Grade (40%) Contact Detail Dean/Program Head Amie P. Matalam, MM (09953860989) SCP-GECC103 | 2  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Course Map GECC103- Simplified Course Pack (SCP) SCP-Topics: Prelim Period SCP- Topics: Midterm Period SCP- Topics: Final Period Patterns and Numbers in Nature Simple and Compound Week 1 Week 7 Data Management Week 13 and the World Interest Patterns and Numbers in Nature Measures of Central Tendency Week 2 Week 8 Week 14 Voting and the World and Measures of Dispersion Four basic concepts: Sets, Elementary Logic: Week 3 Functions, Relations, Binary Week 9 Probability Week 15 Connectives, Quantifiers, Operations Negation, Variables Four basic concepts: Sets, Week 4 Functions, Relations, Binary Geometric designs: recognizing Logic statements ,Truth tables Week 10 Week 16 Operations (Continuation) and analyzing geometric shapes and Tautologies Inductive and Deductive Geometric designs: Week 5 Week 11 Week 17 Symbolic Arguments Reasoning / Problem solving Transformation Week 6 Preliminary Examination Week 12 Midterm Examination Week 18 Final Examination Course Outcomes 1. Discuss and argue the nature of mathematics, what it is, how it is expressed, represented and used; 2. Use different types of reasoning to justify statements and arguments made about mathematical concepts; 3. Use a variety of statistical tools to process and manage numerical data; 4. Use mathematics in other areas such as finance, voting, health and medicine, business, environment, arts and design, and recreation; and, 5. Appreciate the nature and uses of mathematics in everyday life. SCP-GECC103 | 3  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Welcome Aboard! This course deals with nature of mathematics, appreciation of its practical, intellectual, and aesthetic dimensions, and application of mathematical tools in daily life. This course begins with an introduction to the nature of mathematics as an exploration of patterns (in nature and 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 rather as a source of aesthetics in patterns of nature, for example, and a rich language in itself ( and of science) governed by logic and reasoning. 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 resources fairly. These aspects will provide opportunities for actually doing mathematics in a broad range of exercises that brings out the various dimensions of mathematics as a way of knowing, and test the students’ understanding and capacity. (CMO NO. 20, series of 2013) SCP-GECC103 | 4  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached SCP-TOPICS: PRELIM PERIOD TOPICS Week 1 Lesson Title Patterns and Numbers in Nature and the World Identify patterns in nature; convey the importance of Learning Outcome(s) mathematics in our lives At SJPIICD, I Matter! LEARNING INTENT! Terms to Ponder This section provides the meaning and the definition of terminologies that are significant for better understanding of the terms used throughout the simplified course pack of the subject GECC103 – Mathematics in the Modern World. As you go through the labyrinth of learning, in case you will be confronted with difficulty of the terms, refer to the defined terms for you to have a clear picture of the learning concepts. Mathematics is a study of numbers and arithmetic operations. It is a science of logical reasoning, drawing conclusions from assumed premises or strategic reasoning. Mathematics is the study of topics such as quantity (numbers), structure, space and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. Pattern is a regular, repeated or recurring form or design. Symmetry describes rules for moving things around without changing their pattern. Symmetry means that one shape becomes exactly like another when you move it in some way. SCP-GECC103 | 5  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Fractals are infinitely self-similar, iterated mathematical constructs having fractal dimension. Spiral is the winding or circling around a central or pole and gradually receding from or approaching it. Wave is a shape or outline having successive curves. Dunes may form a range of patterns including crescents, very long straight lines, stars, domes, parabolas, and longitudinal or Seif (‘sword’) shapes. A soap bubble forms a sphere, a surface with minimal area — the smallest possible surface area for the volume enclosed. A foam is a mass of bubbles; foams of different materials occur in nature. Tessellations are patterns formed by repeating tiles all over a flat surface. Cracks are linear openings that form in materials to relieve stress. Essential Content Mathematics is the study of topics such as quantity (numbers), structure, space and change. Mathematics is a science of pattern and order. Mathematics is widely used to understand the world around us. It does not only revolve around arithmetic and geometry but also deals with quantitative and qualitative data, inference, deduction and proof, and mathematical models of natural phenomena, of human behaviour, and of social systems. Mathematics can be seen in our surroundings, in our nature. Mathematics is present in the patterns found in our nature. These patterns are very common to us. Patterns in nature are visible regular forms found in the natural world. The patterns can sometimes be modelled mathematically and they include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes. SCP-GECC103 | 6  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached TYPES OF PATTERN 1. Symmetry. Symmetry is pervasive in living things. Animals mainly have bilateral or mirror symmetry, as do the leaves of plants and some flowers such as orchids. Plants often have radial or rotational symmetry, as do many flowers and some groups of animals such as sea anemones. Five-fold symmetry is found in the echinoderms, the group that includes starfish, sea urchins, and sea lilies. Among non-living things, snowflakes have striking six-fold symmetry: each flake’s structure forming a record of the varying conditions during its crystallization, with nearly the same pattern of growth on each of its six arms. 2. Fractals. Fractals are infinitely self-similar, iterated mathematical constructs having fractal dimension. Fractal-like patterns occur widely in nature, in phenomena as diverse as clouds, river networks, geologic fault lines, mountains, coastlines, animal coloration, snowflakes, crystals, blood vessel branching, and ocean waves. 3. Spirals. Spirals are common in plants and in some animals, notably molluscs. For example, in the nautilus, a cephalopod mollusc, each chamber of its shell is an approximate copy of the next one, scaled by a constant factor and arranged in a logarithmic spiral. Plant spirals can be seen in phyllotaxis, the arrangement of leaves on a stem. 4. Waves/Dunes. Waves are disturbances that carry energy as they move. Mechanical waves propagate through a medium – air or water, making it oscillate as they pass by. As waves in water or wind pass over sand, they create patterns of ripples. When winds blow over large bodies of sand, they create dunes. 5. Bubbles/Foam. A soap bubble forms a sphere, a surface with minimal area — the smallest possible surface area for the volume enclosed. A foam is a mass of bubbles; foams of different materials occur in nature. 6. Tessellations. Tessellations are patterns formed by repeating tiles all over a flat surface. The cells in the paper nests of social wasps, and the wax cells in honeycomb built by honey bees are well-known examples. 7. Cracks. Cracks are linear openings that form in materials to relieve stress. SCP-GECC103 | 7  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached 8. Spots/Stripes. These patterns have an evolutionary explanation: they have functions which increase the chances that the offspring of the patterned animal will survive to reproduce. SELF-SUPPORT: You can click the URL Search Indicator below to help you further understand the lessons. Search Indicator https://ecstep.com/natural-patterns/ https://services.math.duke.edu/undergraduate/Handbook96_97/node5.html http://www.patternsinnature.org/Book/PatternsContainSymmetry.html https://study.com/academy/lesson/what-is-symmetry-in-math-definition- lesson-quiz.html https://blogs.glowscotland.org.uk/glowblogs/mbfeportfolio/2017/10/31/the- nature-of-mathematics/ SCP-GECC103 | 8  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached LET’S INITIATE! Activity 1. Identify the type of pattern presented in the following pictures/clips. 1. 2. 3. 4. 5. 6. 7. SCP-GECC103 | 9  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached 8. 9. 10. 11. 12. 13. 14. 15. SCP-GECC103 | 10  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached I LET’S NQUIRE! Activity 1. Do as indicated. 1. Using Mnemonic, define the word MATHEMATICS. [A mnemonic device is a word, phrase, or sentence that is used to remember a number of separate objects, elements, ideas, etc. that make up a group.] Example: (Do not use the following words in your Mnemonics.) Mentally difficult Arduous Thorough Historic Effortful Material All over Time-consuming Important Complicated Scientific ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 2. Answer completely and thoughtfully. How did Mathematics affect your daily life? ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ _________________________________________________________________ SCP-GECC103 | 11  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached LET’S INFER! Activity 1. Write a short response to the following statements. Briefly explain briefly your ideas. Three to five sentences only. 1. What causes the cracks in our nature? ____________________________________________________________________ ____________________________________________________________________ How is mathematics present in the patterns found in our nature? ____________________________________________________________________ ____________________________________________________________________ 2. Mathematics is widely used to understand the world around us. ____________________________________________________________________ ____________________________________________________________________ SCP-GECC103 | 12  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Week 2 Lesson Title Patterns and Numbers in Nature and the World Recognize and solve sequences and Fibonacci sequence by understanding the rules; Establish understanding on the Learning Outcome(s) natural occurrence of Fibonacci sequence in our surroundings LEARNING INTENT! Terms to Ponder Sequence is an ordered list of numbers. Fibonacci sequence is formed by adding the preceding two numbers, beginning with 0 and 1. Logic Patterns are meaningful patterns in strange and unpredictable situations. Essential Content A sequence is an ordered list of numbers that often follow a specific pattern or function. Sequences can be both finite and infinite. The individual elements in a sequence are called terms. The following are examples of a sequence: 2, 4, 6, 8, 10 5, 9, 13, 17, 21 1, 3, 9, 27, 81 In 1202, Leonardo of Pisa (Leonardo Fibonacci) introduced the Fibonacci number sequence. It turns out that simple equations involving the Fibonacci numbers can describe most of the complex spiral growth patterns found in nature. The Fibonacci sequence runs 1, 1, 2, 3, 5, 8, 13… (each subsequent number being the sum of the two preceding ones). SCP-GECC103 | 13  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached The Fibonacci sequence was illustrated from the reproduction of rabbits. Fibonacci posed the following question: If a pair of rabbits is placed in an enclosed area, how many rabbits will be born there if we assume that every month a pair of rabbits produces another pair, and that rabbits begin to bear young two months after their birth? The most famous and beautiful examples of the occurrence of the Fibonacci sequence in nature are found in a variety of trees and flowers, generally associated with some kind of spiral structure. For instance, leaves on the stem of a flower or a branch of a tree often grow in a helical pattern, spiralling around the branch as new leaves form further out. Spirals arise from a property of growth called self- similarity or scaling - the tendency to grow in size but to maintain the same shape. Not all organisms grow in this self-similar manner. Here is where Fibonacci comes in – we SCP-GECC103 | 14  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached can build a squarish sort of nautilus by starting with a square of size 1 and successively building on new rooms whose sizes correspond to the Fibonacci sequence: The Fibonacci sequence can be written as a "Rule". The Rule is xn = xn−1 + xn−2 where: xn is term number "n" xn−1 is the previous term (n−1) xn−2 is the term before that (n−2). Example: term 9 is calculated like this: x9 = x9−1 + x9−2 = x8 + x7 = 21 + 13 = 34 SCP-GECC103 | 15  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached When we take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio "φ" which is approximately 1.618034... Logic Patterns are meaningful patterns in strange and unpredictable situations. SELF-SUPPORT: You can click the URL Search Indicator below to help you further understand the lessons. Search Indicator https://mathigon.org/course/sequences/fibonacci https://ecstep.com/natural-patterns/ SCP-GECC103 | 16  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached https://math.temple.edu/~reich/Fib/fibo.html https://www.mathsisfun.com/numbers/fibonacci-sequence.html https://www.reddit.com/r/lossedits/comments/9c3k8d/found_this_logic_pattern_tes t_online/ http://www.graduatewings.co.uk/how-to-improve-at-logical-reasoning https://www.wikijob.co.uk/content/aptitude-tests/test-types/logical-reasoning At SJPIICD, I Matter! LET’S INITIATE! Activity 1. Give the next three terms of the following sequences. State the rulings used in every sequences. 1. -21, -18, -15, -12, __, __, __ 2. 2, 10, 50, 250, __, __, __ 3. 40960, 10240, 2560, __, __, __ 4. 2, 6, 10, 14, 18, __, __, __ 5. 3, 7, 11, 15, __, __, __ 6. 34, 37, 40, 43, __, __, __ 7. 1, 3, 15, 75, __, __, __ 8. 1/27, 1/9, 1/3, __, __, __ 9. 25, 36, 47, 58, __, __, __ 10. 8, -16, 32, -64, __, __, __ SCP-GECC103 | 17  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached LET’S INQUIRE! Activity 1. Do as indicated. 1. Find an interesting pattern you can find in the Fibonacci sequence. 2. Give two scenarios/objects that you can find in the nature or in the surroundings that follow the Fibonacci sequence. LET’S INFER! Activity 1. Answer the following logic patterns. 1. Find the missing box. 2.. SCP-GECC103 | 18  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached 3.. 4.. SCP-GECC103 | 19  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached 5. SCP-GECC103 | 20  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Find the missing box. 6. Find the next box. 7. Find the missing box. SCP-GECC103 | 21  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached 8. Find the next box. 9. Identify the odd one out.. SCP-GECC103 | 22  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Week 3 Four basic concepts: Sets, Functions, Relations, Binary Lesson Title Operations Learning Outcome(s) Perform operations on mathematical operations correctly LEARNING INTENT! Terms to Ponder A set is a collection of distinct objects, called elements of the set. The symbol ∈ means “is an element of”. A set that contains no elements, { }, is called the empty set and is notated ∅. A subset of a set A is another set that contains only elements from the set A, but may not contain all the elements of A. An ordered pair is a set of inputs and outputs and represents a relationship between the two values. A relation is a set of inputs and outputs, and a function is a relation with one output for each input. Essential Content A set is a collection of objects that have something in common or follow a rule. The objects in the set are called its elements. Capital letters are used to denote sets. Lowercase letters are used to denote elements of sets. Curly braces { } denote a list of elements in a set. There are two notations of set: Set-roster notation and set-builder notation. A Set-roster notation is simply done by listing all the elements, separated by commas, inside two braces. Set-builder notation describes a set by saying what properties its members have. SCP-GECC103 | 23  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Illustrations: Set-roster Notation Set-builder Notation A subset of a set A is another set that contains only elements from the set A, but may not contain all the elements of A. If B is a subset of A, we write B ⊆ A. A proper subset is a subset that is not identical to the original set—it contains fewer elements. If B is a proper subset of A, we write B ⊂ A. SCP-GECC103 | 24  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Relations and Functions An ordered pair is represented as (INPUT, OUTPUT). A relation shows the relationship between INPUT and OUTPUT. A function is a relation which derives one OUTPUT for each given INPUT. (All functions are relations, but not all relations are functions). A function is actually a “special” kind of relation because it follows an extra rule. Just like a relation, a function is also a set of ordered pairs; however, every x-value must be associated to only one y- value. The domain is the set of all x or input values. We may describe it as the collection of the first values in the ordered pairs. The range is the set of all y or output values. We may describe it as the collection of the second values in the ordered pairs. SELF-SUPPORT: You can click the URL Search Indicator below to help you further understand the lessons. Search Indicator https://courses.lumenlearning.com/atd-hostos-introcollegemath/chapter/set- theory/ https://www.mathsisfun.com/sets.html https://www.mathgoodies.com/lessons/sets https://us.sofatutor.com/mathematics/algebra-1/functions-and-relations https://www.chilimath.com/lessons/intermediate-algebra/relations-and- functions/ https://byjus.com/maths/relations-and-functions/ SCP-GECC103 | 25  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached I LET’S NITIATE! Activity 1. Write the set-roster notation and set-builder notation of the following sets. 1. C is the set of all the letters in the word Mathematics. 2. L is the set of all the even numbers between 10 and 15. 3. N is the set of all the counting numbers. 4. W is the set of all the whole numbers. 5. A is the set of all the colors in the rainbow. 6. J is the set of all the positive integers between -1 and 1. 7. Y is the set of all the whole numbers between -5 and 10. 8. O is the set of all the factors of 10. 9. Z is the set of all the integers. 10. B is the set of all the letters in the word MISSISSIPPI. I LET’S NQUIRE! Activity 1. Examine the following statements and identify if it is either TRUE or FALSE. A = {dog} DD = {g, o, d} B = {d, o, g} EE = {cat} C = {d, o, g, c, a, t} FF = {cat, dog} 1. A is a subset of C 2. E is a subset of C 3. F is a subset of C 4. A is a subset of B 5. E is a subset F 6. B is a subset of F 7. B is a subset of D 8. B is a subset of C 9. F is a subset of A 10. D is a subset of E SCP-GECC103 | 26  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached I LET’S NFER! Activity 1. (i) Identify whether or not the following relations is a function or not. (ii) Write the domain set and the range set of each relation. (5 items) 1. 2. 3. 4. 5. SCP-GECC103 | 2  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Week 4 Four basic concepts: Sets, Functions, Relations, Binary Lesson Title Operations Learning Outcome(s) Perform operations on mathematical operations correctly LEARNING INTENT! Terms to Ponder An ordered pair is a set of inputs and outputs and represents a relationship between the two values. A relation is a set of inputs and outputs, and a function is a relation with one output for each input. A binary operation on a set is a calculation involving two elements of the set to produce another element of the set. Essential Content A function is a relation that for each input, there is only one output. The domain is the input or the x-value, and the range is the output, or the y-value. Each x-value is related to only one y-value. We write f (x) to mean the function whose input is x. If f(x) = 2x - 3 then f(4) = 2(4) - 3 = 5. Special Functions in Algebra These are some of the important functions as follow: Constant Function F (x) = c The c-value can be any number. SCP-GECC103 | 3  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Identity Function F (x) = x For the identity function, the x-value is the same as the y-value. Linear Function F (x) = mx + b An equation written in the slope-intercept form is the equation of a linear function Absolute Value Function F (x) = ∣x∣ Inverse Functions An inverse function reverses the inputs with its outputs. The inverse of f(x) = 3x - 4 is f-1(x) = (x + 4) / 3 Function Operations You can add, subtract, mutiply, and divide functions. f(x) + g(x) = (f+g) (x) f(x) − g(x) = (f−g) (x) f(x) × g(x) = (f × g) (x) f(x) / g(x) = g /f (x) SCP-GECC103 | 4  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Binary Operations Note: Introduction Only The binary operations * on a non-empty set A are functions from A × A to A. The binary operation, *: A × A → A. It is an operation of two elements of the set whose domains and co-domain are in the same set. Addition, subtraction, multiplication, division and exponential are some of the binary operations. SELF-SUPPORT: You can click the URL Search Indicator below to help you further understand the lessons. Search Indicator https://us.sofatutor.com/mathematics/algebra-1/functions-and-relations https://byjus.com/maths/relations-and-functions/ https://www.chilimath.com/lessons/intermediate-algebra/relations-and-functions/ http://www.ltcconline.net/greenl/courses/152A/functgraph/relfun.htm https://www.mathwarehouse.com/algebra/relation/math-function.php https://www.toppr.com/guides/maths/relations-and-functions/binary-operations/ https://mathbitsnotebook.com/Algebra1/RealNumbers/RNBinary.html SCP-GECC103 | 5  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached I LET’S NITIATE! Activity 1. Identify which type of function is being illustrated in the following functions. (Sample Questions Only) (10 items) 1. F (x) = 5 2. F (x) = x–2 3. F (x) = x 4. F (x) = 0 5. F (x) = |4| I LET’S NQUIRE! Activity 1. Solve the following functions. (Sample Questions Only) (10 items) 1. f (2) = 2x – 2 2. g (-1) = (x + 1)2 3. f (8) = 0 4. g (-5) = -10x – 1 5. f (-3) = |2x| LET’S NFER!I Activity 1. Solve the following operations of functions. (Sample Questions Only) (10 items) 1. f (x) = 8x – 2 ; g (x) = 7x – 3 ; f (2) – g(2) 2. f (x) = 125x ; g (x) = 3x ; f (-3) / g(-3) 3. f (x) = x + 1 ; g (x) = 3 ; f (2) × g(2) 4. f (x) = -5x + 2 ; g (x) = x +1 ; f (-1) + g(-1) 5. f (x) = x – 2 ; g (x) = |5| ; f (0) × g(0) SCP-GECC103 | 6  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Week 5 Lesson Title Deductive and Inductive Reasoning Use deductive and inductive reasoning to justify statements Learning Outcome(s) and arguments, and to solve word problems LEARNING INTENT! Terms to Ponder Deductive reasoning is the process of reaching conclusions based on previously known facts. The conclusions reached by this type of reasoning are valid and can be relied on. Inductive Reasoning is the process of drawing a general conclusion by observing a pattern based on specific instances. This conclusion is called a hypothesis or conjecture. Essential Content The two major types of reasoning, deductive and inductive, refer to the processes by which someone creates a conclusion as well as how they believe their conclusion to be true. Deductive reasoning requires one to start with a few general ideas, called premises, and apply them to a specific situation. Recognized rules, laws, theories, and other widely accepted truths are used to prove that a conclusion is right. The concept of deductive reasoning is often expressed visually using a funnel that narrows a general idea into a specific conclusion. Deductive reasoning is meant to demonstrate that the conclusion is absolutely true based on the logic of the premises. In practice, the most basic form of deductive reasoning is the syllogism, where two premises that share some idea support a conclusion: If A=B and C=A, then B=C. SCP-GECC103 | 7  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Examples: Use deductive reasoning to reach a conclusion in each of the following. All rabbits have long ears. Fluffy does not have long ears. ‘Conclusion’? Find all numbers that satisfy the equation x2 = 4. Inductive reasoning uses a set of specific observations to reach an overarching conclusion; it is the opposite of deductive reasoning. So, a few particular premises create a pattern which gives way to a broad idea that is likely true. This is commonly shown using an inverted funnel (or a pyramid) that starts at the narrow premises and expands into a wider conclusion. All forms of inductive reasoning, though, are based on finding a conclusion that is most likely to fit the premises and is used when making predictions, creating generalizations, and analyzing cause and effect. Inductive arguments are meant to predict a conclusion. They do not create a definite answer for their premises, but they try to show that the conclusion is the most probable one given the premises. Examples: Use inductive reasoning to predict the next two terms in the following sequences. 1, 3, 5, 7, __, __ 2, 3, 5, 7, 11, __, __ 1, 4, 9, 16, 25, __, __ 1, 1, 2, 3, 5, 8, __, __ SCP-GECC103 | 8  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Strategies for Problem Solving Four Step Process for Problem Solving 1. Understand the problem: You cannot solve a problem if you do not understand what you are being asked to find. Read, read, and read again. The problem must be read and analyzed carefully. 2. Devise a plan: There are many ways to attack a problem you are solving. Here are some strategies: Make a table or a chart. If a formula applies, use it. Look for a pattern. Work backwards. Solve a similar simpler problem. Guess and Check. Draw a sketch. Use trial and error. Use inductive reasoning. Use common sense. Write an equation and solve it. Look for a catch if an answer seems too obvious or impossible. 3. Carry out the plan: Once you know how to approach the problem, carry out your plan. 4. Look back and check: Check your answer to see that it is reasonable. Does it satisfy the conditions of the problem? Have you answered the question asked? Example: Find the missing numbers: __ __ 3 + 2 5 __ 4 4 4 I am thinking of a positive number. If I square it, double the result, take half of that result, and then add 12, I get 37. What is my number? SCP-GECC103 | 9  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached SELF-SUPPORT: You can click the URL Search Indicator below to help you further understand the lessons. Search Indicator http://web.mnstate.edu/jamesju/Fall2010/Content/M102LogicIntro.pdf https://www.sparknotes.com/math/geometry3/inductiveanddeductivereasoning/sect ion2/ https://study.com/academy/lesson/inductive-and-deductive-reasoning-in- geometry.html#:~:text=We've%20learned%20that%20inductive,is%20reasoning%20ba sed%20on%20facts.&text=Because%20the%20world%20of%20math,reasoning%20to% 20produce%20correct%20conclusions. https://www.mscc.edu/documents/writingcenter/Deductive-and-Inductive- Reasoning.pdf http://www.ltcconline.net/greenL/courses/102/financeGeometryLogic/deductive_an d_inductive_reasonin.htm https://www.onlinemath4all.com/inductive-and-deductive-reasoning-worksheet.html https://www.nr.edu/chalmeta/151/Mth_151_Chapter_1_notes.pdf At SJPIICD, I Matter! I LET’S NITIATE! Activity 1. Determine whether the reasoning is an example of deductive or inductive reasoning. Identify the premise(s) and the conclusion. 1. If you take your medicine, you’ll feel a lot better. You take your medicine. Therefore, you’ll feel a lot better. 2. Natalie’s first three children were boys. If she has another baby, it will be a boy. 3. If the same number is subtracted from both sides of a true equation, the new equation is also true. I know that 9 + 18 = 27. Therefore, (9 + 18) - 12 = 27 - 12. SCP-GECC103 | 10  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached I LET’S NQUIRE! Activity 1. Do as indicated. 1. Sketch the next figure in the pattern. 2. A wise old owl climbed up a tree whose height was exactly ninety plus three. Every day the owl climbed up 18 and every night climbed down 15. On what day did the owl reach the top of the tree? 3. A list of equations is given. Use the list and inductive reasoning to predict the next equation. (9 × 9) + 7 = 88 (98 × 9) + 6 = 888 (987 × 9) + 5 = 8, 888 (9876 × 9) + 4 = 88, 888 SCP-GECC103 | 11  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached I LET’S NFER! Activity 1. Answer the given problem thoughtfully. Express your answers in an essay form. On a daily basis, we use reasoning, be it inductive or deductive in nature. Put in words the real-life application of reasoning in your day by day living. Re-evaluate how you employ such reasoning. SCP-GECC103 | 12  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached SCP-TOPICS: MIDTERM PERIOD TOPICS Week 7 Lesson Title Data Management Learning Outcome(s) Organize data through listing, inputting in tables or graphing. At SJPIICD, I Matter! LEARNING INTENT! Terms to Ponder Statistics is from the Latin word ‘staticus’, which means “out of state”. Statistics is the study of the methods of collecting, organizing, presenting, analyzing, and drawing conclusions about data, commonly in numerical form. Descriptive Statistics is the branch of statistics that focuses on collecting, summarizing, and presenting a set of data. Inferential Statistics is the branch of statistics that analyzes sample data to draw conclusions about a population. A Population consists of all elements – individuals, items, or objects – whose characteristics are being studied. A sample from a statistical population is a proportion (a subset) of the population selected for study. Raw data is an unorganized data gathered during the collection stage. SCP-GECC103 | 13  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached A Census is when we collect data for every member of the group (the whole "population"). A Sample is when we collect data just for selected members of the group. Essential Content Statistics as a subject provides a body of principles and methodology for designing the process of data collection, summarizing and interpreting the data, and drawing conclusions or generalities. Statistics can be divided into two areas: (a) Descriptive statistics consists of methods for organizing, displaying and describing data using tables, graphs, and summary measures; and, (b) Inferential statistics consists of methods that use sample results to help make decisions or predictions about a population. Population consists of all elements – individuals, items, or objects – whose characteristics are being studied. The population that is being studied is also called the target population. A sample from a statistical population is a proportion (a subset) of the population selected for study. Two Basic Types of Data (Variables) 1. Qualitative or Categorical data – A variable that cannot assume a numerical value but can be classified into two or more non-numeric categories is called a qualitative or categorical variable. The data collected on such a variable are called qualitative data. SCP-GECC103 | 14  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached 2. Qualitative or Numerical data – A variable that can be measured numerically is called a quantitative variable. The data collected on a quantitative variable are called quantitative data. Data Collection Data can be collected in many ways, such as direct observation, survey, census or sample. Data Organization There are a wide variety of ways to summarize, organize, and present data. Most of the common methods are as follows: stem- and-leaf diagrams, frequency distributions, histograms, bar, and other graphs. Stem-and-Leaf Plot A Stem and Leaf Plot is a special table where each data value is split into a "stem" (the first digit or digits) and a "leaf" (usually the last digit). The "stem" values are listed down, and the "leaf" values go right (or left) from the stem values. Stem "1" Leaf "5" means 15 SCP-GECC103 | 15  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Stem "1" Leaf "6" means 16 Stem "2" Leaf "1" means 21 Dot Plot A Dot Plot is a graphical display of data using dots. Frequency Distribution Frequency is how often something occurs. A frequency distribution is a representation, either in a graphical or tabular format, which displays the number of observations within a given interval. The interval size depends on the data being analysed and the goals of the analyst. The intervals must be mutually exclusive and exhaustive. Example: Newspapers. These are the number of newspapers sold at a local shop over the last 10 days: 22, 20, 18, 23, 20, 25, 22, 20, 18, 20. SCP-GECC103 | 16  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Let us count how many of each number there is: Papers Sold Frequency 18 2 19 0 20 4 21 0 22 2 23 1 24 0 25 1 It is also possible to group the values. Here they are grouped in 5s: Papers Sold Frequency 15-19 2 20-24 7 25-29 1 Graphs / Charts 1. A Bar Graph (also called Bar Chart) is a graphical display of data using bars of different heights. 2. Line Graph: a graph that shows information that is connected in some way (such as change over time) 3. Pie Chart: a special chart that uses "pie slices" to show relative sizes of data. 4. A Pictograph is a way of showing data using images. 5. A Histogram is a graphical display of data using bars of different heights. SCP-GECC103 | 17  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached SELF-SUPPORT: You can click the URL Search Indicator below to help you further understand the lessons. Search Indicator https://k101.unob.cz/~neubauer/pdf/stat_lecture7.pdf https://faculty.math.illinois.edu/~castelln/M103/lecture9_math103.pdf https://www.andrews.edu/~calkins/math/edrm611/edrm02.htm https://www.toppr.com/guides/economics/organisation-of-data/raw-data- classification-of-data-and-variables/ https://www.informit.com/articles/article.aspx?p=353170&seqNum=2 https://www.mathsisfun.com/data/stem-leaf- plots.html#:~:text=A%20Stem%20and%20Leaf%20Plot,(usually%20the%20last %20digit). LET’S INITIATE! Activity 1. Answer the following questions. 1. What is the nature of statistics? ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 2. What is the difference between a quantitative data and a qualitative data? ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 3. What are the possible ways that we can use to organize data? ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ SCP-GECC103 | 18  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached I LET’S NQUIRE! Activity 1. Answer the following questions. 1. Discuss the similarities and differences of a stem-and-leaf plot and a dot plot. ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 2. Discuss the reason(s) why we use graphs or charts in organizing data. ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ LET’S INFER! Activity 1. Do as indicated. 1. Subjects in a psychological study were timed while completing a certain task. Given below are the respective times of each of the subject in minutes: 100, 110, 120, 130, 130, 150, 160, 170, 170, 190, 210, 230, 240, 260, 270, 270, 280, 290, 290 Create (a) a stem-and-leaf plot, (b) dot plot, (c) frequency distribution, and (d) a graph/chart of your choice, using the provided data above. SCP-GECC103 | 19  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Week 8 Lesson Title Measures of Central Tendency Solve the central tendency of a given data; identify the basic Learning Outcome(s) measures of central tendency LEARNING INTENT! Terms to Ponder A measure of central tendency is a summary statistic that represents the center point or typical value of a dataset. These measures indicate where most values in a distribution fall and are also referred to as the central location of a distribution. Mean is the sum of measurements divided by their number. Median is defined as the middle value in a distribution, below and above which lie values with equal total frequencies or probabilities (Collins Dictionary of Statistics). Mode is the single measure or score which occurs most frequently. Ungrouped data, which is also known as raw data, is data that has not been placed in any group or category after collection. Data is categorized in numbers or characteristics. Therefore, the data which has not been put in any of the categories is ungrouped. Essential Content SCP-GECC103 | 20  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Measures of Central Tendency In statistics, the three most common measures of central tendency are the mean, median, and mode. Each of these measures calculates the location of the central point using a different method. Arithmetic Average (Mean): The arithmetic average or mean is most widely used and is hence called the ‘Common Average’ or even simply ‘Average’. It is a measure of central value. It provides an accurate description of the sample and indirectly, that of population. The mean of a distribution of scores may be defined as the point on the scale of measurement obtained by dividing the sum of all the scores by the number of scores. If there is a change in any value, the mean also changes also. However, the mean doesn’t always locate the center of the data accurately. The mean is computed by using the formula or. The Median: The median may be defined as the point on the scale of measurement below and above which lie exactly 50 percent of cases. In the case of ungrouped data, the scores are arranged either in ascending order or in descending order then the mid-point is the median. Two cases of computing the median of an ungrouped data: i. If n is odd, then the median is the middle number. ii. If n is even, then the median is the average of the two middle number. SCP-GECC103 | 21  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached iii. The Mode: The mode is defined as the data which occurs most frequently. It is the most common item or score of a series which is usually repeated maximum number of times. If the data have multiple values that are tied for occurring the most frequently, you have a multimodal distribution. If no value repeats, the data does not have a mode. SELF-SUPPORT: You can click the URL Search Indicator below to help you further understand the lessons. Search Indicator https://statisticsbyjim.com/basics/measures-central-tendency-mean-median- mode/#:~:text=These%20measures%20indicate%20where%20most,mean%2C%20med ian%2C%20and%20mode https://www.yourarticlelibrary.com/education/statistics/measures-of-central- tendency-mean-median-and-mode-statistics/91992 https://people.richland.edu/james/lecture/m113/central_tendency.html https://www.mathsisfun.com/data/central-measures.html http://www.differencebetween.net/language/words-language/difference-between- grouped-data-and-ungrouped-data/ https://www.slideshare.net/LilianneSoriano/measures-of-central-tendency- ungrouped-data SCP-GECC103 | 22  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached LET’S INITIATE! Activity 1. Answer the following questions. 1. Discuss the measure of central tendency. ___________________________________________________________________ ___________________________________________________________________ __________________________________________________________________ 2. Discuss the process of computing the mean, median and mode. ___________________________________________________________________ ___________________________________________________________________ __________________________________________________________________ 3. Show the difference(s) and similarities of mean, median and mode. ___________________________________________________________________ ___________________________________________________________________ __________________________________________________________________ I LET’S NQUIRE! Activity 1. Solve the following problems. Show the complete solution by using the appropriate operations. 1. The math grades of ten students are 85, 80, 88, 83, 87, 89, 84, 80, 94 and 90. Find the mean, median and mode. 2. Find the mean, median and mode of the following set of data. The data below show the score of 20 students in a Math quiz: 25, 33, 35, 45, 34, 26, 29, 35, 38, 40, 45, 38, 28, 29, 25, 39, 32, 27, 47, 45. SCP-GECC103 | 23  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached I LET’S NFER! Activity 1. Do as indicated. 1. The data below show the score of 30 students in the 2012 Division Achievement Test (DAT). Analyze the given data and answer the questions. 35 16 28 43 21 17 15 16 20 18 25 22 33 18 32 38 23 32 18 25 35 18 20 22 36 22 20 14 39 22 a) What score is typical to the group of students? b) What score appears to be the median? How many students fail below that score? c) Which score frequently appears? SCP-GECC103 | 24  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Week 9 Lesson Title Measures of Dispersion / Variation Solve the variation of a given data; identify the process of Learning Outcome(s) solving the measures of variation LEARNING INTENT! Terms to Ponder Dispersion is the state of getting dispersed or spread. Statistical dispersion means the extent to which a numerical data is likely to vary about an average value. Measure of Dispersion tells the variation of the data from one another and gives a clear idea about the distribution of the data. The range is the difference between the largest and the smallest observation in the data. Standard deviation (SD) is the most commonly used measure of dispersion. It is a measure of spread of data about the mean. SD is the square root of the sum of squared deviation from the mean divided by the number of observations. Essential Content The measure of dispersion shows the scatterings of the data. It tells the variation of the data from one another and gives a clear idea about the distribution of the data. The measure of dispersion shows the homogeneity or the heterogeneity of the distribution of the observations. SCP-GECC103 | 25  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Types of Measures of Dispersion There are two main types of dispersion methods in statistics which are: Absolute Measure of Dispersion An absolute measure of dispersion contains the same unit as the original data set. Absolute dispersion method expresses the variations in terms of the average of deviations of observations like standard or means deviations. The types of absolute measures of dispersion are: 1. Range: It is simply the difference between the maximum value and the minimum value given in a data set. Example: 1, 3,5, 6, 7 Range = 7 -1= 6 2. Variance: Deduct the mean from each data in the set then squaring each of them and adding each square. Finally divide them by the total number of values in the data set. σ2 = [∑(X−μ)2}] / N 3. Standard Deviation: The square root of the variance is known as the standard deviation i.e. S.D. = √σ2 = σ. 4. Quartiles and Quartile Deviation: The quartiles are values that divide a list of numbers into quarters. The quartile deviation is half of the distance between the third and the first quartile. SCP-GECC103 | 26  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached 5. Mean and Mean Deviation: The average of numbers is known as the mean and the arithmetic mean of the absolute deviations of the observations from a measure of central tendency is known as the mean deviation (also called mean absolute deviation). Relative Measure of Dispersion The relative measures of depression are used to compare the distribution of two or more data sets. This measure compares values without units. Common relative dispersion methods include: 1. Co-efficient of Range 2. Co-efficient of Variation 3. Co-efficient of Standard Deviation 4. Co-efficient of Quartile Deviation 5. Co-efficient of Mean Deviation SELF-SUPPORT: You can click the URL Search Indicator below to help you further understand the lessons. Search Indicator https://www.toppr.com/guides/business-mathematics-and-statistics/measures- of-central-tendency-and-dispersion/measure-of-dispersion/ https://byjus.com/maths/dispersion/ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3198538/#:~:text=Standard%2 0deviation%20(SD)%20is%20the,an%20easier%20formula%20is%20used. https://revisionmaths.com/advanced-level-maths-revision/statistics/measures- dispersion At SJPIICD, I Matter! SCP-GECC103 | 27  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached I LET’S NITIATE! Activity 1. Answer the following questions. 1. What is a measure of dispersion? __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ 2. What is the difference of the measures of dispersion with the measures of central tendency? __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ 3. What is a standard deviation? __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ I LET’S NQUIRE! Activity 1. Answer the following questions. 1. Why is the Measures of Dispersion important in Statistics? __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ 2. How do you Calculate Dispersion? __________________________________________________________________________ __________________________________________________________________________ __________________________________________________________________________ SCP-GECC103 | 28  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached I LET’S NFER! Activity 1. Answer the following problems in a complete statement. Show your solution. 1. Find the following measurements for the given numbers: 1, 3, 5, 5, 6, 7, 9, 10. a) What is the range? b) What is the variance and standard deviation? SCP-GECC103 | 29  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Week 10 Lesson Title Probability Learning Outcome(s) Solve for the probability of a simple event LEARNING INTENT! Terms to Ponder Probability is the science of how likely events are to happen. The sample space is the set of all possible outcomes in an experiment. An event is defined as some specific outcome of an experiment. An event is a subset of the sample space. Essential Content Probability is simply how likely something is to happen. Probability is the measure of uncertainty of any event (any phenomenon that happened or bound to happen). Basic Probability The probability that an event will occur is a number between 0 and 1. In other words, it is a fraction. It is also sometimes written as a percentage because a percentage is simply a fraction with a denominator of 100. If an event is most likely, it to happen has a probability of 1. An event that will not definitely happen has a probability of 0. Sample Space and Events The sample space is the set of all possible outcomes in an experiment. SCP-GECC103 | 30  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Example: If a die is rolled, the sample space S is given by S = {1, 2, 3, 4, 5, 6} Example: If two coins are tossed, the sample space S is given by S = {HH, HT, TH, TT}, where H = head and T = tail. We define an event as some specific outcome of an experiment. An event is a subset of the sample space. Example: A die is rolled. Let us define event E as the set of possible outcomes where the number on the face of the die is even. Event E is given by E = {2, 4, 6}. The probability (P) that an event will happen is: Number of outcomes that will lead to that event P (E) = Total number of possible outcomes Example: Suppose that you are going to throw a standard dice, and you want to know what your chances of throwing a 6. Solution: In this case, there is only one outcome that leads to that event (i.e. you throw a 6), and six possible outcomes altogether (you might throw 1, 2, 3, 4, 5 or 6). The probability of throwing a six is P (6) = 1 / 6 Number of outcomes that will lead to that event P (6) = Total number of possible outcomes SCP-GECC103 | 31  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Example: Suppose that you want to know what your chances of throwing 1 or 6. There are two favorable outcomes, 1 and 6, but still six possible outcomes. Solution: The probability is therefore 2/6, which can be reduced to 1/3. SELF-SUPPORT: You can click the URL Search Indicator below to help you further understand the lessons. Search Indicator https://www.skillsyouneed.com/num/probability.html#:~:text=Probability%20an%20I ntroduction,- See%20also%3A%20Estimation&text=Probability%20is%20the%20science%20of,the% 20cards%20in%20a%20game.&text=Probability%20is%20used%2C%20for%20exampl e,cost%20of%20your%20insurance%20premiums. https://www.analyzemath.com/statistics/introduction_probability.html https://www.khanacademy.org/math/statistics-probability/probability-library/basic- theoretical-probability/a/probability-the-basics https://www.toppr.com/guides/maths/probability/probability-of-an-event/ SCP-GECC103 | 32  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached I LET’S NITIATE! Activity 1. Answer the following questions. 1. Define probability. ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 2. Define sample space and give an example. ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 3. Define an event and provide an example. ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ I LET’S NQUIRE! Activity 1. Answer the following problems. 1. Give the sample space of the suits in a deck of cards. ___________________________________________________________________ ___________________________________________________________________ 2. Give the sample space of two dice. ____________________________________________________________________ ____________________________________________________________________ SCP-GECC103 | 33  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached I LET’S NFER! Activity 1. Answer the given word problem. Show the complete solution using the formula. 1. There is a bag full of colored balls with the colors, red, blue, green and orange. Balls are picked out and replaced. John did this 1000 times and obtained the following results: Number of blue balls picked out: 300 Number of red balls: 200 Number of green balls: 450 Number of orange balls: 50 a) What is the probability of picking a green ball? b) What is the probability of picking an orange ball? c) What is the probability of getting a non-red ball? SCP-GECC103 | 34  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Week 11 Geometric designs: Recognizing and analyzing geometric Lesson Title shapes Learning Outcome(s) Identify geometric designs; Create their own geometric design LEARNING INTENT! Terms to Ponder A Geometric design is a motif, pattern, or design depicting abstract, nonrepresentational shapes such as lines, circles, ellipses, triangles, rectangles, and polygons. Essential Content Geometric designs are one of the most visually appealing forms in graphic design. They can appear in a wide variety of styles and fulfill a huge range of roles. A geometric pattern is a kind of pattern formed of geometric shapes and typically repeated like a wallpaper design. Geometric patterns consist of a series of shapes. Patterns made from shapes are similar to patterns made from numbers because the pattern is determined by a rule. The rectangle, square, triangle and circle are the basic shapes in the geometrical system. So, many geometrical designs are made with the help of these basic shapes. Example(s) of Geometric Patterns SCP-GECC103 | 35  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Geometric patterns created with Artlandia SymmetryWorks. SELF-SUPPORT: You can click the URL Search Indicator below to help you further understand the lessons. http://studyjams.scholastic.com/studyjams/jams/math/algebra/geomet ric-patterns.htm https://artlandia.com/wonderland/glossary/GeometricPattern.html https://en.wikipedia.org/wiki/Pattern#Art_and_architecture https://www.designwizard.com/blog/70-ways-to-create-amazing- geometric-designs/ https://www.google.com/url?sa=i&url=https%3A%2F%2Fwww.freepatter nsarea.com%2Fdesigns%2Frepeating-seamless-geometric-pattern-design- vectors%2F&psig=AOvVaw3rzCuG4JF_dg8KNXNTmNiv&ust=1603573909 741000&source=images&cd=vfe&ved=0CA0QjhxqFwoTCIiv4cfQy- wCFQAAAAAdAAAAABAK SCP-GECC103 | 36  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached I LET’S NITIATE! Activity 1. Answer the following questions. 1. What is a geometric design? ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 2. What shapes are commonly used in a geometric design? ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 3. How do we create a geometric design? ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ I LET’S NQUIRE! Activity 1. Do as indicated. 1. Research a geometric design and explain the components of it (shapes, colour). I LET’S NFER! Activity 1. Do as indicated. 1. Create your own geometric design. SCP-GECC103 | 37  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached SCP-TOPICS: FINAL PERIOD TOPICS Week 13 Lesson Title Simple Interest Solve real –life problems involving simple interest; Apply the Learning Outcome(s) correct formula in a specific problem At SJPIICD, I Matter! LEARNING INTENT! Terms to Ponder Interest refers to how much is paid for the use of money (as a percent, or an amount). The accrued amount of an investment is the original principal P plus the accumulated simple interest, I = Prt. Simple interest is an interest paid or computed on the original principal only of a loan or on the amount of an account Essential Content Simple interest refers to the money that can be earned by initially investing some money (the principal). A percentage (the interest) of the principal is added to the principal, making the initial investment grow. This type of interest is applicable for a short-term duration, usually in days, weeks, months or even a few years, with not so large amounts of money. SCP-GECC103 | 38  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Factors of Simple Interest There are only 3 common factors to be considered with regards to simple interest. 1. Principal. This is the amount of money being borrowed. This could be loaned from a bank or any loaning establishment or borrowed from a person. This will be the basis of how much will be paid with the additional compensation for borrowing. 2. Rate of Interest. This is the percent to be used to calculate the additional amount to be paid along with the principal. Common rates of interest ranges from 1 to 10% but it can also be higher depending on the agreement between the parties. 3. Time. This is the period from the beginning when the money was borrowed to the period that when the money should be returned with the additional amount (interest). This can also be called a term or deadline. This should properly and strictly be observed especially in huge amount of loans. Simple Interest Equation (Principal + Interest) A = P (1 + rt ) where: A = Total Accrued Amount (principal + interest) P = Principal Amount I = Interest Amount r = Rate of Interest per year in decimal; r = R/100 R = Rate of Interest per year as a percent; R = r * 100 t = Time Period involved in months or years SCP-GECC103 | 39  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached From the base formula, A = P (1 + rt ) derived from A = P + I and since I = Prt then A = P + I becomes A = P + Prt which can be rewritten as A = P(1 + rt ). Note that rate r and time t should be in the same time units such as months or years. Rate r should be in decimal form; r = R /100 Accrued value is also known as ‘future value’ DERIVED FORMULA Calculate Interest, solve for I o I = Prt Calculate Total Amount Accrued (Principal + Interest), solve for A o A = P( 1 + rt ) Calculate Principal Amount, solve for P o P = A / (1 + rt ) Calculate rate of interest in decimal, solve for r o r = (1/t) (A / P - 1) Calculate rate of interest in percent o R = r * 100 Calculate time, solve for t o t = (1/r)(A/P - 1) Example. A 2-year loan of Php500 is made with 4% simple interest. Find the interest earned. SCP-GECC103 | 40  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Solution. Identify the values given in the problem. ✓ Given: Time is 2 years: t = 2 Initial amount is Php500: P = 500 The rate is 4%. Write this as a decimal: r = 0.04 ✓ Now apply the formula: I = Prt = 500 (0.04) (2) = 40 Answer: The interest earned is Php40. Example. A business takes out a simple interest loan of Php10,000 at a rate of 7.5%. What is the total amount the business will repay if the loan is for 8 years? Solution. The total amount they will repay is the future value, A. ✓ Given t=8 r = 0.075 P = 10000 ✓ Using the simple interest formula for future value: A = P ( 1 + rt ) = 10000 (1 + 0.075 (8) ) = 16000 Answer: The business will pay back a total of Php16,000. SELF-SUPPORT: You can click the URL Search Indicator below to help you further understand the lessons. SCP-GECC103 | 41  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Search Indicator http://www.webmath.com/simpinterest.html https://www.mathsisfun.com/money/interest.html https://www.calculatorsoup.com/calculators/financial/simple- interest-plus-principal-calculator.php https://www.investopedia.com/terms/s/simple_interest.asp#:~:tex t=Simple%20interest%20is%20a%20quick,days%20that%20elapse %20between%20payments. https://www.opploans.com/oppu/articles/simple-interest- definition/ https://www.merriam-webster.com/dictionary/simple%20interest https://www.ipracticemath.com/learn/consumermath/simple- interest-intro At SJPIICD, I Matter! LET’S INITIATE! Activity 1. Answer the following questions. 4. What is an interest? ___________________________________________________________________ ___________________________________________________________________ 5. What is a simple interest? ___________________________________________________________________ ___________________________________________________________________ 6. What is an accrued amount? ___________________________________________________________________ ___________________________________________________________________ SCP-GECC103 | 42  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached I LET’S NQUIRE! Activity 1. Answer the following questions. 3. Discuss the process of solving the interest amount in a simple interest. ___________________________________________________________________ ___________________________________________________________________ 4. Discuss the process of solving the accrued amount of a simple interest. ___________________________________________________________________ ___________________________________________________________________ LET’S INFER! Activity 1. Do as indicated. 1. You are tired at the end of the term and decided to borrow Php5500 to go on a trip to Whatever Land. You go to the bank and borrow the money at 11% for 2 years. a) Find the interest you will pay on the loan. b) How much will you have to pay the bank at the end of the two years? SCP-GECC103 | 43  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Week 14 Lesson Title Compound Interest Solve real –life problems involving compound interest; Apply Learning Outcome(s) the correct formula in a specific problem LEARNING INTENT! Terms to Ponder Interest refers to how much is paid for the use of money (as a percent or an amount). Compound interest (or compounding interest) is interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods on a deposit or loan. Essential Content Compound interest is the concept of adding the accumulated interest back to the principal sum so that interest is earned on top of the interest from that moment on. The act of declaring interest to be principal is called compounding. Compound interest, or 'interest on interest', is calculated by multiplying the principal amount by one plus the annual interest rate to the power of the number of compound periods to get a combined figure for principal and compound interest. Subtract the principal if you just want the compound interest. Interest can be compounded on any given frequency schedule, from continuous to daily to annually. When calculating the compound interest, the number of compounding periods makes a significant difference. SCP-GECC103 | 44  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached The formula used in the compound interest calculator is A = P ( 1 + r/n )(nt) A = the future value of the investment P = the principal investment amount r = the interest rate (decimal) n = the number of times that interest is compounded per period t = the number of periods the money is invested for Example. An investment earns 3% compounded monthly. Find the value of an initial investment of Php5000 after 6 years. Solution. Determine what values are given and what values you need to find. Earns 3% compounded monthly: the rate is r = 0.03 and the number of times compounded each year is n = 12 Initial investment of Php5000: the initial amount is the principal, P=5000 6 years: t = 6 You are trying to find A, the future value (the value after 6 years). Now apply the formula with the known values: A = P (1 + rn)nt A = 5000 (1 + 0.0312)12×6 A ≈ 5984.74 Answer: The value after 6 years will be Php5984.74. Example. What is the value of an investment of Php3500 after 2 years if it earns 1.5% compounded quarterly? SCP-GECC103 | 45  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Solution. As before, we are finding the future value, A. In this example, we are given: Value after 2 years: t = 2 Earns 3% compounded quarterly: r = 0.015 and n = 4 since compounded quarterly means 4 times a year Principal: P = 3500 Applying the formula: A=P (1 + rn)nt A = 3500 (1 + 0.0154)4×2 A ≈ 3606.39 Answer: The value after 2 years will be Php3606.39. SELF-SUPPORT: You can click the URL Search Indicator below to help you further understand the lessons. Search Indicator https://www.mathbootcamps.com/compound-interest-formula/ https://www.mathsisfun.com/money/compound-interest.html http://www.moneychimp.com/calculator/compound_interest_calculator. htm https://www.thecalculatorsite.com/finance/calculators/compoundintere stcalculator.php https://www.investopedia.com/terms/c/compoundinterest.asp#:~:text=C ompound%20interest%20(or%20compounding%20interest)%20is%20inter est%20calculated%20on%20the,on%20a%20deposit%20or%20loan.&text =Interest%20can%20be%20compounded%20on,continuous%20to%20dail y%20to%20annually. SCP-GECC103 | 46  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached At SJPIICD, I Matter! I LET’S NITIATE! Activity 1. Answer the following questions. 1. What is an interest? ___________________________________________________________________ ___________________________________________________________________ 2. What is a compound interest? ___________________________________________________________________ ___________________________________________________________________ I LET’S NQUIRE! Activity 1. Answer the following questions. 1. Discuss the process of solving the compound interest. ___________________________________________________________________ ___________________________________________________________________ LET’S INFER! Activity 1. Do as indicated. 1. What is the compounded amount if Php4500 is invested at 1.5% compounded monthly for 2 years? 2. Calculate the compound amount of Php 3,000 invested at 4.3% compounded semi-annually for 2 years and 6 months. SCP-GECC103 | 47  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached Week 15 Lesson Title Voting Learning Outcome(s) Identify how mathematics works on voting LEARNING INTENT! Terms to Ponder Voting, from a mathematical perspective, is the process of aggregating the preferences of individuals in a way that attempts to describe the preferences of a whole group. (Source: https://brilliant.org/wiki/mathematics-of-voting/) Essential Content Whether or not voting systems are fair also depends on the way in which votes are taken. For instance, one could describe just their top choice, rank all their choices in order, or give scores to each of the possible options. Plurality Method In plurality voting, each voter can only put their support behind one candidate. If we assume that each voter will put their support behind their top choice, then that candidate wins the race. (Source: princeton.edu/~cuff/voting/theory.html) Majority Candidate The allure of the plurality method lies in its simplicity (voters have little patience for complicated procedures) and in the fact that plurality is a natural extension of the principle of majority rule the majority candidate In a democratic election between two candidates, the candidate with a majority (more than half) of the votes, should be the winner. (Source: SCP-GECC103 | 48  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached https://mari nmathcircledotorg.files.wordpress.com/2015/12/mmcadv-20120111- votinglecture-ernestodiaz.pdf) Condorcet's Paradox The scientific study of voting and elections began around the time of the French Revolution. One of the founders of the mathematical theory of voting is The Marquis de Condorcet, a French philosopher, mathematician and political scientist. He discovered a counter- intuitive result now called Condorcet’s paradox. In order for a voting system to be robust to candidates, it must elect the same winner even if any of the non-winners were not in the election. This means, among other things, that the winner must beat every other individual candidate if they were the only two candidates in the race. Thus, a voting system is only robust if it picks a winner who would beat every other candidate in a head-to-head majority vote. This candidate is called the Condorcet winner. (Source: princeton.edu/~cuff/voting/theory.html) The Borda Count Method The Borda count is a system that takes that into account. In this system, each position on the ballot is given a score. If there are three candidates it would be like this: first choice gets 2 points; second choice gets 1 point; third choice gets 0 points. (Source: princeton.edu/~cuff/voting/theory.html) SELF-SUPPORT: You can click the URL Search Indicator below to help you further understand the lessons. Search Indicator https://brilliant.org/wiki/mathematics-of-voting/ https://marinmathcircledotorg.files.wordpress.com/2015/12/mmcad v-20120111-votinglecture-ernestodiaz.pdf SCP-GECC103 | 49  ST. JOHN PAUL II COLLEGE OF DAVAO COLLEGE OF TEACHER EDUCATION DEPARTMENT Physically Detached Yet Academically Attached http://math.hws.edu/eck/math110_f08/voting.html https://www.princeton.edu/~cuff/voting/theory.html https://thatsmaths.com/2016/02/04/the-mathematics-of-voting/