MATM111 Reviewer PDF

MATM111 Coverage: PATTERNS - Are regular, repeated or recurring 1. Nature of Mathematics forms or designs. 2. Language of Mathematics 3...

MATM111 Coverage: PATTERNS - Are regular, repeated or recurring 1. Nature of Mathematics forms or designs. 2. Language of Mathematics 3. Sets, Relation and Functions Fibonacci Sequences 4. Fundamentals of Logic - Leonardo Fibonacci discovered the 5. Problem-solving sequence. Mathematics - The sequence begins with one. Each subsequent number is the sum of the - It is formal system of thought for two preceding numbers. recognizing, classifying, and - Thus the sequence begins as follows: exploiting of patterns. What is Math about? 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…. - Numbers, symbols, Notations - Operations, Equations, functions 1+1 = 2, 1+2 = 3, 2+3 = 5, 3+5 = 8, 5+8 = - Process 13… Where is Math? - It is everywhere. What it is for? - To help us unravel the puzzle of nature, a useful way to think about nature. - Organize patterns and regularities as well as irregularities. - To help us control weather and epidemics - Provides new questions to think about. How is mathematics done? - With Curiosity - With eagerness for seeking patterns Fibonacci Spiral and generalizations - Constructed by placing together - With desire to know the truth rectangles of relative side lengths - With trial and error - Without fear of facing more questions equaling Fibonacci numbers. and problems - Fibonacci numbers have geometric Who uses Math? applications. - Mathematicians: pure and applied - A spiral can then be drawn starting - Scientist : natural and social from the corner of the first rectangle of - Practically EVERYONE side length 1, all the way to the corner Why is it important to know? of the rectangle of side length 13. - It puts order in disorder. - It helps us become better persons - It helps us make the world a better place to live on. 1|Page  MATM111 Golden Ratio Mathematical Language and Ratio Importance of language: - In mathematics and the arts, two - It facilitates communication and quantities are in the golden ratio. clarifies meaning - If the ratio between the sum of those - It allows people to express themselves quantities and the larger one is the and maintains their identity. same as the ratio between the larger - It bridges the gap among people from one and the smaller. varying origins and culture without prejudice to their background and upbringing. - In this case, we refer to a very Characteristics of mathematical Language: important number that is known as the - Precise (able to make very fine golden ratio. distinctions or definitions) - The golden ratio is a mathematical - Concise (able to say things briefly) constant approximately 1.6180339887. - Powerful (able to express complex Recall: thoughts with relative ease) 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597,... EXPRESSION - When we divide one of the Fibonacci - A name given to mathematical object numbers to the previous one, we will of interest get results that are so close to each - Number, set, function, ordered pair, other. matrix - Moreover, after the 13th number in the MATHEMATICAL SENTENCE sequence, the ratio will be fixed at - Must state a complete thought approximately 1.618… - A mathematical sentence can be true, false or sometimes true/sometimes 233 / 144, 377 / 233, 610 / 377, 987 / 610, 1597 false / 987, 2584 / 1597… - 233 yun 13th number, tas pag dinivide mo yon. 233 ÷ 144 = 1.618… lahat ENGLISH VS MATHEMATICS magiging 1.618… yun sagot simula sa 13th number, yun fixed na ratio which is 1.6180339887. Vitruvian Man - This illustrates that the human body is proportioned according to the golden ratio. - Leonardo Da Vinci has long been associated with the golden ratio. - An American researcher, Jay Hambridge, established that the golden ratio can also be found in the human body. 2|Page  MATM111 Examples: if it is English/ mathematical 2. 𝑥𝑥 + 2 < 𝑥𝑥 − 3 = 1 + 2 < 1 - 3 sentence or not x=1 =3 1 = 1 > 1 b. −12 > -1 = 1 > - 1 c. 02 > 0 = 0 > 0 d. 22 > 2 = 4 > 2 - hahanapan mo lng false statement yun true statement or yun nasa given. - Parang ano yan, lagi may x, itry mo sa lahat ng number, like 1, 2, -1, 0 etc… tas pag may nahanap kang false statement or mali na sagot. Yon, tawag B. Making a conjecture doon counterexample. Ex: Complete the below procedure for several different number. Use inductive reasoning to Inductive and Deductive Reasoning make a conjecture about the relationship Note: when you use inductive reasoning, you between the size of the resulting number and have no guarantee that your conclusion is the size of the original number. correct. Consider the following procedure: Deductive Reasoning Pick a number. Multiply the number by 8, add - The process of reaching a conclusion 6 to the product, divide the sum by 2 and by applying assumptions, procedure or subtract 3. principles. - 5, 40, 46, 23, 20 A. Establishing a conjecture - 7, 56, 62, 31, 28 Pick a number. Multiply the number by 8, add - 9, 72, 78, 39, 36 6 to the product, divide the sum by 2 and - The resulting number is four times subtract 3. of the original number. - 5, 40, 46, 23, 20 - Kasi tingnan nyo, 5 x 4 = 20, 7 x 4 - 7, 56, 62, 31, 28 =28. - Tingnan nyo lagi yun relationship - 9, 72, 78, 39, 36 ng original number (or yun unang Use deductive reasoning to show that the number) sa result. following procedure produces a number that is four times the original number. 11 | P a g e  MATM111 - Ang deductive, dba general to specific. So x + y = 35 binigay na saatin yun yun clue/ sagot, ang x + 14 = 35 gagawin lang natin ipakita or ispecify na x = 35 – 14 tama yun magiging sagot nya x = 21 (pigs) - Kung kanina making, ngayon establishing. Si inductive – specific to general, binigay na Check: saatin yun given, so hahanapin nlng natin yun sagot(general) si deductive – general to x + y = 35 specific. Binigay na satin yun clue or yun 21 + 14 = 35 sagot(general), so ispespecify nlng natin kung paano nangyare or paano nakuha yun 2x + 4y = 98 sagot na yon 2(21) + 4(14) = 98 B. Logic Puzzles 42 + 56 = 98 - Same example sa maria, sarah, sean, - Gagawa kayo ng equation dyan para brian. Balikan nyo nlng lol makuha nyo yun sagot. - Tas sa huli dapat icheck kung tama ba tlg yun nakuha nyo - (d ko na explain to, kaya nyo na yan Polya’s Problem solving strategy HAHAHA. Chat nyo nlng ako kung d Example: nyu gets.) The number of ducks and pigs in the field is 35. The total number of legs among them is 98. Assuming each duck has exactly two legs and each pig has exactly four legs. Yon, naka 12 pages ako sa math dahil sa explanation kong mahaba hahaha sana Let: nakatulong yun explanation ko sainyo. Kung d x – ducks nyo pa rin gets yun mga yan, chat nyo nlng ako y – pigs video call nlng tayo hahaha. Kung gusto nyo makabisado ng madali yun truth table, may gawa x + y = 35 akong video! Nasa drive lng natin 2 videos yon 2x + 4y = 98 total of 20mins. Kayaaa natin tooo. Goood Luck!! - Aki 2[ x + y = 35] 2 2x + 4y = 98 2x + 4y = 98 2x + 2y = 70 2y = 28 2𝑦𝑦 28 = 2 2 y = 14 (ducks) 12 | P a g e