GEM 111 Mathematics in the Modern World a. Module 2 Chapter 3: Problem Solving and Reasoning Intended Learning Outcome a) Use different types of reasoning to justify statements and arguments made about mathematics and mathematical concepts. b) Express appreciation for mathematics as human endeavour. Introduction A problem can be solved in a more effective and interesting manner if we approach it from a different point of view. A different approach may yield the answer quickly and more efficiently. It also might reveal some interesting reasoning. This chapter helps you become a better problem solver and to reveal that problem solving can be an enjoyable experience. ACTIVITY Warm Up! Activity 1. Predict the next number. 2, 5, 10, 17, 26, ? Activity 2. Consider the following procedure: Pick a number. Multiply the number by 8, add 6 to the product, divide the sum by 2, and subtract 3. Analysis 1. What do you notice about the result? ____________________________________________________________________. 2. What strategies did you use to solve the problem? ____________________________________________________________________. 3. Did you find any pattern in solving problems? ____________________________________________________________________. GEM 111 Mathematics in the Modern World Chapter 3 3.1 Inductive Reasoning Inductive reasoning is the process of reaching a general conclusion by examining specific examples. Example 1 . Use inductive reasoning to predict a number: a.) 5, 10, 15, 20, 25, ? b.) 1, 3, 6, 10, 15, ? Solution a. Each successive number is 5 larger than the preceding number. Thus we predict that the most probable next number in the list is 5 larger than 25, which is 30. b. The first two numbers differ by 2. The second and the 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 differ by 5, we predict that the next number in the list will be 6 larger than 15, which is 21. Example 2. Consider the following procedure: Pick a number. Multiply the number by 9 Add 15 to the product. Divide the sum by 3. Subtract 5. Solution Suppose we choose number 6. Then the procedure would produce the following results: Original number: 6 Multiply by 9: 6 x 9 = 54 Add 15 : 54 + 15 = 69 Divide by 3: 69 ÷ 3 = 23 Subtract 5: 23 - 5 = 18 GEM 111 Mathematics in the Modern World We started with 6 and followed the procedure to produce 18. Starting with 7 as our original number produces a final result of 21. Starting with 10 produces a final result of 30. Starting with 100 produces a final result of 300. In each of these cases the resulting number is three times the original number. We conjecture that following the given procedure will produce a resulting number that is three times the original number. Practice 1 : Complete the table using example 2 procedure for several different numbers, such as 3, 5, 8,9,13, plus two more numbers of your choice. Enter your result in the table. Number Result 3 5 8 9 13 ___ ____ What did you notice about the result? _______________________________________. 3.2 Deductive Reasoning Deductive reasoning is the process of reaching a conclusion by applying general assumptions, procedures, or principles. In Example 2, you performed the following procedure on several numbers. 1. 2. 3. 4. 5. Pick a number. Multiply the number by 9 Add 15 to the product. Divide the sum by 3. Subtract 5. Use deductive reasoning to show that the following procedure produces a number that is three times the original number. Solution If we let n represent the original number. Multiply the number by 9: Add 15 to the product: Divide the sum by 3: Subtract 5: 9n 9n + 15 9n + 15 = 3n + 5 3 3n + 5 – 5 = 3n Therefore, the resulting number is always three times the original number. GEM 111 Mathematics in the Modern World Practice 2: Use deductive reasoning to show that the following procedure produces a number that is three times the original number. Procedure: Pick a number. Multiply the number by 6 Add 10 to the product Divide the sum by 2, Subtract 5. Let x represent the original number. 3.3 Inductive Reasoning vs. Deductive Reasoning Example 3. Determine whether each of the following arguments is an example of inductive reasoning or deductive reasoning. a. During the past 10 years, a tree has produced plums every other year. Last year the tree did not produce plums, so this year the tree will produce plums. b. All home improvements cost more than the estimate. The contractor estimated that my home improvement will cost $35,000. Thus my home improvement will cost more than $35,000. Solution a. This argument reaches a conclusion based on specific examples, so it is an example of inductive reasoning. b. Because the conclusion is a specific case of a general assumption, this argument is an example of deductive reasoning. Practice 3. Determine whether each of the following arguments is an example of inductive reasoning or deductive reasoning. a. All Janet Evanovich novels are worth reading. The novel Twelve Sharp is a Janet Evanovich novel. Thus Twelve Sharp is worth reading. b. I know I will win a jackpot on this slot machine in the next 10 tries, because it has not paid out any money during the last 45 tries. GEM 111 Mathematics in the Modern World 3.4 Problem Solving Strategies Polya’s Problem Solving Strategy Polya’s Four-Step Problem-Solving Strategy 1. Understand the problem. 2. Devise a plan. 3. Carry out the plan. 4. Review the solution. Polya’s four steps are deceptively simple. To become a good problem solver, it helps to examine each of these steps and determine what is involved. 1. Understand the Problem . You must have a clear understanding of the problem. To help you focus on understanding the problem, consider the following questions. Can you restate the problem in your own words? Can you determine what is known about these types of problems? Is there missing information that, if known, would allow you to solve the problem? Is there extraneous information that is not needed to solve the problem? What is the goal? 2. Devise a Plan Successful problem solvers use a variety of techniques when they attempt to solve a problem. Here are some frequently used procedures. Make a list of the known information. Make a list of information that is needed. Draw a diagram. Make an organized list that shows all the possibilities. Make a table or a chart. Work backwards. Try to solve a similar but simpler problem. Look for a pattern. Write an equation. If necessary, define what each variable represents. Perform an experiment. Guess at a solution and then check your result. GEM 111 Mathematics in the Modern World 3.Carry Out the Plan Once you have devised a plan, you must carry it out. Work carefully. Keep an accurate and neat record of all your attempts. Realize that some of your initial plans will not work and that you may have to devise another plan or modify your existing plan. 4. Review the Solution Once you have found a solution, check the solution. Ensure that the solution is consistent with the facts of the problem. Interpret the solution in the context of the problem. Ask yourself whether there are generalizations of the solution that could apply to other problems. Example 3: A baseball team won two out of their last four games. In how many different orders could they have two wins and two losses in four games? Solution 1. Understand the Problem There are many different orders. The team may have won two straight games and lost the last two (WWLL). Or maybe they lost the first two games and won the last two (LLWW). Of course there are other possibilities, such as WLWL. 2. Devise a Plan We will make an organized list of all the possible orders. An organized list is a list that is produced using a system that ensures that each of the different orders will be listed once and only once. 3. Carry Out the Plan Each entry in our list must contain two Ws and two Ls. We will use a strategy that makes sure each order is considered, with no duplications. One such strategy is to always write a W unless doing so will produce too many Ws or a duplicate of one of the previous orders. If it is not possible to write a W, then and only then do we write an L. GEM 111 Mathematics in the Modern World This strategy produces the six different orders shown below. 1. WWLL (Start with two wins) 2. WLWL (Start with one win) 3. WLLW 4. LWWL (Start with one loss) 5. LWLW 6. LLWW (Start with two losses) 4. Review the Solution We have made an organized list. The list has no duplicates and the list considers all possibilities, so we are confident that there are six different orders in which a baseball team can win exactly two out of four games. Mathematical Problems Involving Patterns Terms of a Sequence An ordered list of numbers such as 5, 14 , 27 , 44 , 65, ... is called a sequence. The numbers in a sequence that are separated by commas are the terms of the sequence. In the above sequence, 5 is the first term, 14 is the second term, 27 is the third term, 44 is the fourth term, and 65 is the fi fth term. The three dots “...” indicate that the sequence continues beyond 65, which is the last written term. It is customary to use the subscript notation a n to designate the nth term of a sequence. Example 4. Predict the next term of a sequence using a difference table. 5, 14 , 27 , 44 , 65, …. Solution Construct a difference table. Source: Aufmann, et al.,2013 GEM 111 Mathematics in the Modern World Notice the first differences are not all the same. In such a situation it is often helpful to compute the successive differences of the first differences. These differences of the first differences are called the second differences. Source: Aufmann, et al.,2013 To predict the next term of a sequence, we often look for a pattern in a row of differences. Using the table above the second differences shown in blue are all the same constant which is 4 . If the pattern continues, then a 4 would also be the next second difference, and we can extend the table to the right as shown. Source: Aufmann, et al.,2013 Now we work upward. That is, we add 4 to the first difference 21 to produce the next first difference, 25. We then add this difference to the fifth term, 65, to predict that 90 is the next term in the sequence. This process can be repeated to predict additional terms of the sequence. LOGIC PUZZLE Logic puzzles, can be solved by using deductive reasoning and a chart that enables us to display the given information in a visual manner.. Example 5: Three children ( Ronald, Artfil , Kim) each have a different favorite color (blue, pink, yellow) and different pet (cat, turtle, fish). The following are true. 1. Ronald, whose favorite color is not pink, has a fish. 2. Artfil’s favorite color is yellow. 3. The kid who likes pink also has a turtle. GEM 111 Mathematics in the Modern World Solution: A table may help you organize your solution. Child Color Pet From Clue #1, Ronald has a fish. Child Ronald Color From Clue # 2,Artfil’s favorite color is yellow. Child Color Ronald Artfil yellow Pet fish Pet fish From Clue # 1, Ronald, whose favorite color is NOT pink, so it must be the favorite color of the last person. From Clue #3, the kid who likes pink also has a turtle. Child Ronald Artfil Color yellow pink Pet fish turtle From the information above, you can fill in the remaining blanks too obtain the following. Child Ronald Artfil kim Color blue yellow pink Pet fish cat turtle GEM 111 Mathematics in the Modern World Application General Instruction: Copy the given exercises and answer in long bond paper. Show your solution. Name: _____________________ Course/Section: ________________ Exercise 1. In numbers 1-8, Find a pattern and write the next three numbers in each sequence. 1.) 6, 10, 14, 18, 24, 26, ____ , ____ , _____ 2.) 5, 11, 17, 23, 29, 35, ____ , ____ , _____ 3.) 7, 9, 13, 19, 27, 37, ____ , ____ , _____ 4.) 1, 8, 27, 64, 125, 216, ____ ,____ , _____ 5.) 16, 22, 31, 43, 58 , 76 , ____ , ____, _____ 6.) 154, 133, 116, 103 , ____ , ____ , ____ 7.) 3, 7, 15, 31, 63, 127, ____ , ____ , ____ 8.) 40, 8 , 50 , 10, 60 , 12, _____, ____ , ____ 9.) Consider the following procedure: Pick a number. Multiply the number by 5, add 20 to the product, divide the sum by 2, and subtract 10. Complete the above procedure for 5 different numbers and enter your result in the table below. Number ___ ____ ___ ___ ____ Result ___ ____ ___ ___ ____ 10. Use inductive reasoning to make a conjecture about the relationship between the size of the result and the size of the original number (Table # 9). 11.) Think of a number and perform the following procedure. STEP 1 : Add 3 to a number STEP 2 : Double the result STEP 3: Subtract 2 STEP 4: Cut the result in half. STEP 5: Subtract your original number GEM 111 Mathematics in the Modern World Complete the above procedure for five different numbers and enter your result in the table. Number ___ ___ ___ ___ ___ Result ___ ___ ___ ___ ___ 12.) Use inductive reasoning to give conjecture of table # 11. 13.) Prove your conjecture using deductive reasoning. Begin by using m as your original number. STEP RESULT 1 2 3 4 5 EXERCISE 2 : Solve the Logic Puzzles 14.) Each of four neighbors, Jay-ar , Lean , Ruby, and Joan has a different occupation (Engineer, Banker, Chef, or Dentist). From the following clues, determine the occupation of each neighbor. a. Lean gets home from work after the banker but before the dentist. b. Ruby, who is the last to get home from work, is not the engineer. c. The dentist and Ruby leave for work at the same time. d. The banker lives next door to Joan. Arrives First Second Third Fourth Person Occupation 15.) Three girls ( Rose, Sheila, and Jessa) and one boy (Joemarie) each like a different color (blue, red, pink, green) and are of different ages (8,9,10,11). Determine their ages and favorite color based on the clues below. a. b. c. d. The oldest is a boy who do not like pink. Jessa is the youngest and likes blue. Sheila is one year older than Rose, The 9 year-old child likes red. GEM 111 Mathematics in the Modern World