MATHEMATICS

pattern

is a recurring sequence that follows a

predictable rule. Identifying and understanding these

rules is a fundamental skill in mathematics and is

essential for making predictions and solving

problems.

Patterns

are the building blocks of logical thought. By

recognizing a pattern, we can understand the

structure of a sequence, whether it's a list of

numbers, a series of events, or a repeating design.

This ability to identify a rule allows us to determine

what comes next and to generalize that rule to other

situations.

Fibonacci Sequence

is a series of numbers

where each number is the sum of the two preceding

ones.

You can find this sequence in the branching of trees,

the arrangements of leaves on a stem, and the

spirals of seeds in a sunflower.

Golden Ratio or ∅ (phi),

is an irrational number

approximately equal to 1.618.

Golden Ratio or ∅ (phi),

It's closely related to the Fibonacci sequence: as you

divide a number in the sequence by the one before

it, the ratio gets closer and closer to the Golden

Ratio (e.g. 8/5=1.6; 13/8=1.625; 21/13≈1.615...)

Golden Ratio or ∅ (phi),

• It appears in the spirals of a nautilus shell, the shape

of certain galaxies, and even the proportions of the

human body.

Symmetry

is a balanced and proportionate

arrangement of parts.

Bilateral Symmetry

An object can be divided into

two mirrored halves.

Examples: Butterflies, beetles, human faces.

Radial Symmetry

An object can be divided into

identical parts around a central point.

• Examples: Starfish, sand dollars, sea

anemones, flower petals.

Symmetry

is a fundamental concept in mathematics and nature,

describing a balanced and proportionate arrangement of parts. It is

essentially a property where an object remains unchanged after being

transformed in some way, such as by rotation, reflection, or scaling.

Leonardo da Vinci’s Vitruvian

Man

showing proportion and

symmetry of human body.

Radial symmetry

occurs when an object has

identical parts arranged around a central axis. This

means the object can be divided into several equal

sections, like slices of a pie.

radial symmetry

Organisms with _______ are often

non-motile or slow-moving and include many

aquatic animals and flowers.

Examples include starfish, jellyfish, and a variety of

flowers.

Angle of Rotation

is the smallest angle that a figure can be rotated while preserving the original formation.

Spirals

are patterns that curve around a central

point.

• They are often linked to the Fibonacci Sequence and

the Golden Ratio because of how they grow.

Logarithmic Spirals

are the most common type

found in nature. As the spiral grows, its shape

remains the same.

• Examples: Nautilus shells, the arrangement of

seeds in a sunflower, hurricanes and

tornadoes

fractal

is a never-ending pattern that is

"self-similar," meaning it looks the same at different

scales.

fractal

You can zoom in on a part of a  _____ , and it will

look like the whole thing.

fractal

This pattern is a perfect way to efficiently pack a lot

of surface area into a small space.

• Examples: Fern fronds, broccoli and

cauliflower, river networks and coastlines

tessellation

is a pattern of shapes that fit

together perfectly without any gaps or overlaps.

• Nature's tessellations are incredibly efficient,

especially for maximizing space and strength.

• Examples: Honeycomb (hexagonal pattern),

the scales on a snake or fish, crystals and

some rock formations

A = Pe^rt

The

formula for exponential growth 

where A is the size

of the population after it grows, P is the initial number of

people, r is the rate of growth and t is the time. The Euler’s

constant e has an approximate value of 2.718.

Golden Angle

As you

divide a number in the sequence by its predecessor (e.g., 8/5=1.6), the result gets

progressively closer to φ≈1.618. This ratio corresponds to a specific angle known

as the

Pingal

The sequence first appeared around 200 BC in the work of the Indian scholar

He was exploring the patterns of Sanskrit poetry and used what we

now call Fibonacci numbers to count the possible ways to combine syllables

of different lengths.

Leonardo

of Pisa,

However, the fibonacci sequence is named after the Italian mathematician

Liber Abaci,

was the

key text that introduced the sequence to Western Europe.

Johannes Kepler

The german astronomer independently discussed

the Fibonacci numbers, noting their connection to the proportions of the pentagon.

recurrence relation

(where each term is the sum of the two

preceding ones)

lucas numbers

fibonnaci sequencey are a complementary pair with the

The Rabbit Problem

Fibonacci considers the growth of an idealized (biologically

unrealistic) rabbit population, assuming that: a newly born

pair of rabbits, one male, one female, are put in a field;

rabbits are able to mate at the age of one month so that at

the end of its second month a female can produce another

pair of rabbits; rabbits never die and a mating pair always

produces one new pair (one male, one female) every month

from the second month on (this means that after 2 months

every pair will produce a new pair of rabbits). The

puzzle that Fibonacci posed was: how many pairs will there

be in one year?

The Golden Ratio

is defined by a simple algebraic relationship. Imagine a

line segment divided into two parts, let's call them a and b. The ratio is

considered "golden" if the ratio of the longer part (a) to the shorter part (b)

is the same as the ratio of the entire segment (a+b) to the longer part (a).

Vitruvian Man

most

famously by Leonardo da Vinci in his drawing of

the

Language

It is the code humans use as a form of expressing

themselves and communicating with others.

It is a system of words used in a particular

discipline.

Language

“a systematic means of

communicating by the use of sounds or

conventional symbols” (Chen, 2010).

Precis

it is able to make very fine distinctions or

definitions.

Concise

or brief i.e., if someone can say things in

long expositions or sentences, a mathematician

can say it briefly.

Powerful

one can express complex thoughts

with relative ease.

Logic

is the science and

art that directs the

reasoning process so

that man may attain

knowledge of the truth in

an orderly way, with

ease, and without error.

Logic

It is the science and art of correct thinking.

Logic

is the systematic study of the form of

valid inference, and the most general laws

of truth

illustrates the importance of precision

and conciseness of the language of

mathematics.

Proposition

is a

declarative sentence that is

either TRUE or FALSE but not

both.

SIMPLE Proposition

▪ a proposition that conveys a single idea

▪ example: 2 is an even number

COMPOUND Proposition

▪ a proposition that conveys two or more

ideas

▪ example: 2 is an even and a prime number

negation

changes the truth value of a proposition.

Law of Detachment

(also known as Modus Ponens) is a fundamental rule of inference in

logic. It states that if a conditional statement is true, and its antecedent (the "if" part) is also

true, then its consequent (the "then" part) must be true.

Law of Detachment

The law can be summarized with the following symbolic notation:

If p→q is true, and p is true, then q is true.

Law of Detachment

● p: The antecedent or hypothesis.

● q: The consequent or conclusion.

● p→q: The conditional statement, which is read as "if p, then q."

In simpler terms, if you have a statement like "If A, then B" and you know for a fact that "A" is

true, then you are logically able to conclude that "B" must also be true.

affirming the consequent.

This fallacy occurs when someone incorrectly

assumes the antecedent is true just because the consequent is true.

Law of Syllogism

is a logical principle that allows you to form a new conditional

statement from two given conditional statements. It's often compared to the transitive property

in mathematics because it creates a chain of reasoning.

Law of Syllogism

The law states that if a first statement implies a second, and that second statement

implies a third, then the first statement must imply the third.

Law of Syllogism

○ If p→q is a true statement, and q→r is also a true statement, then the conclusion p→r

is true.

○ This means you can "cut out the middleman" (q) and connect the first and last parts of

the chain.

REASONING

is the foundation for understanding

mathematical concepts and solving problems.

● It helps us make logical connections and draw

valid conclusions.

REASONING

The process of using existing knowledge to draw

conclusions, make inferences, or construct

arguments.

● Involves forming judgments and making

decisions based on logic and evidence.

Inductive Reasoning

● Starts with specific observations or examples.

● Looks for patterns and regularities in these

observations.

Inductive Reasoning

● Formulates a general conclusion or hypothesis

based on these patterns.

● Often described as making an educated guess or

forming a conjecture.

Inductive Reasoning

○ Useful for generating hypotheses and exploring new ideas.

○ Can lead to the discovery of patterns and relationships.

○ Relatively easy to understand and apply.

Inductive Reasoning

● Weakness:

○ Conclusions are not guaranteed to be true (can be incorrect

even with many observations).

○ A single counterexample can disprove an inductive conclusion.

○ The sample size of observations can influence the strength of

the conclusion.

inductive reasonin

Scientists often use

Galileo Galilei

used inductive

reasoning to discover that the time required for

a pendulum to complete one swing, called the

period of the pendulum, depends on the length

of the pendulum.

did not have a clock, so he

measured the periods of the

pendulum in “heartbeats”.

● The period of the pendulum is the

time it takes for the pendulum to

swing from left to right and back to

its original position.

Inductive Reasoning

● Also called as the “Bottom-Up” Approach

Inductive Reasoning

It is the process of reaching a general conclusion by

examining specific examples

● Starts with facts and details and moves to a general

solution

● It is probabilistic, may be strong or weak, and can be

proved false

● Key Idea: Observing specific instances to infer a

general rule.

Inductive Reasoning

● Note that conclusions based on inductive reasoning

may be incorrect.

● Remember that statement is TRUE provided that it can

be proven to be true in ALL cases.

● One case that makes the statement incorrect falsifies

the conclusion or the conjecture. This is called as a

counter example.

Deductive Reasoning

● Starts with general principles, rules, or premises

that are assumed to be true.

● Applies these general principles to specific cases

to reach a logically certain conclusion.

● If the premises are true, the conclusion must be

true.

● Relies on logical rules and structures.

Deductive Reasoning

Mathematical proofs rely heavily on ______

● Starting with axioms (fundamental truths) and

definitions.

● Using logical steps and previously proven

theorems to arrive at a conclusion.

● Ensures the certainty and rigor of mathematical

results.

Deductive Reasoning

● Strengths:

○ Leads to logically certain and irrefutable conclusions if the premises

are true.

○ Provides a strong foundation for mathematical knowledge.

○ Ensures accuracy and precision in reasoning.

Deductive Reasoning

● Weakness:

○ The validity of the conclusion depends entirely on the truth of the initial

premises. If the premises are false, the conclusion may also be false.

○ Can only work with information that is already known or assumed.

Doesn't generate new knowledge in itself (though it can reveal

implications of existing knowledge).

Deductive Reasoning

● Also called as the “Top-Down” Approach

● It is the process of reaching a conclusion by applying general

assumptions, procedures, or principles.

● Starts with a conclusion and then explains the facts , details

and examples

Deductive Reasoning

● It links premises with conclusions

● If all premises are true and clear, the conclusion must also be

true

● Key Idea: Applying general rules to specific instances to

reach a definite conclusion.

Logical Fallacies

are errors in reasoning that can arise in both

inductive and deductive reasoning (e.g., hasty generalization,

false cause, appeal to emotion).

Ad Hominem

Attacking the person making the argument instead of

addressing the argument itself.

Example: Mayor John is proposing a new public transport

system, but we shouldn't listen to him. He was caught in a

corruption scandal a few years ago. (Instead of evaluating

the merits of the transport plan, the argument attacks the

mayor's character.)

Pathos (Appeal to Emotion)

○ Manipulating an emotional response in place of a valid or

compelling argument.

○ Example: Think of all the poor families who will suffer if

this new business doesn't get approved. We must approve

it to help the less fortunate. (While the concern for families

might be genuine, the argument doesn't provide logical

reasons for the business's approval.)

Appeal to Authority (False Authority)

○ Claiming something is true because an unqualified

authority figure says it.

○ Example: My favorite celebrity said that drinking this

brand of bottled water will make you healthier, so it must

be true. (A celebrity might be popular but may not have

expertise in health and nutrition.)

Hasty Generalization

○ Drawing a broad conclusion based on limited or

insufficient evidence.

○ Example: I went to a market in Iloilo City yesterday, and

the prices of vegetables were very high. Therefore, all

prices of goods in Iloilo City are too expensive. (One visit

to one market doesn't represent the overall cost of living.)

Straw Man Fallacy

● Misrepresenting an opponent's argument to make it easier to attack.

○ Drawing a broad conclusion based on limited or insufficient

evidence.

○ Example: Person A says, "We should invest more in public

transportation in Iloilo City." Person B replies, "So you want to

bankrupt the city by spending all our money on buses and trains

instead of building more roads?" (Person B misrepresents A's

argument as being solely about eliminating road construction.)

False Dilemma (False Dichotomy)

○ Example: Either we build this new shopping mall, or the

city will never progress. (This presents only two extreme

options, ignoring other potential ways for the city to

develop.)

Slippery Slope

○ Claiming that one action will inevitably lead to a series of

negative consequences without sufficient evidence.

○ Example: If we allow more tricycles in this area, soon the

traffic will become completely unbearable, and no one will

be able to get around. (This exaggerates the potential

consequences of allowing more tricycles without

providing concrete evidence.)

Appeal to Popularity (Bandwagon)

○ Claiming something is true because many people believe

it.

○ Example: Everyone in my neighborhood is saying that the

new mayor is doing a great job, so it must be true.

(Popular opinion isn't always a reliable indicator of truth.)

Post Hoc Ergo Propter Hoc (False Cause)

○ Assuming that because one event followed another, the

first event caused the second.

○ Example: Since the new public market opened in Iloilo

City, there has been an increase in the number of tourists.

Therefore, the new market caused the increase in tourism.

(Correlation does not equal causation; other factors could

be influencing tourism.)

Critical Thinking:

Recognizing fallacies helps you think more

critically about arguments you encounter.

Effective Communication:

Avoiding fallacies makes your own

arguments more sound and persuasive.

Identifying Manipulation:

It helps you identify when others

might be trying to manipulate your thinking.

Making Informed Decisions:

By recognizing flawed reasoning,

you can make more informed decisions based on solid evidence.