MMWModule- Chapter 1 PDF

Preview

Summary

Download MMWModule- Chapter 1 PDF

Description

FM-AA-CIA-15 Rev. 0 10-July-2020

Study Guide in Mathematics in the Modern World

GE 7 Mathematics in the Modern World

Module 1 : Mathematics in our World

MODULE 1 MATHEMATICS IN OUR WORLD MODULE OVERVIEW

This module consists of two lessons: Mathematics as the Study of Patterns and Fibonacci Sequence and Golden Ratio. Each lesson was designed as a self-teaching guide. Definitions of terms and examples had been incorporated. Answering the problems in “your turn” will check your progress. You may compare your answers to the solutions provided at the later part of this module for you to be able to measure your achievement and as well as the effectiveness of the module. Individual and group activities were prepared to apply what you had learned. Exercises were prepared as your assignment to measure your understanding about the topics. MODULE LEARNING OBJECTIVES

At the end of the module, you should be able to: • Identify patterns in nature and regularities in the world • Articulate the importance of mathematics in one’s life • Argue about nature of mathematics, what it is, how it is expressed, represented, and used • Express appreciation for mathematics as a human endeavor. LEARNING CONTENTS (Mathematics as the Study of Patterns)

Introduction Look around you, do you notice anything that repeats or occur in a similar form? In your life, are there any things that you tend to do over and over again? In this lesson we will investigate patterns and regularities in nature and even in life and how mathematics come into play. At times, consciously or unconsciously you are using mathematics in some routine transactions like buying food, paying bills and even computing how much time do you need to come to class on time. And you can do all of these routine effective and efficiently using your knowledge in mathematics. You as a student taking this course, what is Mathematics for you?

Discussion Lesson 1. Mathematics as the Study of Patterns 1.1 What is Mathematics? Mathematics is defined as the study of numbers and arithmetic operations. Others describe mathematics as a set of tools or a collection of skills that can be applied to questions of “how many” or “how much”. Still, others view it as a science which involves logical reasoning, drawing conclusions from assumed premises, and strategic reasoning based on accepted rules, laws, or probabilities, Mathematics is also considered as an art which deals with form, size, and quantity. In examining the development of mathematics from historical perspective, it can be seen that much has been directed towards describing patterns of relationship that are of interest of various individuals. Patterns arouse curiosity because they can be directly related to common human experience. The focused of this section is, mathematics as a study of patterns. PANGASINAN STATE UNIVERSITY

1

Study Guide in Mathematics in the Modern World

GE 7 Mathematics in the Modern World

FM-AA-CIA-15 Rev. 0 10-July-2020 Module 1 : Mathematics in our World

A Study of Patterns Pattern is an arrangement which helps observers anticipate what they might see or what happens next . Or just simply are regular, repeated, or recurring forms or designs. We see patterns around us – layout of the floor, design of our clothes, butterflies’ wings, and even to the way we say things. Recognizing patterns is natural to us as a rational creature because our brain is hardwired to recognize them. Studying patterns help you in identifying relationships and finding logical connections to form generalizations to make predictions. Below are examples of various patterns:

Logic Patterns. Logic patterns are usually the first to be observed. Classifying things , for example comes before numeration. Being able to tell which things are blocks and which are not precedes learning to count blocks. One kind of logic pattern deals with the characteristics of various objects while another deals with order. These patterns are seen on aptitude tests in which takers are shown a sequence of pictures and asked to select which figure comes next among several choices.

Example 1

What comes Next ?

Solution: PA1 The base figure rotates at an angle of 45° in the counterclockwise direction. Hence choice C is the perfect match.

PANGASINAN STATE UNIVERSITY

2

Study Guide in Mathematics in the Modern World

GE 7 Mathematics in the Modern World

Your turn 1

FM-AA-CIA-15 Rev. 0 10-July-2020 Module 1 : Mathematics in our World

. What comes next ?

Number Patterns. Another class of patterns is the patterns of numbers. The two most common are the arithmetic and geometric patterns. While arithmetic sequence is formed by adding or subtracting a constant number to consecutive terms , geometric sequence needs to be multiplied or divide with same value each time we want to get the next term in the sequence . Examples : 3,7,11,15,…is an arithmetic sequence because it is formed by adding 4 to each term in the sequence which is called the common difference(d) . On the other hand , 3,6,12,24,…is geometric sequence because it is formed by multiplying each term by 2 to get the succeeding terms which is called the common ratio(r). Give the next three terms in the pattern. Identify which is arithmetic and geometric sequence. 1. 1 , 3, 5, 7, 9, ___, ____, ____ 2. 10, 30, 90, ____,_____,____

Example 2

Solution : a. Looking at the given numbers, the sequence is increasing, with each term being two more than the previous term : 3 = 1 + 2 ; 5 = 3 + 2; 7 = 5 + 2; 9 = 7 + 2. Therefore the answer is n arithmetic sequence and the next three terms should be 11, 13, 15. b. The sequence is a geometric sequence because it is formed by multiplying each term by 3 to the succeeding terms, so the next three terms are : 270, 810, 2430. Your tur turn n2

Give the next three terms in the pattern. Identify which is arithmetic and geometric sequence.

a. 4, 2,1,0.5, ___, _____, _____ 3 5 1 3

b. , , ,

4 8 2 8

, ____, ____, _____

Let us investigate more number patterns. Take a look at these examples retrieved from a video on

PANGASINAN STATE UNIVERSITY

3

Study Guide in Mathematics in the Modern World

GE 7 Mathematics in the Modern World

FM-AA-CIA-15 Rev. 0 10-July-2020 Module 1 : Mathematics in our World

youtube. 1×8+1=9 12 × 8 + 2 = 98 123 × 8 + 3 = 987 1234 × 8 + 4 = 9876 12345 × 8 + 5 = 98765 123456 × 8 + 6 = 987654 1234567 × 8 + 7 = 9876543 12345678 × 8 + 8 = 98765432 123456789 × 8 + 9 =?

1×1= 1 11 × 11 = 121 111 × 111 = 12321 1111 × 1111 = 1234321 11111 × 11111 = 123454321 111111 × 111111 = 12345654321 1111111 × 1111111 = 1234567654321 11111111 × 11111111 = 123456787654321 111111111 × 111111111 =?

Have you seen the pattern? If yes, without doing calculation what do you think are the answers on the last row?

Maybe you will agree that mathematics is the science of patterns and it’s all around us. Recognizing number patterns is an important problem –skill. That is one reason why those who use patterns to analyze and solve problems often find success.

Geometric Patterns. Geometric pattern is a motif or design that depicts abstract shapes like lines, polygons, and circles, and typically repeats like a wallpaper. Visual patterns are observed in nature and in art. In art, patterns present objects in a consistent, regular manner.

Example 3

Which of the figures below can be used to continue the series?

Solution:

Since it adds up two squares horizontally and vertically on each term, the correct answer is Figure 1. Your turn 3

PANGASINAN STATE UNIVERSITY

4

FM-AA-CIA-15 Rev. 0 10-July-2020

Study Guide in Mathematics in the Modern World

GE 7 Mathematics in the Modern World

Module 1 : Mathematics in our World

Draw a figure to continue the series below.

Word Patterns. Patterns can also be found in language like morphological rules in pluralizing nouns or conjugating verbs for tense, as well as the metrical rules of poetry. Each of these examples supports mathematical and natural language understanding. The focus here is patterns in form and in syntax , which lead directly to the study of language in general and digital communication in particular. Example 4

Fill in the blank.

knife:knives

life:lives

wife:______

Solution : The pattern is taking the plural form of the words involved, so wife is wives.

Your turn 4

Fill in the blank.

meet :met

lead: led

feed: ____

1.2 Patterns in Nature Patterns in nature are the regular arrangement of objects in any form found everywhereplants, animals, humans, earth formations, and many others. These include symmetries, spirals, waves, arrays, cracks, stripes, etc. Some of these patterns which recur in different context can be modelled mathematically. So, let us start looking for more patterns in nature.

Symmetric Patterns A figure has symmetry if there is a non-trivial transformation that maps the figure onto itself or you can draw an imaginary line across the object and the resulting parts are mirror images of each other. For example, a square has a vertical line symmetry. That is , the reflection about this line maps the square onto itself.

PANGASINAN STATE UNIVERSITY

5

Study Guide in Mathematics in the Modern World

GE 7 Mathematics in the Modern World

FM-AA-CIA-15 Rev. 0 10-July-2020 Module 1 : Mathematics in our World

Notice that left and right portion of the square are exactly the same. The type of symmetry, known as line, or bilateral symmetry, which is evident in most animals, including humans . Example is the figure of the butterfly and Leonardo da Vinci’s Vitruvian Man.

Leonardo da Vinci’s Vitruvian Man is showing the proportions and symmetry of human body

In fact there are other types of symmetry depending on the number of sides or faces that are symmetrical. Take a look at the images below.

PANGASINAN STATE UNIVERSITY

6

FM-AA-CIA-15 Rev. 0 10-July-2020

Study Guide in Mathematics in the Modern World

GE 7 Mathematics in the Modern World

Spiderwort with a three-fold symmetry

Module 1 : Mathematics in our World

Starfish has a five-fold symmetry

If a figure or object can be rotated less than 360° about a point so that the image and the pre-image are indistinguishable , then the figure has rotational symmetry. The smallest angle that a figure can be rotated while still preserving the original formation is called angle of rotation. Angle of rotation can be computed using the following formula: 360° 𝐴𝑛𝑔𝑙𝑒 𝑜𝑓 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 = 𝑛 where 𝑛 is the number of rotational symmetry or order of rotation of a particular object. For the spiderwort, the angle of rotation is 120° while the angle of rotation of the starfish is 72°. Consider the image of a snowflake.

It can be observed that the patterns of snowflake repeat six times. So, what is the angle of rotation of the snowflake?

Let us try to see more patterns in nature by watching this video.

PANGASINAN STATE UNIVERSITY

7

Study Guide in Mathematics in the Modern World

GE 7 Mathematics in the Modern World

FM-AA-CIA-15 Rev. 0 10-July-2020 Module 1 : Mathematics in our World

Watch this Video Why do honey bees love hexagons - by Zack Patterson and Andy Peterson https://www.youtube.com/results?search_query=why+do+honey+bees+love+hexagons,

What you’ve watch is another interesting pattern in nature , the honeycomb. According to MerriamWebster dictionary,”a honeycomb is a mass of hexagonal wax cells built by honeybees in their nest to contain their brood and stores honey. “, But why build hexagonal cells? Why not squares or any other polygons?

The video had explained it well. They love it because more area will be covered using hexagon compared to other polygons. Hexagonal formations are more optimal in making use of avail space. These referred to as packing problem. Packing problems involve finding the optimum method of filling up a given space such as cubic or spherical container. The bees have instinctively found that the best solution, evident in the hexagonal construction of their hives. Let us illustrate this mathematically. Suppose you have circles of radius 1 cm, each of which will then have an area of 𝜋 𝑐𝑚2 . We are then going to fill a plane with these circles using square packing and hexagonal packing. Anna

For square packing, each square will have an area of 4 𝑐𝑚2 . Note from the figure that for each square, it can fit only one circle (4 quarters). The percentage of the square’s area covered by circles will be 𝜋𝑐𝑚2 area of the circles × 100% = × 100% = 78.54% area of the square 4𝑐𝑚2 Anna

PANGASINAN STATE UNIVERSITY

8

Study Guide in Mathematics in the Modern World

GE 7 Mathematics in the Modern World

FM-AA-CIA-15 Rev. 0 10-July-2020 Module 1 : Mathematics in our World

Now, for the hexagonal packing, we can think of each hexagonal as composed of six equilateral triangles with side equal to 2 cm.

The area of each equilateral triangle can be computed using the formula 𝐴 = 𝐴=

(𝑠𝑖𝑑𝑒)2 ⋅ √3 (2𝑐𝑚)2 ⋅ √3 4𝑐𝑚2 ⋅ √3 = √3𝑐𝑚2 = = 4 4 4

𝑠2 ⋅√3 4

, so

This gives the area of the hexagon as 6√3𝑐𝑚2 (multiplying the area of the equilateral triangle to 6 as the number of sides of the hexagon). Looking at figure , there are 3 circles that could fit inside one hexagon (the whole circle in the middle, and 6 one thirds of a circle), which gives the total areas of 3𝜋𝑐𝑚2 . The percentage of the hexagon’s are covered by circles will be area of the circles 3𝜋𝑐𝑚2 × 100% = × 100% = 90.69% area of the hexagon 6√3𝑐𝑚2 Comparing the two percentages, we can clearly see that using hexagons will cover a larger area than when using squares.

1.3. World Population Rapid population growth has been a temporary phenomenon in many countries. As of 2017, it is estimated that the world population is about 7.6 billion. World leaders, sociologists, anthropologist are interested in studying population, including its growth.

PANGASINAN STATE UNIVERSITY

9

FM-AA-CIA-15 Rev. 0 10-July-2020

Study Guide in Mathematics in the Modern World

GE 7 Mathematics in the Modern World

Module 1 : Mathematics in our World

The United Nations World Population Prospects 2019 report paints a demographic picture of a very different world by the end of the 21st century

Mathematics can be used to model population growth. The formula for exponential growth is where

𝐴 = 𝑃𝑒 𝑟𝑡

A = the size of the population after it grows. P = initial number of people r = is the rate of growth t = time 𝑒 = Euler’s constant with an approximate value of 2.718

Example 5

The exponential growth model of 𝐴 = 30 𝑒 0.02𝑡 describes the population of a city in the Philippines in thousands, 𝑡 years after 1995.

a. What was the population of the city in 1995 ? b. What will be the population in 2017?

Solution a. Since our exponential growth model describes the population 𝑡 years after 1995, we consider 1995 as 𝑡 = 0 and then solve for 𝐴, our population size. 𝐴 = 30𝑒 0.02𝑡 𝐴 = 30𝑒 (0.02)(0)

Replace 𝑡 with 𝑡 = 0.

𝑒0 = 1 𝐴 = 30𝑒 0 𝐴 = 30(1) 𝐴 = 30 Therefore, the city population in 1995 was 30,000. b. We need to find 𝐴 for the year 2017. To find 𝑡 , we subtract 2017 and 1995 to get 𝑡 = 22, which we then plug in to our exponential growth model. 𝐴 = 30𝑒 0.02𝑡

PANGASINAN STATE UNIVERSITY

10

FM-AA-CIA-15 Rev. 0 10-July-2020

Study Guide in Mathematics in the Modern World

GE 7 Mathematics in the Modern World

Module 1 : Mathematics in our World

0.02(22)

𝐴 = 30𝑒 Replace 𝑡 with 𝑡 = 22. 0.0.44 𝐴 = 30𝑒 𝐴 = 30(1.55271) 𝑒 0.0.44 ≈ 1.55271 𝐴 = 46.5813 Therefore, the city population would be about 46,581 in 2017

Your turn 5

The exponential growth model 𝑨 = 𝟓𝟎𝒆𝟎.𝟎𝟕𝒕 describes the population of a city in the Philippines in thousands, t years after 1997.

a. What is the population after 20 years? b. What is the population in 2037?

LEARNING POINTS Mathematics as the Study of Patterns Pattern is an arrangement which helps observers anticipate what they might see or what happens next . Or just simply are regular, repeated, or recurring forms or designs. Example of various patterns are : logic patterns, number patterns, geometric patterns, word pattern. Patterns in nature are the regular arrangement of objects in any form found everywhere-plants, animals, humans, earth formations, and many others Exponential Growth Model Population can be modeled by the exponential growth formula 𝐴 = 𝑃𝑒 𝑟𝑡 LEARNING ACTIVITY 1

1. Select a suitable figure from the four alternatives that would complete the figure matrix. Encircle the letter corresponding to the missing pattern.

a.

b.

PANGASINAN STATE UNIVERSITY

11

FM-AA-CIA-15 Rev. 0 10-July-2020

Study Guide in Mathematics in the Modern World

GE 7 Mathematics in the Modern World

Module 1 : Mathematics in our World

c.

d.

e.

f.

2. Calculate 2 + 4 + 6 + ⋯ + 2𝑛 for 𝑛 = 1,2, . .6. 3. Calculate 1 + 3 + 5 + ⋯ + (2𝑛 − 1) for 𝑛 = 1,2, . . ,6. 4. Calculate 1 + 3 + 7 + ⋯ + (2𝑛 − 1) for 𝑛 = 1,2, . . ,6. 5. What is the missing number in each of these sequences? a) _______, 17 , 15, 13, … b) 8, 11, ______, 17 , … c) 5, ______, 27, 38, …. d) 84, _____, 76, 72,… e) 98, 109, ______, 131, … 6. Determine the pattern and find out the numbers which will complete the sequence. a. 58, 68, 57, 67, 56 , __________ b. 3, 4, 6, 10, 18, ___________ c. 10, 54, 98, 1312, 1716 _________ 7. Draw Fig.5 following the given pattern.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

a. If the length of the side of each triangle is 1 unit, what is the perimeter of each figure in the pattern?Complete the table below. Figure 1 2 3 4 5 6 7 PANGASINAN STATE UNIVERSITY

12

FM-AA-CIA-15 Rev. 0 10-July-2020

Study Guide in Mathematics in the Modern World

GE 7 Mathematics in the Modern World

Module 1 : Mathematics in our World

Perimeter 3 b. If this pattern continues; what is the perimter of Fig. 9. c. What is the perimeter of the 𝑛𝑡ℎ figure . 8. Fill the table below to complete the power of 3. 31 32 33 34 35 36 37 38 3 9 a. Enumerate all the numbers in the power of 3 which are in the ones digit, and then find the next 7 terms in the sequence without getting the power of 3. Figure Ones digit

31

32

3

9

33

34

35

36

37

38

39

310

311

312

313

314

315

b. Have you seen the pattern? Find the number in the ones digit for the following power of 3. 316 = ____________; 317 = ____________; 318 = ____________; 319 = ____________; 320 = ____________; Explain the pattern. 9. The population of a certain localit...