Problem Solving Strategies or Hueristics.pdf

PROBLEM SOLVING STRATEGIES OR HUERISTICS Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Learning Outcomes: 1. Define what a problem is. 2. Enumerate and discuss th...

PROBLEM SOLVING STRATEGIES OR HUERISTICS Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Learning Outcomes: 1. Define what a problem is. 2. Enumerate and discuss the families of the problem. 3. Apply the different problem-solving strategies or heuristics. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  What is a problem? What is the difference between problem and exercise? Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Problem Is a condition that challenges the learner to find a resolution, and for which the route to the answer is not instantly known. A question is a problem if the process of solution is not directly known but requires the student to apply imagination and prior knowledge in new and unfamiliar circumstances. To have a problem means to find actions appropriate to attain clearly conceived but not immediately attainable solutions. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Exercise If the answer or even the process of solving the problem is evident, it is no longer a problem but an exercise. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Families of Problems 1. Recreational Problems- This is also known as brain teasers; these problems are usually complex to little formal of mathematics, but instead rely on the creative use of basic strategic principles. They are excellent to work on because no specialized knowledge is needed, and any time spent thinking about a recreation problem will help us later with more mathematically complex issues. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Families of Problems Examples: 1. Who makes it but has no need for it? Who buys it but has no use for it? Who uses it but can neither see nor feel it? What is it? Coffin Answer: 2. What has head and tail but no body? Answer: Coin 3. What has an eye but cannot see? Answer: Needle Prepared by: Roy I. Branzuela MMWORLD: Mathematics in Math 11n-Mathematics in the Modern the Modern World World  Families of Problems 2. Contest problems- They are usually encountered during formal exams with limits. It often requires specialized tools and/or ingenuity to solve. Examples: quiz bee, term exams, and quizzes Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Families of Problems 3. Open-ended problems. These are mathematical equations that are sometimes vaguely worded and possibly have many solutions. Example: Arrange all 12 numbers from 1- 12 in the box below without being close to the numbers that follow or precede them. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  4 7 6 9 1 3 2 12 5 11 10 8 Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Common Types of Problems 1. Arrangement Problems Examples: Word Puzzle RWAET - WATER KEROJ - JOKER Image retrieved from https://apps.apple.com/us/app/jigsaw- PLODAAPP - puzzle/id495583717 PAPAPDOL Jigsaw Puzzle Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Common Types of Problems 2. Transformation Problems Examples: Image retrieved from https://apps.apple.com/us/app/jigsaw- puzzle/id495583717 Image retrieved from https://www.stemlittleexplorers.com/en/make-and-solve-tower-of- hanoi/ Maze Puzzle Tower of Hanoi Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Common Types of Problems 3. Structure Problems- These problems are with words or symbols, e.g. series problems in math, requires sequence/ transformation. These require recognition, mental and cognitive processes. Example: 2x+3=5, find the value of x. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Common Types of Problems 4. Insight Problems-These problems are seemed impossible to figure out, but then an alternative suddenly arise, and the problem is solved. These problems may have different solutions. Example: The product of two whole numbers is 96, and their sum is less than 30. What are the possibilities for the two numbers? Answer: 4 and 24, 6 and 16, 8 and 12 Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Problem Solving Strategies or Heuristics A problem-solving strategy is a plan of action to find a solution. Different techniques have different action plans associated with them The process of applying prior knowledge, experience, skills, and understandings to new and unfamiliar situations to complete tasks, make decisions, or achieve goals. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  1. Finding a Pattern It is a strategy in which the students look for patterns in the data to solve the problem. 1. What is the next shape? 2. Mike wants to know the next two numbers of the following sequence: 4.00, 4.25, 4.50, 4.75, 5.00, 5.50 5.25, 5.75 ____, ____. 36 3. What is the next number? 1, 4, 9, 16, 25, ____. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  2. Logical Reasoning It is also known as “If and then” approach, a conditional statement in solving problems, using rational, systematic series of steps based on sound mathematical procedures and given statements to conclude. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  2. Logical Reasoning Examples: 1. Some of the months have 30 days. Some months have 31 days. How many months have 3011 days? 2. Jason wears socks of two colors, blue and yellow. He has 20 blue socks and 20 yellow socks in a drawer altogether. Assuming he has to take out socks in the dark, how many must he take out to be sure 2he has a pair of socks. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  3. Guess and Check It is a problem-solving approach that students can use to resolve mathematical problems by predicting the answer and then inspecting that the guess fits the conditions of the problem. It requires students to guess a solution, test its exactness, and improve the guess using logical reasoning. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  3. Guess and Check Example: Which of the 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 Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  4. Divide and Conquer It is dividing the significant concentrations into small ones and then rules them. If the problem is relaxed, answer it directly. If the problem cannot be solved as is, divide it into smaller parts and solve one by one. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  4. Divide and Conquer Example: The furniture in a classroom consists of tables and chairs. The teacher is making a seating plan. If two students sit at each table, eight students will be left without a place. If three students sit at each table, four tables will be left empty. How many students are there in the room? Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  4. Divide and Conquer Example: Solution: Choose a variable: Let x be the number of tables. a. Write an expression for the number of students when they sit in 2s: 2x+8 b. Write an expression for the number of students when they sit in 3s: 3(x-4) c. Write an equation: Expressions in (a) and (b) both give the number of students. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  4. Divide and Conquer Example: Solution: Thus, 3(x-4)=2x+8 2x+8=2(20)+8=48 students Solve the equation: 3(x-4)=3(20-4)=3(16)=48 3(x-4)=2x+8 students 3x-12=2x+8 3x-2x=8+12 Therefore, there are 48 X=20 (number of tables) students in the classroom. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  5. Working Backward This strategy entails starting with the results and reversing the steps needed to get those results, to figure out the answer to the problem. This strategy is tremendously useful in dealing with a condition or a sequence of events. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  5. Working Backward Example: Erick walked from Palo to Tolosa. It took 1 hour and 25 mins to walk from Palo to Tanauan. Then it took 25 mins to walk from Tanauan to Tolosa. He arrived at Tolosa at 2:45 pm. At what time did he leave Palo? Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  5. Working Backward Example: Solution: What do you need to find? You need to find the specific time when Erick had left from Palo. How can you solve the problem? You can work backward from the time Erick reached VSU Tolosa. Subtract the time it took to walk from Tanauan to VSU Tolosa. Then subtract the time it took to walk from Palo to Tanauan. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  5. Working Backward Example: Solution: Start at 2:45. This is the time Erick reached Tolosa. Subtract 25 minutes. This is the time it took to get from Tanauan to Tolosa. So, the time is 2:20 Subtract 1 hour 25 minutes. This is the time it took to get from Palo to Tanauan. Therefore, Erick left Palo at 12:55PM. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  6. Organizing Data Constructing an organized list, table, chart, or graph helps students establish their intelligence about a problem. It is also an essential step in investigating any set of data. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Ways of Organizing Data 1. By drawing a picture or diagram. Example: Look at the star pattern. One star has five pattern pieces. If your pattern has three stars, how many pieces will it have in all? Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Ways of Organizing Data Solution: 1 star- = 5 pattern pieces 3 star- = 15 pattern pieces Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Ways of Organizing Data 2. Making a chart, table, list, or graph Example: Tomorrow is the first day of school, and Sharmae is choosing her outfit to wear. She has black and green slacks; 3 blouses (red, flowers, plain) and two sweaters (blue and cream). How many different outfits can she make consisting of one pair of slacks, one blouse and one sweater? Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Ways of Organizing Data Solution: Make a list. Slacks: (B-black, G-green) Blouses: (R-red, F-flowers, P-plain) Sweaters: (B-blue, C-cream) BRB GRB BRC GRC BFB GFB BFC GFC BPB GPB BPC GPC Therefore, Sharmae can have 12 different outfits. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  7. Act It Out Making yourselves the character in the problem Example: Three adventurers and three supports wish to cross a river. There is a boat that can carry up to three people and either adventurers or supports can operate the boat. However, it is never permissible for supports to outnumber the adventurers either in a boat or on the shore. How are the adventurers and supports going to get to the other side? Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  7. Act It Out Making yourselves the character in the problem Example: Three adventurers and three supports wish to cross a river. There is a boat that can carry up to three people and either adventurers or supports can operate the boat. However, it is never permissible for supports to outnumber the adventurers either in a boat or on the shore. How are the adventurers and supports going to get to the other side? Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  POLYA’S PROBLEM SOLVING STRATEGY Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Four basic principles of Problem Solving Polya identifies four basic principles of problem Solving. The four steps are: 1. Understanding the Problem 2. Devising a plan 3. Carrying out the plan 4. Looking back Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Understand the Problem: Comprehend all the words. What is asked for us to show? What is the data? Is there enough data to help us arrive at the solution? What is the condition? Can we restate the problem into our own arguments? What image or diagram can we think that will help us understand the problem? Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Devise a Plan Discover the connection between the data and the unknown. Have we gotten it before? Have we realized the same problem in a somewhat different form? Do we know an associated problem? Do we know a proposition that could be valuable? Look at the unidentified Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Carry out the Plan This step is frequently informal than devising a plan. In general, all you need is attention and persistence, given that you have the necessary skills. Persist with the strategy that you have selected. If it remains not to work, reject it, and choose another. Work carefully by checking each step. Can we see evidence that the step is right? Can we demonstrate that it is truthful? Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Look back Examine the solution obtained. Can we check the outcome? Can we check the dispute? Can we originate from the solution inversely? Can we see it at a glimpse? Can we use the product, or the technique for some other problem? Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Example 1 In three bowling games, Alma scored 138, 141, and 144. What score will she need in a fourth game to have an average of 145 for all four games? Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Example 1 1. Understand the problem. What should be the fourth score? Given data: 138- score in the first game. 141- score in the second game 144- score in the third game. 145- average of all the four games Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Example 1 2. Devise a plan. Formula for finding the average: 𝑠𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑑𝑎𝑡𝑎 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑎𝑡𝑎 𝑁1 + 𝑁2 + 𝑁3 … 𝑁𝑛 𝑥ҧ = 𝑁 Let 𝑁4 be the fourth score Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Example 1 3. Carry out the plan. Solve for 𝑁4 𝑁1 + 𝑁2 + 𝑁3 + 𝑁4 𝑥ҧ = 𝑁 𝑁4 = 𝑥ҧ 𝑁 − 𝑁1 + 𝑁2 + 𝑁3 𝑁4 = 145 4 − 138 + 141 + 144 𝑁4 = 580 − 423 𝑁4 = 157 Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Example 1 4. Look back Compute if the average is 145 given that the final score is 157. 𝑁1 + 𝑁2 + 𝑁3 + 𝑁4 𝑥ҧ = 𝑁 138 + 141 + 144 + 157 145 = 4 1880 145 = 4 Prepared by: Roy I. Branzuela 145 = 145 MMWORLD: Mathematics in the Modern World  Your Turn 1. Jay has many animals on his farm. He has 72 chickens, which make up 60% of his total animals, and the rest of his animals are sheep. How many legs in total do his animals have? 2. Twice the difference of a number and 1 is 4 more than that number. Find the number. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Your Turn 3. Sally was having a party. She invited 20 women and 15 men. She made 1 dozen blue cupcakes and 3 dozen red cupcakes. At the end of the party there were only 5 cupcakes left. How many cupcakes were eaten? Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  QUIZSHO W! Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  1. Find the 3rd and 6th term in the series: 1, 2, __, 42, 1806, __ Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  1. Find the 3rd and 6th term in the series: 1. 2. 6. 42. 1,806. 3,263,442. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  2. Find the next two terms of the given sequence: 2, 3, 6, 11, 18, __, __ Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  2. Find the next two terms of the given sequence: 2, 3, 6, 11, 18, 27, 38 Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  3. Suppose that you and your brother both play baseball. Last season, you had 12 more hits than 3 times the number of hits that your brother had. If you had 159 hits, could you figure out how many hits your brother had? Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Solution: 3𝑥 + 12 = 159 3𝑥 = 159 – 12 3𝑥 147 = 3 3 𝑿 = 𝟒𝟗 Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  4. Which among the following numbers: 5, 8, 14, is the solution to the given linear equation? (𝑥 + 5) 2 − 𝑥 = −20 2 Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Solution: Guess and Check (5 + 5) − 52 = −20 2 10 − 52 = −20 2 5 − 25 = −20 −20 = −20 Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  5. Lynn’s mom gave Lynn some lunch money. After spending $3 on a sandwich and $1 on milk, she still has $2. How much was her lunch money? Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Solution: Working backwards $2 – remaining money $2 + $1 (milk) + $3 (sandwich) $6 – Total lunch money Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  6. Sanika delivered a total of 126 papers last week. If she delivers twice as many papers on each day of the weekend as she does on each day of the week, how many papers does she deliver on Sunday? Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Solution: 5𝑥 + 2𝑥 + 2𝑥 = 126 9𝑥 = 126 9𝑥 126 = 9 9 𝑥 = 14 Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Solution: Since 2x represents the number of newspapers delivered on Saturday and Sunday therefore, 2(14) = 28 newspapers where delivered on Sunday. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  7. How many different outfits can we make out of these given set of clothes: T-shirt - (B-lack, G-rey, W-hite) Pants - (B-lack, C-ream, A-sh grey, Bl-ue) Sneakers - (C-ortez, A-ir force 1, S- amba) Watch - (R-olex, C-artier, P-atek Philippe Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Solution: 1 T-shirt = 36 combinations 36 x 3 (number of T-shirt) = 108 total combinations Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  8. In 2017, Venus’ Birthday fell on Saturday, June 3. On what day of the week does Venus’ birthday fall in 2020? Note that the year 2020 is a leap year. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Solution: 2018 – June 3 (Sunday) 2019 – June 3 (Monday) 2020 – June 3 (Wednesday) Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  9. Consider 8 x 8 checkerboard below. How many squares of all sizes appear on this checkerboard? Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Solution: 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 = 204 squares. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  10. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  10. Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World  Thank you for Listening! Prepared by: Roy I. Branzuela MMWORLD: Mathematics in the Modern World