Mathematics in Modern World Chapter 3 - Problem Solving and Reasoning, Papers of Mathematics

Download Mathematics in Modern World Chapter 3 - Problem Solving and Reasoning and more Papers Mathematics in PDF only on Docsity! MATHEMATICS IN THE MODERN WORLD CHAPTER3: Problem Solving and Reasoning Objectives: a. Differentiate inductive and deductive reasoning b. Understand the Polya’s Problem Solving Strategy c. Apply different problem solving strategy in solving patterns, and recreational activities. Lesson 1:Inductive and Deductive Reasoning Inductive Reasoning Is the process of getting a general conclusion by observing the specific examples or set. Example 1: Use inductive reasoning to predict a next number. 3,6,9,12,15,? Solution: Each successive numberis 3 larger than the preceding number. Thus, we predict that the next number that 3 larger than 15is 18. Example 2: Use inductive reasoning to predict a next number. 1,3,6,10,15,? Solution: The first two numbers differ by 2. The second and third numbers differ by 3. It appears that the difference between any two numbers is always 1 more than the preceding difference. Since 10 and 15 is differ by 5 we predict that the next in the list willbe 6 larger than 15 which is 21. Example 3: Use inductive reasoning to make a conjecture. Consider the following procedure: Pick anumber. Multiply the number by 8, add 6 to the product, divide the sum by 2, and subtract 3. Complete the above procedure for several differentnumbers. Use inductive reasoning to make a conjecture about the relationship between the size of the resulting number and the size of the original number. Solution: Suppose we pick 5 as our original number. Then the procedure would produce the following results: Original Number: 5 Multiply by 8: 8x5=40 Add 6: 40+6=46 Page 1 MATHEMATICS IN THE MODERN WORLD Divide by 2: 46+2=23 Subtract 3: 23-3=20 We started with 5, and by following the procedure we got 20 as the result. Starting at 6 as our original number produce a result of 24. Starting with 10 produces a final result of 40. Starting with 100 produces a final result of 400. In each of these cases the resulting number is 4 times the original number. We conjecture that by following the given procedure produces 4 times the original number. Try to discuss the following scenarios as an example of Inductive reasoning: Scenario 1: Jennifer always leaves for school at 7:00 a.m. Jenniferis always on time. Jennifer assumes, then, that if she leaves at 7:00 a.m. for school today, she will be on time Scenario 2: The cost of goods was $1.00. The cost of labor to manufacture the item was $0.50. The sales price of the item was $5.00. So, the item always provides a good profit for the stores selling it. players on the high school team weigh more than 170 pounds. ‘3 Scenario 3: Ray is a football player. All the other football ’ Therefore, Ray must weigh more than 170 pounds. Deductive Reasoning - Is the process of reaching aconclusion by general assumption, procedures or principle. It is distinguish from inductive reasoning since deductive reasoning is finding conclusion by applying general principle and procedure in the observation. Example 1: Use deductive reasoning to establish a conjecture. Use deductive reasoning to show that the following procedure produces a number that is four times the original number. Procedure: Pick a number. Multiply the number by 8, add 6 to the product, divide the sum by 2, and subtract 3 Page 2 MATHEMATICS IN THE MODERN WORLD ao Scenario: When you're a student, professors and books claim to > prove things. But they don't say what's meant by "prove." You have to an catch on. Watch what the professor does, then do the same thing. Then you become a professor, and pass on the same "know-how" without "knowing what" that your professor taught you. Two interesting Assertions Assertion 1: Logicians don'ttell mathematicians whatto do. They make a theory out of what mathematicians actually do. Assertion 2: Any correct practical proof can be filledin to be a correct theoretical proof. - "If you can do it, then do it!" "It would take too long. And then it would be so deadly boring, no one would read it." Assertion B is commonly accepted. Yet I've seen no practical or theoretical argumentfor it, other than absence of counter-examples. It may be true. It's a matter of faith. Example: "Two points determine a line, and two lines determine a point, unless the lines are parallel." Projective geometry brings in ideal points at infinity —one pointfor each family of parallel lines. The axiom becomes: "Two points determine a line, and two lines determine a point." This is "right." Mathematical Intuition - Mathematical intuition is coming across a problem, glancing atit, and using your logical instinct to pull out an answer without asking further questions. Example: If you look at a number set with a range between 53 and 73 , comprising of the numbers 53, 64, 63, 61, 65, 73and were asked to calculate the average: your first guess would be around the 63-65 mark. How? There is onenumber below the 60s = 53 There is one number above the 60s = 73 These two numbers virtually cancel each other out. Page 5 MATHEMATICS IN THE MODERN WORLD Every other numberis in the 60s, with 61,63, 64 and 65, so the average mustbe arounda 63. All of this happens in less than a second - this is mathematical intuition. Certainty - In math, this one is something thatis accurate and absolute. Even if it's granted that the need for certainty is inherited from the ancient past, andis religiously motivated, its validity is independent of its history andits motivation. Example: We take three examples. First, good old 242=4 Second, The angle sum of any triangle equals two right angles." Finally, a more sophisticated example: a convergentinfinite series. Label the first example FormulaA: 2+2=4 By the associative law of addition, Formula A then is: 1+14+14+1=14+1+14+1 For more knowledge about Intuition, Proof and Certainty, please click the link provided. Q w - https://www.uni-siegen.de/fb6/phima/lehre/phima13/quellentexte/seminar_- beet hersh/hersh-chapter4, pdf Page 6 MATHEMATICS IN THE MODERN WORLD ata f=) atat e Inductive Reasoning is the process of getting a general a conclusion by observing the specific examples or set. 2 © Deductive Reasoning is the process of reaching a conclusion wey by general assumption, procedures or principle. 7 e Proof The old, colloquial meaning of "prove" is: Test, try out, . determine the true state of affairs. Re e Mathematical intuition is coming across a problem, glancing at it, and using your logical instinct to pull out an answer ee without asking further questions. e Certainty is something thatis accurate and absolute. oe JBN a lerale)s) ACTIVITY: Written in a Picture In this activity, students get to look at various pictures of real-life things and draw a general conclusion based off of the observations they make. Procedure: 1.) Students work individually or with a partner. 2.) Students make observations based off of pictures they see to come to a general conclusion. Page 7 MATHEMATICS IN THE MODERN WORLD Problem Solving Strategies - These are the different problem solving strategies that you can use in Mathematics. Problem Solving Strategies - Look for a pattern . Make an organized list . Guess and Check . Make a table . Work backwards . Use logical reasoning - Draw a diagram . Solve a simpler problem - Read the problem carefully : Greate problem solving journals Look fora pattern Example: Findthe sum of the first 100 even positive numbers. Solution: The sum of the first 1 even positive numbersis 2 orl(1+1) =1(2). The sum of the first 2 even positive numbersis 2 + 4 = 6 or2(2+1) =2(3). The sum of the first 3 even positive numbers is 2+ 44 6 = 120r3(3+1) =3(4). The sum of the first 4 even positive numbers is 2+44+6+48 = 200r4(4+1) =4(5). Look for a pattern: The sum of the first 100 even positive numbersis 2+4+6+..= ? or 100(100+1) =100(101) or 10,100. Page 10 MATHEMATICS IN THE MODERN WORLD ¢ Make an organized list Example: Find the median of the following test scores: 73, 65, 82, 78, and 93. Solution: Make a list from smallest to largest: 65 73 78 Since 78 is the middle number, the median is 78. 82 93 ¢ Guess and check Example: Which ofthe numbers 4, 5, or 6 is a solution to (n + 3)(n — 2) = 36? Solution: Substitute each number for “n” in the equation. Six is the solution since (6 + 3)(6 — 2) = 36. ¢ Make a table Example: How many diagonals does a 13- gon have? Solution: Make a table: Number of sides Number of diagonals 3 0 4 2 5 5 6 9 7 14 8 20 Look for a pattern. Hint: If n is the number of sides, then n(n-3)/2is the number of diagonals. Explain in words 13(13-3) why this works. A 13— gon wouldhave = 65 diagonals. ¢ Work backwards Page 11 MATHEMATICS IN THE MODERN WORLD Example: Fortune Problem: a man died and left the following instructions for his fortune, half to his wife; = ofwhat was left wentto his son; = of whatwas left wentto his butler; the man’s pet pig got the remaining $2000. How much money did the man leave behind altogether? Solution: The pig received $2000. sor? = $2000 2 = $6000 Sot ? = $6000 2= $7000 5 of 2 = $7000 2 = $14,000 e Use logical reasoning Example: At the Keep in Shape Club, 35 people swim, 24 play tennis, and 27 jog. Of these people, 12 swim and play tennis, 19 play tennis and jog, and 13 jog andswim. Nine people do all three activities. How many members are there altogether? Solution: Hint: Draw a Venn Diagram with 3 intersecting circles. ‘Swim “Ay ea e« Draw adiagram Example: Fortune Problem: a man died and left the following instructions for his fortune, half to his wife; + ofwhat was left wentto his son; = of whatwas left wentto his Page 12 MATHEMATICS IN THE MODERN WORLD 4 124+ 16+21+26 75 5 124+ 16+21+ 26+ 31 106 6 124+ 16+21+ 26+ 31+ 36 142 7 12+ 164214 26+ 31+ 36+ 41 183 8 124+ 164214 26+ 31+ 36+41+46 229 a) The number of dots for a pattern with 6 hexagons in the first column is 142 b) If there are 229 dots then the pattern has 8 hexagons in the first column. Example 2: Each member of a club shook hands with every other member who came for a meeting. There were a total of 45 handshakes. How many members were present at the meeting? = > = aw e e e e e e e e e e yay e e e e e Hcn s e e e e e Hm. e e e e e e ye e e e e e e e ye e e e e e e e M@|ipmso 8 7 6 5 4 3 2 °1 Solution: Total =9+8+7+6+5+4+4+3+2+1= 45 handshakes There were 10 members. & For more puzzles in Mathematics, please click the link provided: é wy -_~—sChttps://www.mathsisfun.com/puzzles/index. html Page 15 MATHEMATICS IN THE MODERN WORLD Recreational problems using Mathematics Puzzles and riddles are perhaps the most well-known activities within recreational math. Math puzzles and riddles are fun andinteresting, and they help improve problem solving skills and thinking capacity! Puzzles and riddles are also an important area of research for many mathematicians. The Puzzle: Alphabet Numbers Using any letter only once, what are the largest andsmallest 1p? numbers that you can write down in words? Example: EIGHTY But not NINETY as N is used twice Bonus 1: allow negatives such as MINUS TWO Bonus 2: allow calculations such as TWO SQUARED we The Puzzle: Alphabet Numbers (SOLUTION) Largest: FIVE THOUSAND Smallest: ZERO or NOUGHT Bonus 1: allow negatives Smallest: MINUS FORTY Bonus 2: allow calculations Largest: SIXTY CUBED (=216,000) from Steven Nguyen The Riddle 1: How do you go from 98 to 720 using just one letter? The Riddle 1: Answer Add an "x" between "ninety" and "eight". Ninety x Eight= 720 The Riddle 2: There is a three digit number. The second digitis fourtimes as big as the third digit, while the first digitis three less than the second digit. What is the number? 141 The Riddle 2: Answer Page 16 MATHEMATICS IN THE MODERN WORLD 42 & For more riddles in Mathematics, please click the link provided: 3 ¥ hag? https://www.brainzilla.com/brain-teasers/riddles/math/ ‘A APPLICATION 1. 2. a, e Polya’s 4 step problem solving strategies are: es - Understand the problem. - Devisea plan z , - Carry outthe problem =a - Look back Wie ACTIVITY: You got me. How can you add eight 8's to get the number 1,000? (only use addition). Choose your partner, and try to answerthese 5 riddles How do you make the number 7 an even number without addition, subtraction, multiplication or division? How can you take 2 from 5 and leave 4? | am a number with a couple of friends, quarter a dozen, and you'll find me again. What am |? https://nzmaths.co.nz/problem-solving-strategies http://web.mnstate.edu/peil/M110/Worksheet/PolyaP roblemSolve.pdf https://study.com/academy/lesson/inductive-and-deductive-reasoning- Page 17