Math in the Modern World - Module 1 (Sequences) PDF

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GE 1 – Mathematics in the Modern World Author: Mary Jane B. Calpa Section 1: The Nature of Mathematics 1 Mathematics in our World (A Study of Patterns) OverviewWelcome to the first module of GE 1 (Mathematics in the Modern World)!This course begins with an introduction to the nature of mathematics a...

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Module Section 1: The Nature of Mathematics

GE 1 – Mathematics in the Modern World Author: Mary Jane B. Calpa

1.1 Mathematics in our World (A Study of Patterns) Overview

Welcome to the first module of GE 1 (Mathematics in the Modern World)! This course begins with an introduction to the nature of mathematics as an exploration of unseen patterns in nature and environment, a rich language in itself governed by logic and reasoning, and an application of inductive and deductive reasoning. Section 1 is composed of the following: 1.1 Mathematics in our World; 1.2 Mathematics Language and Symbols; and 1.3 Problem Solving and Reasoning. These topics will allow students to go beyond the typical understanding of mathematics as purely a bunch of memorized formulas and duplicated mathematical computations, but as a powerful tool used to understand better the world around us. Moreover, we will discuss and argue about the nature of mathematics, what it is, and how it is expressed, represented, and used. We will study mathematics as a language in order to read and write mathematical texts and communicate ideas with precision and conciseness. We will also justify statements and arguments made about mathematics and mathematical concepts using different methods of reasoning. Mathematics has always been perceived as a study of numbers, symbols, and rules. It is an art of geometric shapes and patterns, a tool in decision-making and problem solving. It has a language that differs from the ordinary speech. It is done with curiosity, with a penchant for seeking patterns and generalities, with the desire to know the truth, with trial and error, and without the fear of facing more questions and problems to solve. The following diagram shows the very nature of mathematics. study of patterns

art

language

Mathematics is a/an … set of problemsolving tools

process of thinking

Nocon, R. & Nocon, E.

In this module, we will focus on Lesson 1.1 - Mathematics in our World (A Study of Patterns). The lesson is anchored by the following core idea: Mathematics is a useful way to think about nature and the world. Our intention is to observe things, in both in nature and the world, through pattern-seeking, understand the substantial interconnection and relationship of the mathematics and the world, and appreciate mathematics as a discipline full of essence and beauty

Department of Mathematics, College of Science, University of Eastern Philippines

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Module Section 1: The Nature of Mathematics

GE 1 – Mathematics in the Modern World Author: Mary Jane B. Calpa

Learning Outcomes After working on this module, you will be able to: 1. identify patterns in nature and irregularities; 2. articulate the importance in mathematics in one’s life; 3. argue about mathematics, what it is, how it is expressed, represented, and used; and 4. express appreciation for mathematics as a human endeavor.

Activities To Do 1.) Watch the video “Nature by Numbers” by Cristóbal Vila (link: https://vimeo.com/9953368) and write one (1) sentence that describes your impression after watching the video. 2.) Identify pattern/s observed in the pictures. (a)

(b)

(d)

(e)

(c)

Sources: (a) https://www.library.illinois.edu/mtx/2018/10/09/mathematics-in-nature/; (b); https://www.weareteachers.com/teacherdresses-ms-frizzle/ (c) https://www.smithsonianmag.com/science-nature/science-behind-natures-patterns-180959033/;(d) and (e) http://mustafacil-online.blogspot.com/2015/08/manmade-patterns.html

Questions To Pon Ponder der The video and the pictures leave us questions to think about. • What are the different kinds and forms of patterns you have seen in the video and/or pictures? • How does these patterns help us understand the connection between our world and mathematics?

Department of Mathematics, College of Science, University of Eastern Philippines

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GE 1 – Mathematics in the Modern World Author: Mary Jane B. Calpa

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Module Section 1: The Nature of Mathematics

Patterns and Numbers in Nature and the World When we buy clothes, accessories, furniture, house decorations, and other similar objects, we tend to look for beautiful geometric designs or patterns. We appreciate the patterns seen in the colorful wings of butterflies, the arrangement of flowers and leaves, the reflection of the mountain tops to the clear waters of lakes, the different shapes of clouds in the skies, and other patterns seen in the nature. In the busy streets of the cities, we are impressed by the intricate but well-designed modern homes and high-rise buildings. We are wowed by nature and man-made creations because of these repeated designs of geometric visuals. Repeated ways or occurrences that happens or was done are also considered as patterns. For example, the cycle of the moon, the changing seasons, and even the transmission pattern of the COVID 19 pandemic. Patterns surround us. It is everywhere and are in every people’s task or activity.

Mathematics, developed by human mind and culture, is a formal system of thought for recognizing, classifying, and exploiting patterns. (Stewart, I.). Mathematics is indeed a study of patterns. Results in mathematics are brought by the generalizations of patterns. The study of patterns allows us to observe and identify relationships, discover logical connections, and make generalizations. Moreover, the use and study of patterns allows us to be logical thinkers and better problem solvers. Now, let us take a look of some of these patterns.

Example

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Logic Pattern Choose the figure that completes the pattern. 1.

?

_________ A

B

C

D

A

B

C

D

2.

Sources: (1) and (2) hhtp://www.jobtestprep.co.uk;

Solution 1. D The sketch is being built stage by stage. A new line is added in each stage and it never touches the last line added in the previous stage. 2. B Each figure consists of 3 shapes; namely: external shape, middle shape, and inner shape. Notice that the external shape appears to be the middle shape of the next figure. The middle shape disappears in the next figure. While the inner shape appears to be as the external shape of the figure two steps forward. For example: The external shape of the 4th figure is a circle, The middle shape is a pentagon, and the inner shape is a hexagon. The circle becomes the 4th figure middle shape of the next figure, the pentagon disappears in the next figure, and the hexagon becomes the external shape of the figure two steps forward from the 4th figure. Department of Mathematics, College of Science, University of Eastern Philippines

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Module Section 1: The Nature of Mathematics

GE 1 – Mathematics in the Modern World

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Number Pattern Find the next number in the sequence.

Example

Author: Mary Jane B. Calpa

1. 3, 8, 13, 18, 23, ? 2. 1, 4, 9, 16, 25, 36, ? Solution 1. For this sequence, the difference between each term is 5. 3, 8, 13, 18, 23, __ ⋁ ⋁ ⋁ ⋁ ⋁ 5 5 5 5 5 Thus, the next number is 28 (23 + 5). 2. The next number in the sequence can be determined in two ways: (1) The sequence 1, 4, 9, 16, 25, 36 can be written as 12, 22, 32, 42, 52, 62. Thus, the next number is 72, that is, 49 49. (2) For this sequence, the difference 1, 4, 9, 16, 25, 36, __ ⋁ ⋁ ⋁ ⋁ ⋁ ⋁ 3 5 7 9 11 13 3 + 2 = 5; 5 + 2 = 7; 7 + 2 = 9; 9 + 2 = 11; 11 + 2 = 13 Thus, the next number is 49 .

Examples 1 and 2 are usually seen on aptitude tests. Before we determine the next shape or number, we have to observe the objects, look into their properties, and their relationship on other objects. In such a way, we are allowed to hypothesize, predict, and construct generalizations based on the observed patterns. Patterns, such as geometric and word patterns, are also very common to us. Word patterns focused on the morphological rules in pluralizing nouns, conjugating verbs for tense, and metrical rules of poetry. Examples: baby: babies buy: bought trolley: trollies bring: brought ally: ? catch: ?

answer: allies

answer: caught

https://newsinfo.inquirer.net/941295/bat ok-tattooing-tattooing-mambabatok

https://www.our7107islands.com/basey-samarthe-new-banig-capital-of-the-philippines/

http://alvicsbatik.weebly.com/ mindanao-accessories--page2.html

While geometric patterns are designs that depict geometric shapes like lines, circles, and polygons. Geometric patterns are observed in nature. These patterns are also associated to the identification of a particular country and culture. Below are samples of geometric patterns that are associated to Philippine ethnic groups and local regions.

(1)

(3)

(1) Tattoos in the Cordillera (2) Woven mat “banig” in Basey, Samar (3) T’boli belt made of beads (2)

Department of Mathematics, College of Science, University of Eastern Philippines

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Module Section 1: The Nature of Mathematics

GE 1 – Mathematics in the Modern World Author: Mary Jane B. Calpa

Self-Assessment Activity 1 I. For each set of figures, what comes next? 1.

A

B

C

B

C

D

2.

A

D

3.

A

B

C

D

II. What is the next number in the series? 4. 3, 6, 12, 24, 48, ? 5. 1, 4, 10, 22, 46, ? 6. 4, – 1, – 11, – 26, – 46, ? Sources: (1) and (2) hhtp://www.jobtestprep.co.uk; (3) www.psychometric-success.com

5 Answers to SAA 1: 1. C, 2. B, 3. C, 4. 96, 5. 94, 6. –71

Department of Mathematics, College of Science, University of Eastern Philippines

GE 1 – Mathematics in the Modern World Author: Mary Jane B. Calpa

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Module Section 1: The Nature of Mathematics

The world and the universe are full of beautiful patterns and designs that are mathematical in nature. Let us take a closer look on some of these patterns in nature and the world. Symmetry and Ang Angle le of Rotation Consider the figure below.

Leonardo da Vinci’s Vitruvian Man is an image of proportion and symmetry of the human body.

https://sotafoundations9.wordpress.com/2018 /03/08/transferring-your-design-using-radialsymmetry/#jp-carousel-89

http://www.bio.miami.edu/dana/226/226F 09_22.html

ssions/clabaugh/today/health.html

https://leonardodavinci.stanford.edu/submi

When you draw an imaginary line and the resulting parts are mirror images of each other, we have shown a symm symmetry etry etry. The A figure above is symmetric about the axis indicated by the broken line. This is called as line or bilateral symmetry and is common to animals and humans. Here are other images showing symmetry.

This flower has a three-fold symmetry.

This starfish has a five-fold symmetry.

Images Source: https://biologydictionary.net/radialsymmetry/

Observe that if we rotate the flower and the starfish by several degrees, we can still have the same appearance as the original position. This is called the rotation otational al symmetry symmetry. The smallest angle an object can be rotated while it is preserving its original formation is called the angle of rotation rotation. 1 A figure has a rotational symmetry of order 𝑛 (𝑛-fold rotational symmetry) if of a complete 𝑛 turn leaves the figure unchanged. To compute for the angle of rotation, we use 360° 𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 = 𝑛 For example, This three-leaf clover has a 3-fold 120o symmetry. The angle of rotation is 120O. 120o 90o

120o

This four-leaf clover has a 4-fold symmetry. The angle of rotation is 90O.

Department of Mathematics, College of Science, University of Eastern Philippines

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Module Section 1: The Nature of Mathematics

GE 1 – Mathematics in the Modern World Author: Mary Jane B. Calpa

https://listverse.com/2013/04/21/10-beautifulexamples-of-symmetry-in-nature/

Snow Snowflakes flakes and Honeyc Honeycombs ombs Look into a microphotographed snowflake below.

https://www.benefits-of-honey.com/honeycombpattern.html#:~:text=Studies%20on%20the%20geomet ry%20of,and%20square%20makes%20smaller%20area.

examples-of-symmetry-in-nature/

https://listverse.com/2013/04/21/10-beautiful-

Notice that it exhibits a pattern on each arm that repeats six times. This snowflake indicates a six-fold symmetry. However, many snowflakes are not perfectly symmetric due to the effects of the different atmospheric conditions such as temperature and humidity on the ice crystals as it forms when they descend from the skies. The angle of rotation for the snowflake with a 6-fold symmetry is 60O. Humans are also marveled with the almost perfect hexagonal shape arrangements in honeycombs.

Peacock’s Tail The patterns exhibited in animal’s external appearance has to do with their growth; their survival; and even with their chances to attract their mates. Symmetric and repeated patterns, enhanced with bright, beautiful colors, on the feathers of a peacock’s tail are used to attract their mates.

https://www.benefits-of-honey.com/honeycombpattern.html#:~:text=Studies%20on%20the%20geomet ry%20of,and%20square%20makes%20smaller%20area.

The image on the right explains why mathematicians believed that hexagon is the most effective way of storing honey. The hexagonal formation allows bees to store the largest possible amount of honey with the use of the least amount of wax.

Department of Mathematics, College of Science, University of Eastern Philippines

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Module Section 1: The Nature of Mathematics

GE 1 – Mathematics in the Modern World Author: Mary Jane B. Calpa

https://www.benefits-of-honey.com/honeycombpattern.html#:~:text=Studies%20on%20the%20geomet ry%20of,and%20square%20makes%20smaller%20area.

Sunflower Nature has gifted us with beautiful flowers. The brilliant colors, fragrant odors, petal arrangements, and different sizes and number of petals make flowers very appealing. If we closely observe these flowers, we can find interesting patterns. For example, let us take a closer look on the orderly arrangement of sunflower seeds. We can see clockwise and counterclockwise spirals extending outward from the center of the flower. Moreover, the sunflower seed arrangement displays a very interesting numerical sequence called the Fibonacci sequ sequence ence ence. The Fibonacci sequence is 1, 1, 2, 3, 5, 8, 13, 21, … and so on. Each number on the sequence is determined by adding the two preceding numbers. The number of seeds spirals in a sunflower adds up to a Fibonacci number. Spirals of many plants such as pineapple and pinecones also add up a Fibonacci number.

https://www.bigwalls.net/climb/camf/index.html

https://www.benefits-of-honey.com/honeycombpattern.html#:~:text=Studies%20on%20the%20geomet ry%20of,and%20square%20makes%20smaller%20area.

Nautilus Shell Another example that shows how nature seems to follow a certain set of rules governed by mathematics is spiral patterns seen in a shell of a nautilus.

As the mollusk grows inside the shell, the shell also expands and attempts to maintain the same proportional shape as it grows outward. This growth pattern results to refined spirals on the shell which is very evident when it is sliced. The image on the right is called the logarithmic spiral, also known as equiangular spirals. The image shows a mathematical curve which has the property of maintaining a constant angle between the radius and the tangent to the curve at any point on the curve. Equivalently, the property states that as the distance from the spiral center increases (radius), the amplitudes of the angles formed by the radii to the point and the tangent to the point remain constant.

Department of Mathematics, College of Science, University of Eastern Philippines

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Module Section 1: The Nature of Mathematics

GE 1 – Mathematics in the Modern World Author: Mary Jane B. Calpa

World Population According to the estimates of Worldometers (https://www.worldometers.info), the Philippines will be rank 13 on the Most Populous Countries in year 2050, with population of 144, 488, 158. Questions like “How is this estimate being computed?”; “What are the factors used in the computation of the estimate?” may arise. These questions can be answered by the mathematical model of population growth. The formula for exponential growth is, 𝐴 = 𝑃𝑒 𝑟𝑡 where: 𝐴 = size of the population after it grows; 𝑃 = initial number of people; 𝑟 = rate of growth; 𝑡 = time; 𝑒 ≈ 2.718 (This is the Euler’s constant with an approximate value of 2.718)

Example

2

Determine what is being asked in each problem. 1. Substitute the given values in the formula 𝐴 = 𝑃𝑒𝑟𝑡 to find the missing quantity. a. 𝑃 = 680,000; 𝑟 = 12% per year; 𝑡 = 8 years b. 𝐴 = 731,093; 𝑃 = 525,600; 𝑟 = 3% per year 2. In the midyear of 2020, a country’s population is 109,581,078 with a growth rate of approximately 1.35% per year. What will be the country’s population in 2050? 3. The exponential growth model 𝐴 = 25𝑒 0.02𝑡describes the population of a town in Northern Samar in thousands, 𝑡 years after 1998. What was the population of the town in 1998? Solution 1. a. To find the missing quantity, 𝐴, we substitute the given values to the formula: 𝐴 = 𝑃𝑒 𝑟𝑡 = (680,000) 𝑒 (0.12)(8) we let 𝑟 = 12% = 0.12 0.96 𝐴 = (680,000) 𝑒 𝐴 = (680,000) (2.611696) 𝑒 0.96 ≈ 2.611696 𝑨 = 1,775,953 b. To find the missing quantity, 𝑡, we use the formula of 𝐴 to derive a formula for 𝑡. 𝐴 Dividing both of the formula by P to isolate 𝑒 𝑟𝑡 . 𝐴 = 𝑃𝑒 𝑟𝑡 ⇔ 𝑃 = 𝑒 𝑟𝑡 731,093 = 525,600

𝑒(0.03)𝑡

1.390968 = 𝑒 (0.03)𝑡 𝑙𝑛 1.390968 = 0.03𝑡 0.330000= 0.03𝑡 11 = 𝒕

Substitute the given values and let 𝑟 = 3% = 0.03 731,093

≈ 1.390968 Definition of logarithm 𝑙𝑛 1.390968 ≈ 0.330000 Divide both sides by 0.03

525,600

Checking of the solution by substitution of the given values and the obtained answer will serve as your exercise.

2. Given are the following quantities: 𝑃 = 109,581,078; 𝑟 = 1.35% = 0.0135; 𝑡 = 30 years (Subtract: 2050 – 2020) 𝐴 = 𝑃𝑒 𝑟𝑡 = (109,581,078)𝑒 (0.0135)(30) = 164,295,239. Thus, the population of the country in year 2050 is estimated to be 164, 64,295, 295, 295,239 239 . 3. Since the exponential growth model describes the population 𝑡 years after 1998, we consider 1998 as 𝑡 = 0 year and solve for the population size. 𝐴. 𝐴 = 25𝑒 0.02𝑡 = 25𝑒 0.02(0) = 25𝑒 0 = 25(1) = 25 Therefore, the population of the town in 1998 is 25,000. Department of Mathematics, College of Science, University of Eastern Philippines

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GE 1 – Mathematics in the Modern World Author: Mary Jane B. Calpa

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Module Section 1: The Nature of Mathematics

Self-Assessment Activity 2 Answer completely. (Use 6 significant digits in the approximated values) 1. Find the missing quantity in the formula 𝐴 = 𝑃𝑒 𝑟𝑡 by substitution of the given values: a. 𝑃 = 505,050; 𝑟 = 5% per year; 𝑡 = 1 year b. 𝑃 = 240,100; = 11% per year; 𝑡 = 10 years c. Find 𝑟 correct to 4 significant digits. 𝐴 = 786,000; 𝑃 = 247,000; 𝑡 = 17 years 2. The exponential growth model 𝐴 = 45𝑒 0.19𝑡 describes the population of a city in the Philippines in thousands, t years after 1995. a. What is the population of the city in...