Module 1/Week 1
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Program
Mathematics in the Modern World
Semester of A.Y. 2020-2021
Introduction
Mathematics in the Modern World is about mathematics as a system of knowing or
understanding our surroundings. It is similar to an English literature textbook, an introduction to
philosophy textbook, or perhaps an introductory psychology textbook. Each of those books provides
glimpses into the thoughts and perceptions of some of the world’s greatest writers, philosophers, and
psychologists. Reading and studying their thoughts enables us to better understand the world we
inhabit.
COURSE MODULE
Rationale
Mathematics in the Modern World provides glimpses into the nature of mathematics and how it
is used to understand our world. The general purpose of this subject is to explore those facets of
mathematics that will strengthen your quantitative understandings of our environments. I hope you
enjoy the journey.
Intended Learning Outcomes
A. Identify pattern and understand the rule behind its construction.
B. Extend a pattern and also construct a new pattern using various objects.
C. Recognize examples of the Fibonacci sequence in nature.
Activity
What‘s the Rule?
Name:
Direction: Read each pattern, figure out the rule and write it down.
Example: 2, 4, 6, 8, 10, 12 the rule is add 2
Pattern
1. 650, 640, 630, 620, 610
2. 13, 11, 9, 7, 5
3. 20, 26, 32, 38, 44
4. 29, 23, 17, 11, 5
5. 16, 24, 32, 40, 48
6. 5, 9, 13, 17, 21, 25
7. 23, 34, 45, 56, 67
8. 42, 35, 28, 21, 14
9. 28, 30, 32, 34, 36,
10. 1, 1, 2, 3, 5, 8
Rule
Bonus: Figure out the rule and write it down and continue the pattern
5, 10, 20, 40, 80, 160, ___, ___, ___
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Paul John B. Panganiban, LPT
Module 1/Week 1
Nature of Mathematics
The emergence of digital technology has sparked a monumental rise in the rate at which we
consume and produce data. Most people do the same routine tasks every day and the fundamental
concepts that make these activities possible are often overlooked. Do you also notice patterns around
you? What other examples can you think of?
As rational creatures, we tend to identify and follow patterns, whether consciously or
subconsciously. Recognize patterns feel natural, like our brain is hardwired to recognize them. Early
humans recognized the repeating interval of day and night, the cycle of the moon, the rising and falling
of tides, and the changing of the seasons. Awareness of these patterns allowed humans to survive. In a
similar fashion, many flora and fauna also follow certain patterns such as the arrangement of leaves and
stems in a plant, the shape of a snowflake, the flowers’ petal, or even the shape of a snail’s shell.
Check Your Progress 1
1. Do you also notice patterns around you?
2. What other examples can you think of?
COURSE MODULE
Patterns and Numbers in Nature and the World
We see patterns every day – form the layout of floor tiles, designs of
skyscrapers, to the way we tie our shoelaces. Studying patterns help
students in identifying relationships and finding logical connections to
form generalizations and make predictions.
Patterns are regular,
repeated, or recurring
forms or designs.
Let’s take a look at this pattern
?
A
B
What do you think will be the next figure in the pattern above?
Solution
Looking at the given figures, the lines seem to rotate at 90 – degrees intervals in a counter
clockwise direction, always parallel to one side of the square. Hence, either A or B could be the answer.
Checking the other patterns, the length of the lines inside the square follow a decreasing trend. So
again, either A or B could be the answer. Finally, looking at the number of the lines inside the box, each
succeeding figure has the number of lines increase by 1. This means that the next figure should have
five lines inside. This leads to option A as the correct choice.
Check Your Progress 2
1. What number comes next in 3, 5, 7, 9, ___?
Explain your answer
2. What number comes next in 1, 4, 9, 16, 25, ____?
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Paul John B. Panganiban, LPT
Module 1/Week 1
Snowflakes and Honeycombs
Recall that symmetry
indicates that you can
draw an imaginary line
across an object and the
resulting parts are mirror
images of each other.
The figure above is symmetric about the axis indicated by the dotted line. Note that the left and
right portions are exactly the same. This type of symmetry known as line or bilateral symmetry is
evident in most animals, including humans. Look in a mirror and see how the left and right sides of your
face closely match.
COURSE MODULE
Examples of symmetry:
Order of Rotation
A figure has a rotational symmetry of order n(n-fold rotational symmetry) if 1/n of a complete
turn leaves the figure unchanged. To
compute for the angle of rotation, we
The smallest angle that a
use the following formula:
Angle of Rotation =
figure can be rotated
while still preserving the
original formation
360°
𝑛
World Population
As of 2017, it is estimated that the world population is about 7.6 billion. World leaders,
sociologist, and anthropologists are interested in studying population, including its growth. Mathematics
can be used to model population
growth.
A is the size of the
population
P is the initial number
of people
r is the rate of growth
t is time
Formula for Exponential
Growth
𝐴 = 𝑃𝑒 𝑟𝑡
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Paul John B. Panganiban, LPT
Module 1/Week 1
Example1. Population Growth
The exponential growth model A = 3
thousands, t years after 1995.
0 02
describes the population of a city in the Philippines in
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 = 3 𝑒 0 02𝑡
𝐴=3 𝑒
Replace t with t = 0
(0 02)(0)
𝑒 is understood as Euler number
𝐴 = 3 𝑒0
𝑒0 = 1
COURSE MODULE
𝐴 = 3 (1)
𝐴=3
Therefore, the city population is 1995 was 30,000
b. We need to find A for the year 20-17. To find t, we subtract 2017 and 1995 to get t = 22, which we
then plug in to our exponential growth model.
𝐴 = 3 𝑒 0 02𝑡
Replace t with t = 2017 – 1995 = 22
𝐴 = 3 𝑒 (0 02)(22)
𝐴 = 3 𝑒 0 44
𝑒 0 44 = 1 55271
𝐴 = 3 (1 55271)
𝐴 = 46 5813
Therefore, the city population would be about be about 46, 581 in 2017.
Check Your Progress 3
The exponential growth model
=5
0 07
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?
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Paul John B. Panganiban, LPT
Module 1/Week 1
The Fibonacci sequence
In mathematics we can generate patterns by performing one or several mathematical
operations repeatedly. Suppose we choose the number 3 as the first number in our pattern. We then
choose to add 5 to our first number, resulting in 8, which is our second number. Repeating this process,
we obtain 13, 18, 23, and 28 … as the succeeding numbers that form our pattern. In mathematics, we
call these ordered lists of numbers a sequence.
Sequence
A sequence is an ordered list of numbers, called terms, which may have repeated values. The
arrangement of these terms is set by a definite rule.
Example2. Generating a Sequence
Analyze the given sequence for its rule and identify the next three terms.
a. 1, 10, 100, 1000
b. 2, 5, 9, 14, 20
COURSE MODULE
Solution
a. Looking at the set of n umbers, it can be observed that each term is a power of 10: 1 =1 0 , 10
=1 1 , 100 =1 2 , and 1000 =1 3 . Following this rule, the next three terms are: 1 4 = 10,000, 1 5 =
100,000, and 1 6 = 1,000,000.
b. The difference between the first and second terms (2 and 5) is 3. The difference between the
second and third terms (5 and 9) is 4. The difference between the third and fourth and the fifth terms
is 6. Following this rule, it can be deduced that to obtain the next three terms are 20 + 7 = 27, 27 + 8 =
35, 35 + 9 = 44.
Check Your Progress 4
Analyze the given sequence for its rule and identify the next three terms.
a. 16, 32, 64, 128
b. 1, 1, 2, 3, 5, 8
Fibonacci sequence
It is named after the Italian mathematician Leonardo of Pisa, who was better known by his
nickname Fibonacci.
Fibonacci sequence has many interesting properties. Among these is that this pattern is very
visible in nature. Some of nature’s most beautiful patterns, like the spiral arrangement of sunflower
seeds, the number of petals in a flower, and the shape of a snail’s shell – things that we looked at
earlier in this module all contain Fibonacci numbers. It is also interesting to note that the ratios of
successive Fibonacci numbers approach the number ɸ (Phi), also known as the Golden Ratio.
The Fibonacci sequence is the series of numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, ...
The next number is found by adding up the two numbers before it:
- the 2 is found by adding the two numbers before it (1+1),
- the 3 is found by adding the two numbers before it (1+2),
- the 5 is (2+3),
- and so on!
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Paul John B. Panganiban, LPT
Module 1/Week 1
Test Yourself
Name: _______________________________________________
Course/year: __________________________________________
*************************************************************************************
A. Determine what comes next in the given patterns?
1. A, C, E, G, I, _______
2. 15 10 14 10 13 10 ______
3. 3 6 12 24 48 96 ________
4. 27 30 33 36 39 ________
5. 41 39 37 35 33 ________
B. Substitute the given values in the formula
=
to find the missing quantity.
COURSE MODULE
6. P 680,000; r = 12% per year; t = 8 years
7. A = 1,240,000; r = 8% per year; t = 30 years
8. A = 786, 000; P = 247, 000; t = 17 years
9. A = 731,093; P = 525, 600; r = 3% per year
C. Let Fib (n) be the nth term of the Fibonacci sequence, with Fib (1) = 1, Fib (2) = 1, Fib (3) = 2, and so
on.
10. Find Fib (8).
11. Find Fib (19)
12. If Fib (22) = 17,711 and Fib (24) = 46,368, what is Fib (23)?
13. Fib (1) + Fib (2) = ______
14. Fib (1) + Fib (2) + Fib (3) = _______
15. Fib (1) + Fib (2) + Fib (3) + Fib (4) = ________
Reflection
1. In this fast – paced society, how often have you stopped to appreciate the beauty of the things
around you?
2. Have you ever paused and pondered about the underlying principles that govern the universe?
3. How about contemplating about the processes and mechanisms that make our lives easier, if not
more comfortable?
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Paul John B. Panganiban, LPT
Module 1/Week 1
Math Problem Solving Rubric
NOVICE - 1
Understands
the Problem
COURSE MODULE
Uses
Information
Appropriatel
y
Applies
Appropriate
Procedures
Uses
Representati
ons
Answers the
Problem
APPRENTICE - 2
PROFICIENT - 3
DISTINGUISHED - 4
Doesn't
understand
enough to get
started or make
progress
Uses
inappropriate
information
Understands enough
to solve part of the
problem or to get
part of the solution
Understands the
problem
Identifies special factors
that influences the
approach before
starting the problem
Uses some
appropriate
information correctly
Uses all appropriate
information
correctly
Explains why certain
information is essential
to the solution
Applies
inappropriate
procedures
Uses a
representation
that gives little or
no significant
information
about the
problem
No answer or
wrong answer
based upon an
inappropriate
plan
Applies some
appropriate
procedures
Uses a
representation that
gives some important
information about
the problem
Applies completely
appropriate
procedures
Uses a
representation that
clearly depicts the
problem
Explains why procedures
are appropriate for the
problem
Uses a representation
that is unusual in its
mathematical precision
Copying error,
computational error,
partial answer for
problem with
multiple answers, no
answer statement,
answer labelled
incorrectly
Correct solution
Correct solution of
problem and made a
general rule about the
solution or extended the
solution to a more
complicated solution
Score
Resources and Additional Resources
2018, Rex Book Store, Inc. Mathematics in the Modern World,
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Paul John B. Panganiban, LPT