Mathematics in the Modern World (Patterns and Sequences).pptx

Patterns exist in different variety of forms. The petals of a flower, arrangement of leaves reveals a sequential pattern. Natures are bounded by different colors and shapes – the rainbow mosaic of a butterfly’s wings, the undulating ripples of a desert dune. But these miraculous creations not only delight the imagination, they also challenge our understanding. How do these patterns develop? What sorts of rules and guidelines, shape the patterns in the world around us? Some patterns are molded with a strict regularity. At least superficially, the origin of regular patterns often seems easy to explain. Thousands of times over, the cells of a honeycomb repeat their hexagonal symmetry. The honeybee is a skilled and tireless artisan with an innate ability to measure the width and to gauge the thickness of the honeycomb it builds. Although the workings of an insect's mind may baffle biologists, the regularity of the honeycomb attests to the honey bee's remarkable architectural abilities. A pattern is something which helps us anticipate what we might see or expect to happen next. It may also help us know what may have come before or what we are seeing currently. There are four types of patterns; (1) logic patterns, (2) number patterns, (3) geometric patterns and (4) word patterns. 4. Rippled pattern observed on the desert sand. 5. Honeycomb structure A pattern has symmetry. Isometry of the plane that preserves the pattern. It is a way of transforming the plane that preserves geometrical properties such as length. There are four types of isometries according to Euclidian isometry of plane transformation (1) Translation (2) Reflection (3) Rotation (4) Dilation. Moreover, we have to consider sometimes the combination of Reflection, translation and rotations makes another isometry called rigid transformation which leave the dimensions of the object and its image unchanged. A propositional variable represented by a lowercase or capital letter in the English alphabet denotes an arbitrary proposition with an unspecified truth value. An assertion which contains at least one proposition variable is called a propositional form. In the preposition: “if I study the lesson, Then I will pass the test” In a condition hypothetical proposition, the truth does not rest on the truth of every statement taken singly. Rather, it depends on valid sequence between members of the proposition. In the example given, we don’t assert, “I study the lesson” nor we assert, “I will be able to pass the test”. We ae going to simply declare the fact that the statement “I study the lesson” is dependent on the other statement which is “I will pass the test” and vice versa. Note: The word “then’ as part of the consequent maybe omitted. “IF I study the lesson, I will pass the test” The consequent may also be written ahead of the antecedent and the word “then” is omitted. “I will pass the test, IF I study t

1. Mathematics in the Modern World

2. Patterns It is defined as an organization of shapes and symbols distributed in regular interval.

3. Symmetry Symmetry occurs when dividing a figure by two and will produce the two similar figures. Symmetry can be reflective or rotational. For reflective symmetry, the right half and left half of the figure should be mirror reflection of each other. For rotational symmetry, if the figure rotates at its center, the figure should be the same as the original.

4. Reflective symmetry

5. Rotational symmetry

6. Order of rotational symmetry - it is defined by the number of times one can rotate a figure to get the same figure. - if the shape is a regular polygon the order depends on the number of sides for example: A rectangle has an order of 2 but a square has an order of 4

7. fractals Pattern of infinite iterations of the figure going in a loop.

8. Spiral Coiled pattern revolving about a center point.

9. Waves Disturbances that carry energy

10. Tessallations Formed by repeating shapes over a plane.

11. Cracks Openings formed to release stress

12. Spots and stripes

13. sequence A list of numbers or elements arranged in a certain pattern. Each elements will be called terms It can be finite or infinite It can be ascending or descending order Each term can be evaluated through direct formula, relations from the preceding term or table of values

14. Arithmetic sequence Each terms have a common difference between their preceding terms. 1,2,3,4,5… The sequence has a common difference of 1 2,4,6,8,10… The sequence has a common difference of 2 1,3,5,7,9… The sequence has a common difference of 2 Equation for nth term A 1 is the first term and d is the common difference  

15. Geometric sequence Each terms have a common ratio between their preceding terms. 1,2,4,8,16… The sequence has a common ratio of 2 1,4,16,64… The sequence has a common ratio of 4 r is the common ratio  

16. Harmonic Sequence Each term is the reciprocal of the terms in an arithmetic sequence 1,1/2,1/3,1/4,1/5…(from 1,2,3,4,5) 1,1/4,1/7,1/10,1/13…(from 1,4,7,10,13)

17. Quadratic sequence The difference between the terms are arranged in arithmetic sequence 1 3 6 10 15 +2 +3 +4 +5 +1 +1 +1

18. Triangle Number Sequence - the pattern of these sequence form equilateral triangles Square Number Sequence - the pattern of these sequence form squares Cube Number Sequence - the pattern of these sequence form cubes

19. Fibonacci sequence The next term of the sequence is the sum of the two preceding terms F0 F1 1 F2 1 F3 2 F4 3 F5 5 F6 8 F7 13 F8 21 F9 34 F10 55 F11 89 F12 144 F13 233

20. Golden ratio The limit of the ratio of two consecutive terms of the Fibonacci Sequence. Approximately φ = 1.618034 Seen usually in nature Mathematical ratio describing aesthetically pleasing For Fibonacci Sequence, for the Nth term,  

21. Example For the 5 th number = 5 For the 11 th number = 89  

22. Fibonacci series spiral Formed by drawing a spiral guided by stacking squares with dimensions from the Fibonacci Sequence. Known for its aesthetic and symmetrical appearance

23. Mathematical language and symbols The study on Mathematical Language involves how to convey the ideas and solutions of mathematics among others. The communication lies with how the idea is presented. The presentation should be precise, concise and compelling. Mathematical symbols are the components used in forming these Mathematical Language.

24. Mathematical expression vs mathematical sentences Both are layout of mathematical symbols but mathematical expression is simply a part of a mathematical sentence. The mathematical expression does not give answers or confirms the solution but mathematical sentence can.

25. Symbols and its interpretations Symbols Interpretations + Addition(“plus”, ”add”, “increased”, “sum”, “more than”) or Positive number - Subtraction(“minus”, “less”, “less than”, “decreased”, “difference”) or Negative number or / Division(“divide”, “ratio”, “quotient”) Multiplication(“times”, “multiply”, “product”) = Equal(“equal”, “the same”) >,<, > and < Inequality(“greater than”, “less than”, “greater than or equal” and “less than or equal”) and Approximately equal and congruent to respectively % Percent(number times 100) n √ nth root { }, [ ] and ( ) Braces, Brackets and Parenthesis respectively A b or A^b Exponent(“raised to”, “power of”) |x| Absolute value + Plus - Minus ! Factorial ∑ Summation Sign Symbols Interpretations + Addition(“plus”, ”add”, “increased”, “sum”, “more than”) or Positive number - Subtraction(“minus”, “less”, “less than”, “decreased”, “difference”) or Negative number Division(“divide”, “ratio”, “quotient”) Multiplication(“times”, “multiply”, “product”) = Equal(“equal”, “the same”) >,<, > and < Inequality(“greater than”, “less than”, “greater than or equal” and “less than or equal”) Approximately equal and congruent to respectively % Percent(number times 100) n √ nth root { }, [ ] and ( ) Braces, Brackets and Parenthesis respectively A b or A^b Exponent(“raised to”, “power of”) |x| Absolute value + Plus - Minus ! Factorial ∑ Summation Sign

26. Variables Variables are symbols used to represent some values. Usually in the form of an alphabet.

27. Set A collection of numbers or values written as a set-roster or set-builder ∈ means elements Set-Roster: X = {a, b, c}, X = {a, b, c, …} or X = {a, b, c, {d}} Set-Builder: {X ∈ ℝ | a < X < b } : “X is set with an open interval between a and b” {X ∈ ℝ | a < X < b } : “X is set with an closed interval between a and b”

28. Universal Set -A collection of all elements from different set Empty Set -A set with no elements. Null or Void. Ø Unit Set - A set with one element Complement of a set - A set with elements not found from another set. A C Subset - A set of elements that are part of another set. ⊆ Union - A collection of elements between two or more sets. ∪ Intersection - A collection of the same elements between two or more sets. ∩

29. Venn diagram A Venn Diagram uses circles as representation of groups of elements for easy understanding. Each circles convey a set. Each sets have their own category. The circles will be drawn in a universal set If there are overlapping circles, there is an intersection between the two or more set

30. Function A mathematical expression that convey that for every input, there is one designated output Relations in mathematics establishes a connection between two elements usually for input and output, x and y respectively for most time. y = f(x) Domain is a set of values that can be used as input for the function. It will be inserted to the independent variable, “x”. The domain will result to a real value. Range is a set of values that are the output of the function if the domain is inserted. It will be all possible value of the dependent variable, “y”.

31. Binary operations An action performed on two elements to produce a new elements. A + B = C A – B = C A x B = C A / B = C

32. Logic It is a study on statements or arguments to create valid interpretation Connectives are words or phrases that will connect two sentence to create a new statement. Example: “and”, “or”, “if-then” and “if and only if” Proposition is a statement that can be evaluated as either true or false but it can’t be both.

33. Connectives ~ or ¬ for negation( not true ) ∧ for conjunction ( “and”). ∨ for disjunction (“ either or both”) and ⊻ for exclusive disjunction (“ either but not both”) -> for condition (antecedent and consequent, “if-then”) <-> for biconditional(“if and only if”)

34. Negation

35. Conjunction

36. Disjunction

37. Exclusive disjunction

38. Conditional

39. biconditional

40. Example p: "The sky is cloudy". q: "A triangle has three sides". r: "It's raining". s: "The bell is ringing". t: "The dog is barking". Write the statement: ¬p ∧ q: The sky is not cloudy and a triangle has three sides s →¬( t ∨ r ): If the bell is ringing, then it's not true that the dog is barking or it's raining p ⊻ s → t: If the sky is cloudy or the bell is ringing (but not both), then the dog is barking q ↔¬ t : A triangle has three sides if and only if the dog is not barking

41. Quantifiers These are words that describes the quantity of object Universal Quantifiers – a statement that proposes that a characteristic is true for all elements like “for all” or “for every”. Denoted by ∀ Existential Quantifiers- a statement that proposes that a characteristic is true for a specific element or elements like “there exists” or “some”. Denoted by ∃

42. Example ∀ x ((x < 0) -> (x 3 < 0)): Every negative x has a negative cube ∃ x (x ∈ R -> (x 3 < 0)) : Some value of x has a negative cube

43. Problem solving and reasoning

44. Deductive reasoning vs inductive reasoning The process of deductive reasoning is analyzing the general premises and draw specific conclusion while in inductive reasoning predicts the conclusion Deductive reasoning All catholic priests are men. Padre Damaso is a catholic priest. Therefore, Padre Damaso is a Man. Inductive reasoning Jessie is not using his bike. It is raining. Therefore, Jessie won’t use his bike because it is raining

45. Deductive reasoning 1. All insects have exactly six legs. Spiders have eight legs. Therefore, spiders are not insects. 2. Blue litmus paper turns red in the presence of acid. The blue litmus paper turned red after I dropped some liquid on it. Therefore, the liquid is acidic.

46. Inductive reasoning 1. 90% of the sales team met their quota last month. Pat is on the sales team. Pat likely met his sales quota last month. 2. I get tired if I don't drink coffee. Coffee is addictive. I'm addicted to coffee.

47. Polya’s problem solving technique Understand the Problem Devise a Plan Carry out the Plan Look Back

48. Kayla is 24 years. Kayla is twice as old as Erin when Kayla was as old as Erin is now. How old is Erin now? Understand the Problem Given: Kayla’s age now = 24 Kayla is 2x older than Erin when Kayla was the same age as Erin now Asked: The current age of Erin

49. Devise a Plan Use: K = Kayla’s age now E = Erin’s age Y = Years past when Kayla’s age was the same as Erin’s age Y years ago, K – Y = E K – Y = 2(E – Y) Now K = 24

50. Carry out the plan K = 24 K – Y = E K – Y = 2(E – Y) E = 2E – 2Y 2Y = 2E – E E = 2Y K – Y = E 24 – Y = 2Y 3Y = 24 Y = 8 E = 2Y = 2(8) = 16 y/o

51. Look Back 8 years ago, Kayla’s age = 24 – 8 = 16 same as Erin’s current age Erin’s age = 16 – 8 = 8 so Kayla is twice as old

52. What time after 3 o’clock will the hands of the clock are together for the first time? Given: It is after 3 o clock Asked: What time will both clock arms be together for the first time after 3 o clock Both clock arms move at different rate. The minute hand completes a revolution for 60 minutes while the hour arm completes a revolution for 720 minutes. To determine the position of the arms, we use angles as in radians. So one revolution is 2 π . The rates for the two arms are: π /30 for th e minute arm π /360 for the hour arm

53. The rates will be multiplied to how many minutes have passed. The next consideration is the initial positions of each arms. Let us use the 12 th hour mark as reference. The minute will start at the reference but the hour arm started at the 3 rd hour mark so an initial position of π /2 is considered for the hour arm. The difference between the positions of both arms presents the angle between them. Use: T = time passed in minutes Pm = minute arm position Ph = hour arm position B = angle between the two arms  

54. Since the two arms should be together B = 0 16.36 mins have passed from 3:00 when the two arms were together. So the time is 3:16.36  

55. As the two clocks show here the incident occurs between 3:16 to 3:17

56. A pump can pump out water from a tank in 11 hours. Another pump can pump out water from the same tank in 20 hours. How long will it take both pumps to pump out water in the tank? Given: Pump 1 can do its job for 11 h Pump 2 can do its job for 20 h Asked: How long can the two pump do its job together

57. The rate of pump 1 is 1 job for 11 hour(1/11) while the rate of pump 2 is 1 job for 20 h(1/20). Combining the two rates will lessen the time. The solution will be the sum of the two rates multiplied to the time consumed by the two pumps and equate it to one job. Use: T = time consumed by the two pumps to do one job The time is faster than both pump individually.