Problem Solving: Inductive & Deductive Reasoning PDF

PROBLEM SOLVING What is a -a task that requires the learner to reason Problem? through a situation that will be challenging but not impossible Exercise? -provides practice in using algorithm and maintaining the basic facts Problem...

PROBLEM SOLVING What is a -a task that requires the learner to reason Problem? through a situation that will be challenging but not impossible Exercise? -provides practice in using algorithm and maintaining the basic facts Problem -encompasses exploring, reasoning, strategizing, estimating, conjecturing, testing, explaining, and Solving? proving. Problem Solving Process Goal Obstacle Solution 1. Inductive Reasoning -the process of reasoning to a general conclusion through observations of specific cases - als0 called induction - Often used by mathematicians and scientists to predict answers to complicated problems. 2. Deductive Reasoning -process of reasoning to a specific conclusion from general statement  Examples of Inductive Reasoning 1. Question: What is the next term of this sequence? 1,3,5,7,9,11,… 2. Question: What is the next shape ?  Examples of Inductive Reasoning 3. Q: What is the next shape? 4. Q: What is the next number in the sequence? 1/2, 2/6, 3/18, …. Prove, using deductive reasoning, that the procedure in previous example will always result in twice the original number selected PROBLEM: Each of four siblings ( Edmund, Genalyn, Madelyn, and Sonia) bought four different cars. One chooses Honda, a Mitsubishi, a Toyota, and a Suzuki car. From the following clues, determine which sibling bought which car. 1. Edmund, living alone, stays next door to his sister who bought the Honda car and very far from his sister who bought the Suzuki car. 2. Genalyn, also living alone, is younger than the one who bought the Mitsubishi car and older than her sibling who bought the Toyota car. 3. Madelyn did not like Toyota and Suzuki cars. But she and her sibling, who bought the Toyota car, live in the same house.  SOLUTION: Honda Mitsubis Toyota Suzuki hi Edmund x ok x x Genalyn x x x ok Madelyn ok x x x Sonia x x ok x Routine and Non-routine Problem *Routine problem - using at least one of the four Ex. A sales promotion in a store advertises a jacket regularly priced at Php1255.98 but arithmetic operations now selling for 20% off the regular price. and/or ratio to solve problems The store also waives the tax. You have Php1000 in your pocket. Do you have enough money to buy the jacket? *Non-routine problem -may be solved in many Ex. Mrs. Rivera wishes to tile her classroom different ways or floor with square tiles. She wants to use whole tiles, without cutting any pieces. The rectangular strategies and may floor has dimensions 8.4 meters by 7.2 meters. have more than one What is the minimum number of whole identical answer or solution square tiles required and what are the dimensions of each tile?  George Polya: “The Father of Problem Solving” POLYA’S 4-STEP: 4. Look back (verification) (STEPs 3. Carry out the plan (Insight) IN 2. Devise a plan (thinking time) PROBLEM 1. Understand the problem (Preparation) SOLVING) Problem Solving S T R A T E G I E s Applying the Strategies Fibonacci Sequence: 3. 100 students were Fibonacci, “Son of Bonaccio” interviewed 1+ 5 𝜑= 28 took PE, 2 31 took BIO, 1. Find the 12th 42 took ENG, Fibonacci number: 0, 1,1,2,3,5,8,13,21,34,55,…… 9 took PE and BIO, 10 took PE and ENG, 2. Put the numbers 1 to 9 into 6 took BIO and ENG, the square so that all rows, 4 took all three columns and diagonals add to subjects. the magic number. How many students took none of the three subjects?  Magic Squares A 3 x 3 magic Square Put the numbers 1 to 9 into the square so that all rows, columns and diagonals add to the magic number. 2 6 7 2 3 1 4 1 5 9 8 7 6 5 8 3 4 9 Magic Number = ? 1  A 4 x 4 Magic Square Put the numbers 1 to 16 into the square so that all rows, columns and diagonals add to the magic number. 1 16 3 2 13 5 10 11 8 9 6 7 12 4 15 14 1 34 Magic Number = ? 4. In a survey involving 150 different factories, it was found out that 70 purchased brand A 75 purchased brand B 95 purchased brand C 30 purchased brands A and B 45 purchased brands A and C 40 purchased brands B and C 10 purchased brands A, B, and C How many factories did not purchase the three brands?  A B U 20 15 5 10 35 30 20 15 C 2 brands: B and C, 40 = 10 + 30 A and C, 45 = 10 + 35 A and B, 30 = 10 + 20 1 brand: A= 70 – ( 20 + 10 + 35) = 5 B = 75 – ( 20 + 10 + 30) = 15 C = 95 – ( 35 + 10 + 30) = 20 Problem: Mercy has a certain amount in her bank account on Friday morning. During the day she writes a check for $ 24.50, makes an ATM withdrawal of $80 and deposit a check for $ 235. At the end of the day she sees that her balance is $ 451.25. How much money did she have in the bank at the beginning of the day? We need to find the money in Mercy’s bank account at the beginning of the day on Friday She took $24.50 and $80 and put $235 She ended up $451.25 Start with strategy $451.25 - $235 + $80 + $24.50 = 320.75 Check: Mercy’s starts with $ 320.75 She works a check $320.75 - $24.50 = $296.25 She withdraw $296.25 - $80 = $216.25 She deposits $235 $216.25 + $235 = $451.25 Guess and Check 9. Sofia takes a ribbon that is 48 inches long and cuts it in two pieces. One piece is three times as long as the other. How long is each piece? We need to find two numbers that add up to 48. One number is three times the other number. We guess two random numbers, one three times bigger than the other, and find the sum. If the sum is too small we guess larger numbers, and if the sum is too large we guess smaller numbers. Guess 5 and 15: 5+15=20 sum is too small Guess 6 and 18: 6+18=24 sum is too small 36 and 12 Work Backward Trina’s father is 36. He is 16 years older than four times Trina’s age. How old is Trina? Step 1. Understand: We need to find Trina’s age. We know her father is 16 years older than four times her age, or 4×(Trina's age)+16. We know her father is 36 years old. Step 2. Strategy: To get from Trina’s age to her father’s age, we multiply Trina’s age by four and add 16. Working backwards means we start with the father’s age, subtract 16 and divide by 4. Step 3: Apply Strategy/Solve Start with the father's age36 Subtract 16. 36−16=20 Divide by 4. 20÷4=5 Step 4: Check Trina is 5 years old. Her father’s age is 4(5)+16=36. This is correct. I am thinking of a two digit number……. itit is is odd. odd. 10 20 30 40 50 60 70 80 90 11 21 31 41 51 61 71 81 91 itstens its tensdigit digitisiseven. even. 12 22 32 42 52 62 72 82 92 13 23 33 43 53 63 73 83 93 itit is is prime. prime. 14 24 34 44 54 64 74 84 94 the sum the sum of of its its digits digits 15 25 35 45 55 65 75 85 95 is 11. is 11. 16 26 36 46 56 66 76 86 96 theproduct productof ofits its 17 27 37 47 57 67 77 87 97 the digitsisis24. digits 24. 18 28 38 48 58 68 78 88 98 19 29 39 49 59 69 79 89 99 The answer is 83.  Recreational RECREATION Problems using Math PROBLEM -involves riddles, puzzles, brain teasers and games carried out for leisure rather than application-based professional activity  B CARD E MAGIC CARD CARD CARD A CARD C CARD D -Players will determine who will go first by “rock-paper-scissors”. -Player 1 begins by calling out either number “1” or “2”. -Player 2 will now call out the next number by adding 1 or 2 from the number previously announced by player 1. The cycle goes until one player announces “20” and will be declared winner. Three stamps are to be torn from a sheet of nine stamps as shown below. The three stamps must be intact so that each stamp is joined to another stamp along at least one edge. Find the possible patterns for these three stamps. The figure below shows 9 matchsticks arranged as an equilateral triangle. Rearrange exactly 5 of the matchsticks to form 5 equilateral triangles, without leaving any stray matchsticks. stamps.