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Mathematics in the Modern World Lesson 1: Patterns and Numbers in Nature and the World Patterns in Nature - consists of shapes like polygons and 4. Cube Numbers circles that are repeated to create a - the regularities that we...

Mathematics in the Modern World Lesson 1: Patterns and Numbers in Nature and the World Patterns in Nature - consists of shapes like polygons and 4. Cube Numbers circles that are repeated to create a - the regularities that we see in the 5. Fibonacci Numbers design. forms of the things in the world - Tessellations. Pattern that is formed Natural Patterns: by repeating polygons to cover a 1. Symmetry. If you draw an plane. (e.g. Honeycombs). imaginary line in an object, the result - Regular Tessellations. A regular is mirrors of each other. polygon repeated. 2. Spiral. Curved patterns that focus - Semi-Regular Tessellations. Two on center point. (found in animals and or more polygons being repeated. plants). - Fractals. A never-ending pattern. 3. Meander. Series of sinuous curve, Formed by continuously repeating bends, loops, turns, or windings in the something. Examples: channel of stream, river, and other watercourses. 1. Sierpinski Triangle – named after Polish mathematician 4. Cracks. Linear opening that forms Waclaw Franciszek Sierpinski in materials to relieve stress. 2. Pascal’s Triangle – 5. Stripe. A strip or band that has a contains numerical coefficient of different color from the surface binomial expansions. surrounding it. (e.g. Zebra). 3. Fractal Tree IMPORTANT: 4. Koch Snowflake - some pattern cannot be seen as they are already part of human experience. Word Patterns: (e.g. Water Cycle). - Can be found in the meters of poetry - all of them are closely related to and rhythm of words. mathematics. - example of word patterns: Logical Patterns: 1. Plural of Nouns - logic reasoning and pattern 2. Past Tense of Verbs observing are first 2 math standards. 3. Analogy – compare two - are usually the first to be observed different things, showing relationship since making categories and between them. classification comes before numeration. 4. Rhyme Scheme – the rhymes’ pattern at the line of a poem. - Aptitude Test. Takers are shown a sequence of pictures and ask which Number Patterns: figure comes next. - a list of numbers that follow a - in identifying logical patterns, you particular sequence or order. need to look out four things: - example of number patterns: a. rotating shapes; 1. Arithmetic Sequence – the b. increase or decrease in numbers of difference between consecutive terms shapes; called the common difference. c. alternating patterns, colors, shapes; 2. Geometric Sequence – a and sequence where a term is multiplied by a constant called as common ratio. d. mirror images. 3. Triangular Numbers Geometric Patterns: Page 1 of 3 Lesson 2: The Fibonacci Sequence and The Golden Ratio The Fibonacci Sequence will get the value of the golden ratio - Nouns (object, fixed things) for as you move to higher digits. Numbers. Leonardo Pisano or Leonardo of Pisa Application - Connectiveness/Associations for - Was the one who popularize the operational or grouping symbol. Fibonacci sequence. 1. Golden Spiral: - Verbs (to show comparison) for - Born in Pisa, Italy, 1170 - many artists after Phidias have used relation symbol. Golden Spiral. Like Mona Lisa - Popularly known as Fibonacci. A - Pronouns for variables. shortened word for the Latin term 2. Golden Rectangle: fillius Bonacci which means son of Expression vs Sentence in the - a rectangle whose sides are in the Bonacci. Language of Mathematics proportion of the Golden Ratio. - His father is Guglielmo Bonaccio. - expression is an expression with a - Example is Temple like Parthenon correct arrangement of math symbols Fibonacci Sequence in Greece. (but no relation symbol) 1, 1, 2, 3, 5, 8, 13, 21… 3. Proportion of Human Body; - sentence, it makes sense to ask if the Fibonacci Number in Nature - the closer the proportion is to the sentence is true or false (has relation Golden Ratio, the more beautiful the symbol) 1. In the number of petals of flower. body is. 2. In the number of sections in fruits. Fibonacci Spiral Lesson 3: The Language of Mathematics Mathematical Symbols and Convention in the Language Symbols Operations: +,−,×,/ - Can be found in: nautilus shell, hurricanes and tornadoes, Stand-in for Values: a, b, c,… arrangement of sunflower seeds, and Special Symbols: =, ,… some part of human body. Grouping: ( ), { }, [ ],… Golden Ratio Other: Σ the sum of; - Denoted as the Greek letter phi (Ф/φ). It is named after sculptor ∃ there exists; Phidias. ∀ for every (for any); and - Approximately 1.618033… ∞ infinity - It is a irrational number Letter Conventions - Also known as Divine Ratio or Start of Alphabet: a, b, c,… (for Divine Proportion. constants). - φ = √𝟓 +𝟏 /2 = 2(sin54) From I to N: (for positive integers; Golden Ratio and Fibonacci counting). Sequence End of Alphabet: (for variables or unknown). - Lowercase letters for variables or counting. - Uppercase letters for sets and formulas. - If you divide the next number and The English Language and the preceding number respectively, you Language of Mathematics Page 2 of 3 Lesson 4: The Language of Sets 3. Unit/Singleton Set – a set with only their union is the universal set. one element. Denoted as A’ or Ac. Sets 4. empty or Null Set – a set with no - an organized collection of object. object or element. Denoted as { } or Operations on Sets: A = {1, 2, 3, 4, 5} ∅. Note: don’t combine the two. 1. Union – union of both sets A and Where, A is the name of the set, and 5. Equal Set – two sets containing B is the set containing all elements of 1, 2, 3, 4, 5 are the elements (element same elements are equal. Denoted both sets. can be in any order but not repeated) as A = B. 2. Intersection – the intersection of Set Symbols 6. Equivalent Set – if Sets A and B two sets A and B is the set containing have the same number of elements, or ∈ - is a component of the common element of both sets. same cardinality. Uses symbols “~” ∉ - is not a component of or “≡”. 3. Combination = -equality relation ⊆ - subset Note: Equal sets are always 4. Cross Product – sets A and B is ⊂ - proper subset equivalent. But equivalent sets are the set of all ordered pairs (a, b). ⊄ - not a subset not always equal. where first element “a” is from set A and “b” is from set B. ⊇ - superset 7. Disjoint or Non-Intersection Set – ⊃ - proper superset two sets with no common elements. ⊅ - not a superset Subsets, Supersets, and Power Sets U – universal set |A| or n{A} – cardinality of a set Subsets ∪ - Union - A is a subset of B if every element ∩ - Intersect of A is in B Rules in Writing Sets: - Null set is considered to be a subset 1. Name using capital letter. of another set and every set is a subset of itself. 2. The element should be in small letters. - The number of subsets are computed using the formula 2n, where 3. The elements are separated by n is the cardinality of the set. commas and written inside braces { }. Proper Subsets 4. If an object is an element of the set, - If every element in A is in B, but use the symbol ∈. there is at least one element in B that is not in A. Set of Real Numbers: - Null set is a proper subset of every 1. Natural Number (ℕ) – counting set. But every set is not a proper number, aka positive integers. subset of itself. 2. Integers (ℤ) – natural numbers with - The number of proper subsets are negative and zero. computed using the formula 2n – 1. 3. Rational Numbers (ℚ) – numbers Supersets that can be represented as a/b such that b is not equal to 0. - If A is a subset of B, then we can say that B is a superset of A. 4. Irrational Number (ℚ’) – numbers that cannot be reprersented as ratios. Power Sets Example: pi. - The set of all subsets of set A is 5. Real Numbers (ℝ) – rational and called the power set of A. Denoted as irrational. P(A). * Set Builder Notation: {x | x > -2} Universal Sets Kind of Sets: - The set is the set containing all the possible elements under 1. Finite Set – a set whose elements consideration. are limited or countable Complementary Sets 2. Infinite Set – a set whose elements are unlimited. Denoted with (… aka - Set A and B are complementary if ellipsis) in the end. they have no common elements, and Page 3 of 3