Midterms

what is mathematics?

• defined as study of numbers and arithmetic operations

• set of tools that can ba applied to question of “how many” or “how much”

• a study of pattern

• a language

• a process of thinking

• a set of problem solving tools

• an art

pattern

• an arrangement which helps observes anticipate what they might see or what happens next

• also shows what may have come before

• organizes information so that it becomes more useful

• patterns are studied because they are everywhere; people just need to learn to notice them

various types of pattern

• logic patterns

• number patterns

• geometric patterns

• word patterns

logic patterns

• usually the first to be observed

• classifying things, for example, comes before numeration

• being able to tell which things are blocks and which are not precedes learning to count blocks

• one kind of it deals with characteristics of various objects while another deals with order

number patterns

• number patterns, such as 2,4,6,8,10 are familiar to students since they are among the first patterns encountered in school

• mathematics is especially useful when it helps predict events

geometric patterns

• a motif or design that depicts abstract shapes like lines, polygons, and circles, 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 here is pattern in form and in syntax, which lead directly to the study language in general and digital communication

fibonacci sequence

• named after the Italian mathematician Leonardo of Pisa, who was better known by his nickname Fibonacci

• he is said to have discovered this sequence as he look a hypothesized group of rabbits bred and reproduced

• fibonacci sequence has many interesting properties

• among these is that this pattern is visible in nature

• some beautiful pattern, like the spiral arrangement of sunflower seeds, the number of petals in a flower, and the shape of a snail’s shell - all contain fibonacci numbers

phi

• golden ratio

• 1.618

language

• “a systematic means of communicating by the use of sound of conventional symbols” (chen, 2010, p. 353)

• the code humans use as a form of expressing themselves and communicating with others

• may also be defined as a system of words used in a particular discipline

• facilitates communication and clarifies meaning, it allows people to express themselves and maintains their identity

mathematical language

• 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

sentence

must contain a complete thought. In the English language, an ordinary sentence must contain a subject and a predicate

expression

a name given to the mathematical object of interest

characteristics of mathematical language

• precise

• concise

• powerful

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

proposition

a declarative sentence that can be classified as true or false, but not both

paradox

self-contradictory proposition

simple proposition

• conveys one thought with no connecting words

  • 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.

  • 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, disjunction, conditional, biconditional, negation

conjunction

• ( ^ / and )

China has Xi Jinping AND the Philippines has Ferdinand Marcos Jr as its leader.

disjunction

• ( ∨ / or  )

  • INCLUSIVE (OR) - I will pay the whole hospital bill OR I will get a Philhealth subsidy.

  • EXCLUSIVE - I am either COVID-19 positive or negative.

conditional

• ( → / if-then  )

IF Senator Cayetano gets re-elected THEN every Filipino family will have 10,000 pesos ayuda.

biconditional

• ( ↔ / if and only if )

Two sides of a triangle are congruent IF AND ONLY IF the two angles opposite them are congruent.

negation

•  ~ / not )

The Supreme Court did NOT rule with finality that the Marcos family owned the government P23 billion in estate tax.