Math 3 Handout PDF

GE1707 Problem Solving and Reasoning Inductive and Deductive Reasoning Inductive Reasoning is the process of reaching a general conclusion by examining specific examples. This is the type of reasoning that forms a conclusion based on the examination of speci...

GE1707 Problem Solving and Reasoning Inductive and Deductive Reasoning Inductive Reasoning is the process of reaching a general conclusion by examining specific examples. This is the type of reasoning that forms a conclusion based on the examination of specific examples. The conclusion formed is often called a conjecture, since it may or may not be correct. Example 1: Use Inductive Reasoning to Predict a Number Use inductive reasoning to predict the next number in each of the following lists. a. 3,6,9,12,15, ? b. 1,3,6,10,15, ? Solution: a. Each successive number is 3 larger than the preceding number. Thus, we predict that the next number on the list is 3 larger than 15, which is 18. b. The first two (2) numbers differ by 2. The second and the third numbers differ by 3. It appears that the difference between any two (2) 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: Use Inductive Reasoning to Make a Conjecture 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. Complete the above procedure for several different numbers. 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: 8 × 5 = 40 Add 6: 40 + 6 = 46 Divide by 2: 46 ÷ 2 = 23 Subtract 3: 23 − 3 = 20 We started with 5 and followed the procedure to produce 20. Starting with 6 as our original number produces a final 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 four (4) times the original number. We conjecture that following the given procedure produces a number that is four (4) times the original number. Example 3: Use Inductive Reasoning to Solve an Application Scientists often use inductive reasoning. For instance, Galileo Galilei (1564–1642) used inductive reasoning to discover that the time required for a pendulum to complete one swing, called the period of the pendulum, depends on the length of the pendulum. Galileo did not have a clock, so he measured the periods of pendulums in “heart-beats.” The following table shows some results obtained for pendulums of various lengths. For the sake of convenience, a length of 10 inches has been designated as 1 unit. Length of pendulum, in units Period of pendulum, in heartbeats 1 1 4 2 9 3 16 4 25 5 36 6 Use the data in the table and inductive reasoning to answer each of the following questions: 03 Handout 1 *Property of STI Page 1 of 7  GE1707 a. If a pendulum has a length of 49 units, what is its period? b. If the length of a pendulum is quadrupled, what happens to its period? Solution: a. In the table, each pendulum has a period that is the square root of its length. Thus, we conjecture that a pendulum with a length of 49 units will have a period of 7 heartbeats. b. In the table, a pendulum with a length of 4 units has a period that is twice that of a pendulum with a length of 1 unit. A pendulum with a length of 16 units has a period that is twice that of a pendulum with a length of 4 units. It appears that quadrupling the length of a pendulum doubles its period. Deductive Reasoning is the process of reaching a conclusion by applying general assumptions, procedures, or principles. Example 1: Use Deductive Reasoning to Establish a Conjecture Use deductive reasoning to show that the following procedure produces a number that is four (4) times the original number. Procedure: Pick a number. Multiply it by 8, add 6 to the product, divide the sum by 2, and subtract 3. Solution: Let 𝑛𝑛 represent the original number. Multiply the number by 8: 8𝑛𝑛 Add 6 to the product: 8𝑛𝑛 + 6 8𝑛𝑛+6 Divide the sum by 2: = 4𝑛𝑛 + 3 2 Subtract 3: 4𝑛𝑛 + 3 − 3 = 4𝑛𝑛 We started with 𝑛𝑛 and ended with 4𝑛𝑛. The procedure given in this example produces a number that is four (4) times the original number. 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: Each of the four (4) neighbors, Sean, Maria, Sarah, and Brian, has a different occupation (editor, banker, chef, or dentist). From the following clues, determine the occupation of each neighbor. 1. Maria gets home from work after the banker but before the dentist. 2. Sarah, who is the last to get home from work, is not the editor. 3. The dentist and Sarah leave for work at the same time. 4. The banker lives next door to Brian. Solution: From clue 1, Maria is not the banker or the dentist. In the following chart, write X1 (which stands for “ruled out by clue 1”) in the Banker and the Dentist columns of Maria’s row. From clue 2, Sarah is not the editor. Write X2 (ruled out by clue 2) in the Editor column of Sarah’s row. We know from clue 1 that the banker is not the last to get home, and we know from clue 2 that Sarah is the last to get home; therefore, Sarah is not the banker. Write X2 in the Banker column of Sarah’s row. 03 Handout 1 *Property of STI Page 2 of 7  GE1707 From clue 3, Sarah is not the dentist. Write X3 for this condition. There are now Xs for three of the four occupations in Sarah’s row; therefore, Sarah must be the chef. Place a  in that box. Since Sarah is the chef, none of the other three people can be the chef. Write X3 for these conditions. There are now Xs for three of the four occupations in Maria’s row; therefore, Maria must be the editor. Insert a  to indicate that Maria is the editor, and write X3 twice to indicate that neither Sean nor Brian is the editor. From clue 4, Brian is not the banker. Write X4 for this condition. Since there are three Xs in the Banker column, Sean must be the banker. Place a  in that box. Thus Sean cannot be the dentist. Write X4 in that box. Since there are 3 Xs in the Dentist column, Brian must be the dentist. Place a  in that box. Sean is the banker, Maria is the editor, Sarah is the chef, and Brian is the dentist. 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 any, so this year it 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. 03 Handout 1 *Property of STI Page 3 of 7  GE1707 b. Because the conclusion is a specific case of a general assumption, this argument is deductive reasoning. Polya’s Problem Solving Strategy One of the foremost recent mathematicians to make a study of problem solving was George Polya (1877-1985). He was born in Hungary and moved to the United States in 1940. The basic problem-solving strategy that Polya advocated considered of the following four (4) steps. Polya’s Four-Step Problem Solving Strategy 1. Understand the problem. 2. Devise a plan. 3. Carry out the plan. 4. Review the solution. Understand the Problem. This part of Polya’s four-step strategy is often overlooked. 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 a missing information that, if known, would allow you to solve the problem? Is there an extraneous information that is not needed to solve the problem? What is the goal? Devise a Plan. Successful problem solvers use a variety of techniques when they attempt to solve a problem. The following 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, then check your result. 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. Review 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 if there are generalizations of the solution that could apply to other problems. Example 1: Karl Friedrich Gauss was a scientist and mathematician. His work encompassed several disciplines including number theory, analysis, astronomy, and optics. He is known for having shown mathematical prowess as early as the age of three. It is reported that soon after Gauss entered elementary school, his teacher assigned the problem of finding the sum of the first 100 natural numbers. Gauss was able to determine the sum in a matter of few seconds. The following solution shows the thought process he used. 03 Handout 1 *Property of STI Page 4 of 7  GE1707 Understand the Problem The sum of the first 100 natural numbers is represented by 1 + 2 + 3 + ⋯ + 98 + 99 + 100. Devise a Plan Adding the first 100 natural numbers from left to right would be time consuming. Gauss considered another method. He added 1 and 100 to produce 101. He noticed that 2 and 99 have a sum of 101, and that 3 and 98 have a sum of 101. Thus the 100 numbers could be thought of as 50 pairs, each with a sum of 101. Carry Out the Plan To find the sum of the 50 pairs, each with a sum of 101, Gauss computed 50 × 101 and arrived at 5050 as the solution. Review the Solution Because the addends in an addition problem can be placed in any order without changing the sum, Gauss was confident that he had the correct solution. The sum 1 + 2 + 3 + ⋯ + (𝑛𝑛 − 2) + (𝑛𝑛 − 1) + 𝑛𝑛 can be found by using the following formula. A summation formula for the first 𝑛𝑛 natural numbers: 𝑛𝑛(𝑛𝑛 + 1) 1 + 2 + 3 + ⋯ + (𝑛𝑛 − 2) + (𝑛𝑛 − 1) + 𝑛𝑛 = 2 Example 2: A baseball team won two (2) out of their last four games. In how many different orders could they have two (2) twins and two (2) losses in four (4) games? Understand the Problem There are many different orders. The team may have won two (2) straight games and lost the last two (2) (WWLL). Or maybe they lost the first two (2) games and won the last two (2) (LLWW). Of course there are other possibilities, such as WLWL. Devise a Plan We will make an organized list of all 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. Carry Out the Plan Each entry in our list must contain two (2) Ws and two (2) 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 W, then and only then do we write an L. This strategy produces the six (6) different orders as shown below. 1. WWLL (Start with two (2) wins) 2. WLWL (Start with one (1) win) 3. WLLW 4. LWWL (Start with one (1) loss) 5. LWLW 6. LLWW (Start with two (2) losses) 03 Handout 1 *Property of STI Page 5 of 7  GE1707 Review the Solution We have made an organized list. The list has no duplicates and it considers all possibilities, so we are confident that there are six (6) different orders in which a baseball team can win exactly two (2) out of four (4) games. Problem Solving with Patterns Term 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 fifth term. The three (3) dots “...” indicate that the sequence continues beyond 65, which is the last written term. It is customary to use the subscript notation 𝑎𝑎𝑛𝑛 to designate the 𝒏𝒏𝒏𝒏𝒏𝒏 term of a sequence. That is, 𝑎𝑎1 represents the first term of a sequence. 𝑎𝑎2 represents the second term of a sequence. 𝑎𝑎3 represent the third term of a sequence.... 𝑎𝑎𝑛𝑛 represents the 𝑛𝑛𝑛𝑛ℎ term of a sequence. nth-Term Formula for a Sequence The “nth” term is a formula with “n” in it which enables you to find any term of a sequence without having to go up from one term to the next. “n” stands for the term number, so to find the 50th term, we would just substitute 50 in the formula in place of “n”. Example: 1,4,7,10, … This has a difference which is 3. Find the nth term by using this formula: nth term = 𝑑𝑑𝑑𝑑 + (𝑎𝑎 − 𝑑𝑑) Where 𝑑𝑑 is the difference between the terms, 𝑎𝑎 is the first term and 𝑛𝑛 is the term number. nth term = 3𝑛𝑛 + (1 − 3) Which becomes nth term = 3𝑛𝑛 − 2 Difference Table The difference table shows the differences between successive terms of the sequence. The following is a difference table for the sequence 2, 5, 8, 11, 14, … Each of the numbers in row (1) of the table is the difference between the two (2) closest numbers just above it (upper right number minus upper left number). The differences in row (1) are called the first differences of the sequence. In this case, the first differences are all the same. Thus, if we use the above difference table to predict the next number in the sequence, we predict that 14 + 3 = 17 is the next term of the sequence. This prediction might be wrong; however, the pattern shown by the first differences seems to indicate that each successive term is 3 larger than the preceding term. 03 Handout 1 *Property of STI Page 6 of 7  GE1707 The following is a difference table for the sequence 5, 14, 27, 44, 65, … In this table, 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 are shown in row (2). These differences of the first differences are called the second differences. The differences of the second differences are called the third differences. To predict the next term of a sequence, we often look for a pattern in a row of differences. For instance, in the following table, the second differences are all the same constant, namely 4. If the pattern continues, then a 4 would also be the next second difference, and we can extend the table to the right ad shown. 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. REFERENCES: Aufmannn R., Lockwood J., Nation R., & Clegg D. (2013). Mathematical excursions (3rd ed.). Cengage Learning. The nth term. (2018). Retrieved from: https://www.s-cool.co.uk/gcse/maths/sequences/revise-it/the-nth-term 03 Handout 1 *Property of STI Page 7 of 7