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VARIOUS TYPES OF PATTERNS
LOGIC PATTERNS
OUTLINE
Nature of Mathematics
Mathematics as Study of Patterns
Mathematics in Nature
Fibonacci Sequence
Logic patterns are usually the first to be observed. Classifying
things, for the 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 the
characteristics of various objects while another deals with order.
NATURE OF MATHEMATICS
It is in the objects we create; in the works of art we admire.
Although we may not notice it, mathematics is also present in
the nature that surrounds us, in its landscapes and species of
plants and animals, including the human species. Our attraction
to other humans and even our mobility depends on it.
Sunflowers are stunning, and the way their huge yellow heads
pose against a bright blue sky is classic. And of course, most of
us enjoy munching on the seeds they produce have you ever
stop looking at the pattern of seeds kept in the center of the
center of those rare flowers? Sunflowers are more than just fine
food- they’re mathematical wonders as well the seed pattern in
a sunflower is based on the Fibonacci sequence.
MATHEMATICS
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Defined as study of numbers and arithmetic operations.
Set of tools that can be applied to question of “how many
“of “how much”.
A study of pattern
A language
A process of thinking
A set of problem-solving tools
An art
NUMBER PATTERNS
MATHEMATICS AS A STUDY OF PATTERNS
GEOMETRIC PATTERNS
A pattern is an arrangement which helps observes anticipate
what they might see or what happens next. A pattern also shows
what may have come before. A pattern organizes information so
that it becomes more useful.
Here are some examples of patter- seeking behavior of
humans from childhood to adulthood:
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A toddler separates blue blocks from red blocks.
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A kindergarten student learns to count.
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A first grader does skip counting.
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A third grader creates notices that multiples of two are even
numbers.
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A sixth grader creates patterns that cover a plane.
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A junior high school student learns that a function is
essentially a pattern of how one number is transformed to
another.
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A college biology undergraduate studies the sequence of
DNA and proteins.
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A stock trader studies trends in the stock market.
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A weatherman makes weather forecasts based on
atmospheric pattern.
Patterns are studied because they are everywhere; people just
need to learn to notice them.
Number patterns, such as 2,4,6,8,10 is familiar to students they
are among the first patterns encountered in school. Mathematics
is especially useful when it helps predict events.
Example:
1. 1, 10, 100,1000, 10000, 100000
2. 180, 360, 540, 720, 900, 1080
A geometric pattern is a motif or design that depicts abstract
shapes like lines, polygons, and circle, and typically repeats like
a wallpaper. Visual patterns are observed in nature and in art.
WORD PATTERNS
Patterns can also be found in language like the morphological
rules on 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
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TRANS: Unit Title
here is pattern in form and in syntax, which lead directly to the
study language in general and digital communication.
Example:
Wife: wives: wolf: wolves
Tomato: tomatoes: mango: mangoes
Sun: Earth: Earth: moon
MATHEMATICS IN NATURE
This spiderwort has three-fold symmetry
Vitruvian man of Leonardo da Vinci
Leonardo da Vinci’s Vitruvian man showing the proportion and
symmetry of human body.
This snowflake has six-fold symmetry
People, such as da Vinci, saw mathematics as a universal
constant, with proper proportions repeating themselves across
the universe. It is known as the Vitruvian Man because it is
actually an illustration of concepts describe by the Roman
Vitruvius in the 1st century BC.
Fibonacci sequence is named after Italian mathematician
Leonardo of Pisa, who was better known by his nickname
Fibonacci. He is said to have discovered this sequence as he
looks a hypothesized group of rabbits bred and reproduced.
FIBONACCI SEQUENCE
Pattern indicates a sense of structure and organization that it
seems only humans are capable of producing these intricate,
creative, and amazing formations. It is from perspective that
some people see an “intelligent design” in the way that nature
forms.
Recall that symmetry indicates that you can draw an imaginary
line across an object and 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 of bilateral
symmetry, is evident in most animal, including humans.
Starting with 1 and 1, succeeding term n the sequence can be
generated by adding the two numbers that came before the term
1+1=2
1+2=3
2+3=5
3+5=8
1,1,2
1,1,2,3
1,1,2,3,5,8
1,1,2,3,5,8,…….
Fibonacci sequence has many interesting properties. Among
These is that this pattern is visible in nature. Some beautiful
pattern like the spiral arrangement sunflower seeds, the number
of petals in a flower, and the shape of a snail’s shell- all contain
Fibonacci numbers.
There are other types of symmetry depending on the number of
sides or faces that are symmetrical.
Pinecone showing clockwise and counterclockwise spiral.
This starfish has a five-fold symmetry
It is also interesting to note that the ratios of successive
Fibonacci number approach the number 0 (Phi) , also known as
the Golden Ratio. This is approximately equal to 1.618.
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The Golden Ratio can also be expressed as the ratio between
two numbers, if the latter is also the ratio between the sum and
the larger of the two numbers.
VOCABULARY
of symbols or words.
GRAMMAR
LEONARDO OF PISA
Consisting of rules on the use of these symbols.
COMMUNITY OF PEOPLE
who use and understand these symbols.
RANGE OF MEANING
that can be communicated with these symbols.
Mathematics is a system of communication about objects like
numbers, variables, sets, operations, functions, and equations.
It is a collection of both symbols and their meaning shared by a
global community of people who have an interest in the subject.
Regardless of where in the world learners of math come from or
what language they speak, they will likely understand what
those symbols mean.
Leonardo Pisano Fibonacci (1170-1240 or 1250) was an Italian
number theorist. He introduced the world to such wide-ranging
mathematical concepts as what is now known as the Arabic
numbering system, the concept of square roots, number
sequencing, and even math word problems.
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OUTLINE
Is Mathematics a Universal language?
Characteristics of Math Language
Expressions vs. Sentences
Conventions on the Mathematical Language
Propositions and Symbols
MATHEMATICAL LANGUAGE
LANGUAGE
KEY TAKEAWAYS: WHY MATH IS A LANGUAGE
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In order to be considered a language, a system of
communication must have a vocabulary, grammar, syntax,
and people who use and understand it.
Mathematics meets this definition of a language. Linguists
who don't consider math a language cite its use as a written
rather than spoken form of communication.
Math is a universal language. The symbols and
organization to form equations are the same in every
country of the world.
EXPRESSION VS. SENTENCE
A sentence must contain a complete thought. In the English
language, an ordinary sentence must contain a subject and a
predicate. An expression is a name given to the mathematical
object of interest.
Language is “a systematic means of communicating by the use
of sound or conventional symbols” (Chen, 2010, p. 353). It is the
code humans use as a form of expressing themselves and
communicating with others. It may also be defined as a system
of words used in a particular discipline.
Language facilitates communication and clarifies meaning. It
allows people to express themselves and maintains their
identity.
Likewise, language bridges the gap among people from varying
origins and cultures without prejudice to their background and
upbringing. The study of language teaches and encourages
respect for other people. It is also an avenue to discover
cultures.
We can be living in the same location but have different
languages. Moreover, every field has its own language. For
example, “sin” in Religion is an immoral act however in Math
“sin” is a trigonometric function. An “LP” for a Meteorologist
means “Low Pressure” and for a teacher, it simply means a
“Lesson Plan”.
OMG ILY IRL, JK! Even in the world wide web, humans have
developed their own language. Jejemon and Bekimon are some
other notable trending languages in the Philippines.
LANGUAGE COMPONENTS
These definitions describe language in terms of the following
components:
Figure 1 and figure 2 show comparison and contrast of English
and Mathematics as languages. In English, the subject can be a
noun which could specifically be a person, place, or thing. On
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the other hand, for Mathematics, the subject turns out to be the
numbers, sets, functions, or matrices. While the sentences
formed using the respective expressions in both languages may
either be true, false and sometimes true or sometimes false.
A sentence that is “Always False” is the same as saying that it
is ”Never True”.
Here are some examples of expression for English and Math
languages:
English
Math
Poltician
Philippines
Covid-19 Pandemic
x-y
R
f(x)
And here are some examples of a sentence for English and
Math languages:
English
Math
The Philippines elected
corrupt politicians.
The
COVID-19
pandemic slowed the
economic growth of
most countries.
This
module
is
exercising its freedom
of speech.
x+3=y-2
f(x) = sin x
x ∈R
CHARACTERISTIC OF MATHEMATICAL LANGUAGE
PRECISE
It can make very fine distinctions or definitions among a set of
mathematical symbols.
CONCISE
It can express otherwise long exposition of sentences briefly
using the language of mathematics.
PROPOSITIONS AND SYMBOLS
Logic allows us to determine the validity of arguments in and out
of mathematics. It highlights the importance of precision and
conciseness in the language of mathematics.
A Proposition is a declarative sentence that can be classified as
true or false, but not both. There are other types of sentences
where true or false values cannot be assigned.
PARADOX
There are other types of sentences where true or false values
cannot be assigned. The sentence “This article is false” is one
example. If we assume that it is true, then it is false; if we
assume that it is false, then it is true. Thus, the sentence cannot
be classified as either true or false, so it is not a proposition. A
self-contradictory proposition like this is called a paradox.
COMPONENT STATEMENTS
Words that make up a compound proposition.
SIMPLE PROPOSITION
a proposition that conveys one thought with no connecting
words.
Example:
The Romualdez-Marcos political dynasty is the biggest scam the
Philippines had by far.
COMPOUND PROPOSITION
contains two or more simple propositions that are put together
using connective words.
Example:
Imelda graduated from the University of the Philippines, College
of Law and her younger brother Ferdinand graduated from the
University of Oxford.
TYPES OF COMPOUND PROPOSITION
CONJUNCTION (∧)
E.g., China has Xi Jinping AND the Philippines has Ferdinand
Marcos Jr as its leader.
DISJUNCTION ( ∨ )
POWERFUL
One can express complex thoughts with relative ease.
MATHEMATICAL SYMBOL
There are two ways the word or is used in English.
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Inclusive Disjunction (OR)
The inclusive or is true when at least one of the two propositions
is true.
E.g., I will pay the whole hospital bill OR I will get a
Philhealth subsidy.
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Exclusive Disjunction (XOR)
The Exclusive or is true when exactly one (but not both) of two
propositions is true.
E.g., I am either Covid-19 positive or negative.
CONDITIONAL (→)
E.g., IF Senator Cayetano gets re-elected THEN every Filipino
family will have 10,000 pesos Ayuda (aid)
BICONDITIONAL (⟷)
Mathematics is a branch of science that is composed of several
fields. Several symbols are being used for each field of
mathematics.
The Conjunction of two conditional statements where the
antecedent/premise and consequent/conclusion of the first
statement have been switched in the second statement. E.g.,
Two sides of a triangle are congruent IF AND ONLY IF the two
angles opposite them are congruent.
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NEGATION (∼)
The negation of a given statement is a statement that is false
whenever the given statement is true, and true whenever the
given statement is false. E.g., The Supreme Court ruled with
finality that the Marcos family owned the government P23 billion
in estate tax. Negation: The Supreme Court did NOT rule with
finality that the Marcos family owned the government P23 billion
in estate tax.
finality that the Marcos family owned the government P23 billion
in estate tax. Negation: The Supreme Court did NOT rule with
finality that the Marcos family owned the government P23 billion
in estate tax.
REFERENCES
Notes from the discussion by Jan Mclean Tan
National University MMW Module 1
National University MMW Module 2
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