3. O
B
J
E
C
T
I
V
E
S
Identify patterns in nature and regularities of the
universe;
Determine the importance of patterns in life;
Explain the importance of utilizing mathematical
models;
Solve basic problems involving sequences and
mathematical models
5. Activity 1: Video Watching
https://youtu.be/9mozmHgg9Sk
https://youtu.be/SjSHVDfXHQ4 https://www.youtube.com/w
atch?v=7GiKeeWSf4s
Fibonacci
Sequence
Golden
Ratio
Tesselations
To mathematically explain patterns in the world, watch a video which contains
discussions on the following math concepts:
6. Guide
Questions 3 points
01
What math concept is fully
discussed on each video?
5 points
02
What is about these concepts?
Cite few more real-world cases
or examples that would show
the concepts.
7 points
03
What are your insights about
these mathematical truths or
certainties?
7. Guide
Questions
10 points
04
What did Galileo mean when
he said, “Mathematics is the
alphabet by which God has
written the universe”?
Do you agree on this adage?
Why?
8. Fibonacci Sequence
This sequence begins with the numbers 1
and 1 or 0 and 1, and then each subsequent
number is found by adding the two previous
numbers
Named for the famous mathematician,
Leonardo Fibonacci, this number
sequence is simple, yet profound pattern.
9. Fibonacci Sequence
Illustration
After 1 and 1, the next number is 2, that is,
1+1. The next number is 3, taken from 1+2,
and then 5, taken from 2+3 and so on.
Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, …
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, …
12. Golden
Ratio
The "golden ratio" is a unique
mathematical relationship.
Two numbers are in the golden
ratio if the ratio of the sum of the
numbers (a + b) divided by the
larger number (a) is equal to the
ratio of the larger number divided
by the smaller number (a/b).
19. Golden
Ratio
The ratios of sequential Fibonacci
numbers (2/1, 3/2, 5/3, etc.)
approach the golden ratio.
In fact, the higher the Fibonacci
numbers, the closer their
relationship is to 1.618.
20. Golden
Ratio
The golden ratio is sometimes called the
"divine proportion," because of its
frequency in the natural world.
The number of petals on a flower, for
instance, will often be a Fibonacci
number.
22. Golden
Ratio Even the sides of an unpeeled banana will usually be a
Fibonacci number—and the number of ridges on a
peeled banana will usually be a larger Fibonacci
number.
23. Tesselations
Tessellation (Tiling) is a
shape that repeats to form a
pattern.
It is created when a shape is repeated
over and over again covering a plane
without any gaps or overlaps.
33. S t u d y i n g
P A T T E R N S
Helps in identifying
relationships
Aids in finding logical connections to form generalizations
and make predictions
34. Analyze the given
sequence for its rule and
identify the next three
terms
1. 1, 10, 100, 1000
2. 2, 5, 9, 14, 20
3. 16, 32, 64, 128
4. 1, 1, 2, 3, 5, 8
5.Let Fib(n) be the nth term of a Fibonacci
sequence, with Fib(1) = 1, Fib(2) = 1, Fib(3) = 2.
Find:
a. Fib(8)
b. Fib(19)
Activity 4:
Generating a
Sequence
35. Mathematics can be used to model
population growth. Using the
exponential growth formula, 𝑨 = 𝑷𝒆𝒓𝒕
,
where,
A is the size of population after it grows,
P is the initial number of people,
r is the rate of growth,
t is time, and
e is Euler’s constant, ≈ 2.718
World Population
36. Activity 5: Population Growth
The exponential growth model formula,
𝑨 = 𝟑𝟎𝒆𝟎.𝟎𝟐𝒕
, 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 by the end of 2021?
37. 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 = 30𝑒0.02𝑡
= 30𝑒0.02(0)
= 30𝑒0
= 30 1
𝑨 = 𝟑𝟎
Therefore, the city population in 1995 was 30,000.
38. Solution:
b. We need to find A by the end of 2021. To find t,
we subtract 2021 and 1995 to get t = 26. Hence,
𝑨 = 30𝑒0.02𝑡
= 30𝑒0.02(26)
= 30𝑒0.52
= 30(2.718)0.52
= 30(1.68194)
𝑨 = 𝟓𝟎. 𝟒𝟓𝟖𝟐
Therefore, the city population would be about 50,458
by the end of 2021.
39. Assessment Task #1
Answer the following comprehensively:
1. If Fib(22) = 17,711 and Fib(24) = 46,368, what is
Fib(23)?
2. Evaluate the following sums:
a.Fib(1) + Fib(2) = ______
b.Fib(1) + Fib(2) + Fib(3) = ______
c.Fib(1) + Fib(2) + Fib(3) + Fib(4) = ______
3. Determine the pattern in the successive sums from
the previous question. What will be the sum of
Fib(1) + Fib(2) +…+ Fib(10)?
4. Suppose the population of a certain bacteria in a
laboratory sample is 100. If it doubles in population
every 6 hours, what is the growth rate? How many
bacteria will there be in two days?
40. References
Mathematics in the Modern World by Rex Bookstore, 2018
https://youtu.be/SjSHVDfXHQ4
https://youtu.be/9mozmHgg9Sk
https://www.youtube.com/watch?v=7GiKeeWSf4s
https://www.mathsisfun.com/numbers/fibonacci-sequence.html
