Midterm

• Mathematics is a formal system of thought for recognizing, classifying, and exploiting patterns.

  • Study of Patterns
  • Language
  • Art
  • Set of problem-solving tools
  • Process of thinking
• Nature of Mathematics is the chief driving force that has propelled the evolution of a highly sophisticated world and lifestyle.
• Patterns are core topics in mathematics. It is also known as the science of patterns.
• Two Types of Patterns

  • Numeric Patterns
  • Geometric Patterns
• Numeric Pattern is a sequence of numbers that follow a certain order.
• Fibonacci sequence is a recursive sequence generated by adding the two previous numbers in the sequence.
• Geometric patterns are formed geometric shapes
• Geometry is the branch of mathematics that describes shapes
• sphere is a perfectly round geometrical object in three-dimensional space
• Cone is a three-dimensional geometric

  shape that tapers smoothly from a flat, usually circular base to a point called the apes or vertex.
• Hexagon is a six-sided polygons, closed,

  2-dimensional, many-sided figures with

  straight edges.
• Parallel lines stretch to infinity, neither converging or diverging
• Concentric Circles are circles that are different sizes but share the same center. They circle each other, growing out and getting bigger and bigger.
• Artistry and Abstraction

  • best exhibited in the intricate designs found in the textile products and architectural designs and ornamentations.
  • concrete testaments not only of the rich cultural heritage but also to the mathematical ingenuity of Filipinos
  • established links with mathematics
  • gave birth to ethnomathematics
• Symmetry is when a figure has two sides that are mirror images of one another. It is a major factor in determining physical attraction
• Two Kinds of Symmetry

  • Bilateral Symmetry
  • Radial Symmetry
• Bilateral Symmetry has two sides that are mirror images of each other.
• Radial Symmetry is where there is a center point and numerous lines of symmetry could be drawn
• Propositions

  • Basic building blocks of logic
  • declarative sentence that is either true or false, but not both
• Connectives are logical operators that are used to form new propositions from 2 or more existing propositions.
• Disjunction (OR) - A proposition formed by combining two or more simpler propositions using the words "or", "either...or", "neither...nor", etc. True if at least one of the statements is true.
• Conjunction (AND) - A proposition formed by combining two or more simpler propositions using the word "and". True when both statements are true.
• Conditional (IF-THEN) - A proposition formed by combining two or more simpler propositions using the phrase "if...then..." If p is true, then q must be true. The only way that this can fail (or be false) is when p is true while q is false.
• Negation (NOT) - A proposition formed by combining one or more simpler propositions with the word "not".
• Biconditional (IF AND ONLY IF) - A proposition formed by combining two or more simpler propositions using the phrase "if and only if".
• Logical Matrix is an array of decision value. It displays the relationships between the truth values of propositions. And it gives 2 possible decisions for q, since a proposition may either be T or F.
• Importance of Language

  • To understand the expressed ideas
  • To communicate ideas to others
• Characteristics of the Mathematics Language

  • Precise - able to make very fine distinctions or definitions
  • Concise - able to say things briefly
  • Powerful - able to express complex thoughts with relative ease
• Parts of Speech in Mathematics

  • Numbers - nouns (objects); quantity
  • Operation Symbols - connectives in math sentence (+, ÷, ^, v)
  • Variables - letters that represent quantities & act as pronouns
  • Relation Symbols - used for comparison; acts as verb (=, ≤, ~)
  • Grouping Symbols - used to associate groups of numbers & operators ( ), { }, [ ]
• Mathematical expression refers to objects of interest acting as the subject in the ordinary language.
• Mathematical sentence is a sentence with a complete thought which can be regarded as true or false.
• Grammar of Mathematics is a structural rules governing the use of symbols representing mathematical objects
• Inductive Reasoning uses patterns to arrive at a conclusion. It uses specific observations to reach a general conclusion. Probability
• Deductive Reasoning uses facts, rules, definitions or properties to arrive at a conclusion. It uses a general idea to reach specific conclusions. Certainty
• Inductive Reasoning could be:

  • Generalized
  • Statistical
  • Predictive
  • Causal
  • Analogical
• George Pólya was the Father of Problem Solving. He was a Hungarian mathematician and one of the most remarkable mathematicians of the 20th Century, who made fundamental contributions to a wide range of topics and to the theory of problem solving.
• The four stages of Polya's method are Understanding the problem, Devising a plan, Carrying out the plan, Looking back
• Devising a Plan involves making a diagram, drawing a picture, listing all possible solutions, checking your work, and looking up formulas.
• Understanding the problem involves reading the question carefully, identifying what is given and unknown, determining whether it is an open-ended or closed-ended question, and deciding if there are any assumptions that need to be considered.
• Carrying Out The Plan involves doing calculations, applying formulas, substituting values, simplifying expressions, and checking answers