Mathematics in the Modern World, Lecture notes of Mathematics

Download Mathematics in the Modern World and more Lecture notes Mathematics in PDF only on Docsity! set MATHEMATICS IN-THE MODERN WORLD. ee Ps ! ON ——_ a TT, vO <a . |[Medule 3 . Problern eh Module 3 - PROBLEM SOLVING AND REASONING Introduction The major purpose of learning mathematics is to develop an ability to understand and interpret the world, and to solve the problems that occur in it. We cannot be certain of the role that future technologies will play in the lives of our students when they are adults, nor can we predict all the skills and knowledge they will require in their work. We can be sure, however, that our students, as adults, will be solving problems, and we can help to prepare them for the future by teaching them problem-solving skills that will allow them to use mathematics in meaningful ways. This module is intended to give students a learning experience that will tell them, show them, and let them try to do problem solving. Learning Outcomes Upon completion of the module, the students should be able to: 1. use different types of reasoning to justify statements and arguments made about mathematics and mathematical concepts; 2. write clear and logical proofs; 3. solve problems involving patterns and recreational problems following Polya's four steps; and 4. organize one's methods and approaches for proving and solving problems. Overview What Is Problem Solving? Problem solving is the process of applying prior knowledge, experience, skills, and understandings to new and unfamiliar situations in order to complete tasks, make decisions, or achieve goals. In the mathematics program, problem-solving situations can provide the meaningful and stimulating experiences through which students learn concepts and skills. By involving students in problem-solving experiences, math could be taught in an exciting and effective way. 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: Let us consider 3 as the original number and we proceed with the steps as instructed. Original number : 3 Multiply by 8 : 3x 8=24 Add 6 : 24+6=30 Divide by 2 : 30+2=15 Subtract 3 : 15-3=12 It is observed that when we started with 3 and followed the procedure, we produced 12. When we start with 4, the resulting number will be 16. With 10 as the original number, following the same procedure we will arrive at 40. Starting with 25, we will get 100. For all the cases tried, it was observed that the resulting number is always 4 times the original number. Hence, a conjecture that the resulting number is four times the original number is made, following the given procedure. Counterexamples A statement is a true statement provided that it is true in all cases. If you can find one case for which a statement is not true, called a counterexample, then the statement is a false statement. Example 3. Finding a counterexample Verify that each of the following statements is false statement by finding a counterexample. For all numbers #7: a w>n b. vnF=n Solution: a. If = 1, we will have 17= 1. Since 1? is not greater than 1, so we have found a counterexample. Thus, “for all numbers 72, 7? > 7” is a false statement. b. For 7 = -2, JC2y = /4 = 2. Since -2 is not equal to 2, we have found a counterexample. Therefore, “for all numbers #, Vn? =n” isa false statement. Deductive Reasoning Another type of reasoning is called deductive reasoning. It is distinct from inductive reasoning because the process of arriving at the conclusion develops by applying general principles and procedures.  Deductive reasoning is the process of reaching a conclusion by applying general assumptions, procedures, or principles. Example 4. Using deductive reasoning to establish a conjecture instruction: Pick a number. Multiply the number by 8, add 6 to the product, then divide the sum by 2, and finally subtract 3 from the quotient. Use deductive reasoning to make a conjecture about the relationship between the size of the resulting number and the size of the original number. Solution: Let x be the original number. Original number : x Multiply by 8 : 8x Add 6 : 8x +6 8x+6 Divide by 2 2 =4x+3 Subtract 3 : 4x+3-3=4x Observe that we started with x and ended with 4x. By applying the procedure, we resulted to a number that is four times the original number. Hence, our conjecture is that, “the resulting number is four times the original number.” Inductive Reasoning versus Deductive Reasoning In the following examples we will analyze arguments to determine whether inductive or deductive reasoning is used. Example 5. Determining types of reasoning Determine whether each of the following arguments is an example of inductive or deductive reasoning. a. During the past 8 years, a tree bears fruits every other year. This year the tree did not bear fruits, so next year it will bear fruits. b. All home improvements cost more than the estimate. The Architect estimated that my home improvement will cost P500,000.00. Thus my home improvement will cost more than P500,000.00. Solution: a. The argument demonstrates an inductive reasoning because it reaches a conclusion based on specific examples. b. The conclusion is a specific case of a general assumption. So, the argument is an example of deductive reasoning. Lesson 2. Problem Solving with Patterns Sequence A sequence is an ordered list of numbers. Each number in a sequence is called a term of the sequence. In a sequence, a, is the first term, a, is the second term, and so on, and the a, is used to designate the n‘ term. A formula that can be used to generate all the terms of a sequence is called an n® _ term formula. When we are given a sequence, we usually ask the questions: a. What is the next term? b. What formula or rule can be used to generate the terms? Example 1: Predicting the next term in a sequence Let us consider the sequence 2, 7, 24, 59, 118, ... Here, a, = 2, 2 is the first term of a sequence. @,= 7,7 is the second term of a sequence. While 24, 59, and 118 are the third, fourth, and fifth term of a sequence, respectively. To determine the next term, and the rule used to generate the other terms, it is useful to construct a difference table and look for a pattern. The following illustration is the difference table for the sequence 2, 7, 24, 59, 118, ... Sequence: 2 7 24 59 NUN First differences: 5 17 a oN Second differences: “27 Third differences: 6 6 6 Example 1: Apply Polya’s Strategy — Make an organized list 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: 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 (LLVWV). Of course there are other possibilities, such as WLWL. Devise a plan. We will make an organized list of all the possible orders. An organized fist is an enumeration of all feasible outcomes using a systematic recording that ensures that each of the different orders will be listed once and only once. 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 duplication. We can start writing with two Ws first, followed with one W, then two Ls, and lastly with one L. This strategy will produce all the six different orders, as shown below. Start with two Ws - WWVLL Start with one Wo - WLIWL, WLLW Start with two Ls = - LLWW Start with one L - LWLW, LWWL Look back. 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. Example 2: Apply Polya’s Strategy — Solve a simifar but simpler problem In a basketball league consisting of 10 teams each team plays each of the other teams exactly three times. How many league games will be played? Solution: Understand the problem. There are 10 teams in the league, and each team plays exactly three games against each be of the other teams. The problem is to determine the total number of league games that will played. Devise @ pian. Let us try to work on a similar but simpler problem. Consider a league with only four teams (denoted by A, B, C and D) in which each team plays with each of the other teams only once. The diagram on the next page illustrates that the games can be represented by line segments that connect the points A, B, C, and D. A B Since each of the four teams will play a game against each of the other three, we might conclude that this would result in 4x 3 = 12 games. However, the diagram above shows only six segments. It appears that our procedure has counted each game twice. For instance, when team A plays with team B, team B also plays with team A. To produce the correct result, we must divide our previous result, 12, by 2. Hence, four teams can play each other once in = = 6 games. Carry out the plan. Using the process developed above, we see that 10 teams can play each other once ina total of = 45 games. Since each team plays each opponent exactly three times, the total number games is 45 x 3 = 135. Look back. We could check our work by making an organized list, as in example 1, that includes all 10 teams represented by A, B,C, D, E, F, G, H,1, and J. The organized listing below would illustrate the 45 games that are required for each team to play with each other teams once. AB AC AD AE AF AG AH AI Ad BC BD BE BF BG BH BI BJ CD CE CF CG CH Cl CJ DE DF DG DH ODI DJ EF EG EH El EJ FG FH FI FJ GH Gl GJ Hl HJ ld Notice that first row has nine games, the second row has eight games, and so on. Thus, the 10 teams require 9+8+7+6+5+4+3+2+1 = 45 games if each team plays with every other team once. Since each team is required to play with each of the other teams thrice, then 45 x 3 = 135 games in all will be played in the league. Example 3: Apply Polya’s Strategy — Make a table and fook for a pattern Determine the digit 50 places to the right of the decimal point in the decimal representation of > Solution: Understand the problem. Express the fraction = in decimal form and look for a 7 pattern that will enable us to determine the digit 50 places to the right of the decimal point. Device a plan. Dividing 5 by 27 using a calculator will give us 0.185185185... Since the decimal representation repeats the digits 185 over and over again, we know that the digit located 50 places from the right of the decimal point is either 1, 8, or 5. A table may be useful for us to see the pattern and enable us to determine which one of the three digits is in the 50'" place. Since there are three digits that keeps on repeating and repeating, we use a table with three columns. 10 The first 12 decimal digits of the fraction = are shown in the table below. Column 1 Column 2 Column 3 Location Digit Location Digit Location Digit 1st 1 2nd 8 3rd 5 4th 1 5th 8 6th 5 7th 1 8th 8 9th 5 10th 1 11th & 12th 5 Carry out the plan. Only in Column 3 is each of the decimal digit locations are consistently divisible by 3. From this pattern we can tell that 48th decimal digit is divisible by 3 and must be 5. Since 50" place is two digits away from the 48", then it falls on Column 2 and therefore must be 8. Look back. If we look back to the table above, additional pattern is illustrated. If each of the location numbers is Column 1 is divided by 3, the remainder is 1. If each of the location numbers in Column 2 is divided by 3, the remainder is 2. When we divide 50 by 3, the remainder is 2, so itis in Column 2. Therefore, the 50" digit is 8. Example 4: Apply Polya’s Strategy — Work backwards When Maureen won a certain amount in a raffle draw, she saved half of it, one- third of the other half she spent for a furniture and the remaining amount for a colored television worth P10,000. How much did she win? Solution: Understand the problem. We need to determine the amount that Maureen won in a raffle draw. Devise a plan. We could guess and check, but we might need to make several guesses before we found the correct solution. An algebraic solution might work, but setting up the necessary equation could be challenging. Since we are given the end result (the P10,000 worth of colored television), let us try the method of working backwards. Carry out the plan. Maureen must have P15,000 before buying the furniture: and P30,000 before saving an amount. This means that she won P30,000. Look back. To check our solution, we start with P30,000 and proceed through each event that happened. P30,000 less half of it is P15,000. One-third of P15,000 is P5,000. The remaining amount is P10,000, which was the amount of the colored television. Example 5: Apply Polya’s Strategy — Guess and check The product of the ages, in years, of three teenagers is 4590. None of the teens are of the same age. What are the ages of the teenagers? Solution: Understand the problem: We need to determine three distinct counting numbers. Take note that teenagers are usually of ages 13 to 19. So we need to find out which of these numbers have a product of 4590. 11