MODULE 1
THE NATURE OF MATHEMATICS
Module Introduction
The earliest concepts on mathematics that that gained…
LESSON NO. 1
LESSON TITLE Patterns and Numbers in Nature
DURATION/HOURS 3 hours
Specific Learning During the learning engagements, the students are expected to:
Outcomes:
1. Identify patterns in nature and regularities of the
universe
2. Determine the importance of patterns in life.
TEACHING LEARNING ACTIVITIES
Introduction
Mathematics is all around us. As we discover more and more about our environment and our
surroundings we see that nature can be described mathematically. The beauty of a flower, the
majesty of a tree and mountain, even the rocks upon which we walk can exhibit nature’s sense
of symmetry and patterns. Describe orally the picture below according to how you’re perceived
it.
A B
C D
Processing
Have you ever stopped to look around and notice all the amazing shapes and patterns we see in
the world around us? Mathematics forms the building blocks of the natural world and can be
seen in stunning ways.
Here are a few of my favorite examples of math in nature, but there are many other examples
as well.
Fibonacci Sequence
Named for the famous mathematician, Leonardo Fibonacci, this number sequence is a simple,
yet profound pattern. Based on Fibonacci’s rabbit problem, this sequence begins with the
numbers 1 and 1, and then each subsequent number is found by adding the two previous
numbers. Therefore, after 1 and 1, the next number is 2 (1+1). The next number is 3 (1+2) and
then 5 (2+3) and so on. What’s remarkable is that the numbers in the sequence are often
seen in nature. A few examples include the number of spirals in a pine cone, pineapple or
seeds in a sunflower, or the number of petals on a flower. The numbers in this sequence also
form a unique shape known as a Fibonacci spiral, which again, we see in nature in the form of
shells and the shape of hurricanes.
Hexagons in Nature
Another of nature’s geometric wonders is the hexagon. A regular hexagon has 6 sides of equal
length, and this shape is seen again and again in the world around us.
The most common example of nature using hexagons is in a bee hive.
Bees build their hive using a tessellation of hexagons. But did you know that every snowflake
is also in the shape of a hexagon?
We also see hexagons in the bubbles that make up a raft bubble. Although we usually think of
bubbles as round, when many bubbles get pushed together on the surface of water, they take
the shape of hexagons.
Concentric Circles in Nature
Another common shape in nature is a set of concentric circles. Concentric means the circles all
share the same center, but have different radii. This means the circles are all different sizes, one
inside the other.
A common example is in the ripples of a pond when something hits the surface of the water.
But we also see concentric circles in the layers of an onion and the rings of trees that form as it
grows and ages.
If you live near woods, you might go looking for a fallen tree to count the rings, or look for an
orb spider web, which is built with nearly perfect concentric circles.
Patterns in Outer Space: Moving away from planet earth, we can also see many of these
same mathematical features in outer space.
For instance, the shape of our galaxy is a Fibonacci spiral. The planets orbit the sun on paths
that are concentric. We also see concentric circles in the rings of Saturn.
But we also see a unique symmetry in outer space that is unique (as far as scientists can tell)
and that is the symmetry between the earth, moon and sun that makes a solar eclipse possible.
Every two years, the moon passes between the sun and the earth in such a way that it appears to
completely cover the sun. But how is this possible when the moon is so much smaller than the
sun?
1. Each team leader will discuss the salient points on the topic assigned to them.
2. To mathematically explain further these patterns, assign to each team a video which
contains discussions on certain math concept.
Team A on Fibonacci Sequence at https://youtu.be/SjSHVDfXHQ4
Team B on Golden Ratio at https://youtu.be/9mozmHgg9Sk
Team C on Tessellation at https://www.youtube.com/watch?v=7GiKeeWSf4s
Drill. Determine the next 5 terms in the following sequences.
1. 13, 21, 34, 55, __, __, …
2. 55, 89, 144, 233, __, __, …
3. 2, 3, 5, 8, __, __, …
4. 21, 34, 55, 89, __, __, …
5. 89, 144, 233, 377, __, __, …
6. 34, 55, 89, 144, __, __, …
7. 8, 13, 21, 34, __, __, …
8. 3, 5, 8, 13, __, __, …
FORMATION
Questions for Discussion
1. What math concept is fully discussed on your assigned video?
2. What is about this concept? Cite few more real-world cases or examples that would show
the concept.
3. What are your insights about these mathematical truths or certainties?
4. 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?
SYNTHESIS
Write three important concepts or learning you have from this lesson. Rank them accordingly.
ASSESSMENTS
Answer the following. Write the letter of the correct answer.
1. The first four Fibonacci numbers are 0, 1, 1 and 2, what is the twelfth number?
a. 55 b. 77 c. 88 d. 89
2. If the first three Fibonacci numbers are given as x1 = 1, x2 = 1 and x3 = 2, then what is the
least value of n for which xn > 500?
a. n = 13 b. n = 14 c. n = 15 d. n = 16
3. The Golden Ratio = 1.61803398874989484820... = 1.618 correct to 3 decimal places. If xn
xn +1
are terms of the Fibonacci sequence, then what is the least value of n for which =
xn
1.618 correct to 3 decimal places?
a. n = 8 b. n = 9 c. n = 10 d. n = 11
4. If xn is a Fibonacci number and n = -4, which of the following is true?
a. x-4 = x-5 + x-6 b. x-4 = x-2 + x-3 c. x-4 = -(x5 + x6) d. x-4 = x3 –
x2
5. The partial sums of the first n and n + 1 numbers of the Fibonacci sequence are both
divisible by 11. What is the smallest value of n for which this is true?
a. n = 11 b. n = 10 c. n = 9 d. n = 8
6. The diagram shows how the numbers of the
Fibonacci sequence can be obtained from the
numbers in Pascal's Triangle. Following the same
pattern, which numbers of Pascal's triangle can be
added together to give the next number of the
Fibonacci sequence?
a. 34 = 1 + 8 + 15 + 9 + 1
b. 34 = 1 + 7 + 10 + 10 + 5 + 1
c. 34 = 1 + 2 + 15 + 15 + 1
d. 34 = 1 + 7 + 15 + 10 + 1
ASSIGNMENTS Do it yourself:
1. Think of a number.
2. Describe a second number.
3. Add the two together.
4. Determine the next 12 numbers in your sequence.
5. Name your series of numbers.
6. What are your conclusions about the ratio of
consecutive numbers in your sequence as numbers
get larger?
7. What are the ratios of each of your consecutive
numbers divided by the next larger number in the
sequence?
RESOURCES Dugopolski, Mark. (2000). Algebra for College Students. 2nd
Edition. New York. McGraw Hill Company, Inc.
Poole, Barbara. (1989). Intermediate Algebra. New Jersey. Prentice
Hall.
Rees, Paul K. (1990). College Algebra. New York. McGraw Hill
Company, Inc.
https://youtu.be/SjSHVDfXHQ4
https://youtu.be/9mozmHgg9Sk
https://www.youtube.com/watch?v=7GiKeeWSf4s
https://www.mathsisfun.com/numbers/fibonacci-sequence.html
Slideshare.net/MaryVanDyke/seeing-math-patterns-in-nature
Vectortatu/iStock/Thinkstock
AndreaAstes/iStock/Thinkstock
ThinkstockImages/iStock/ThinkStock
Mathematics in the Modern World. Rex Book Store, Inc. (RBSI)
Mathematical Excursions, 14th Edition, by Richard N. Aufmann,
Joanne S. Lockwood, Richard D. Nation, and Daniel K. Clegg
Discrete Mathematics: An Introduction to Mathematical Reasoning,
1st Edition, by Susanna S. Epp
