Problem Solving: Inductive & Deductive Reasoning PDF

PROBLEM SOLVING INDUCTIVE & DEDUCTIVE  REASONING Prepared by: JOSEPH G. TABAN University of Northern Philippines Students’ Perspective  MMW by Joseph G. Taban , UNP 6 Basic Facial Expression  MMW by Joseph G. T...

PROBLEM SOLVING INDUCTIVE & DEDUCTIVE  REASONING Prepared by: JOSEPH G. TABAN University of Northern Philippines Students’ Perspective  MMW by Joseph G. Taban , UNP 6 Basic Facial Expression  MMW by Joseph G. Taban , UNP Teachers’ perspective  MMW by Joseph G. Taban , UNP  What is a problem?  Generally, it is a situation you want to change! A problem is a situation that conforms the learner, that requires resolution, and for which the path of the answer is not immediately known. There is an obstacle that prevents one from setting a clear path to the answer. MMW by Joseph G. Taban , UNP  What is a Problem Solving   Problem Solving has been defined as higher-order cognitive process that requires the modulation and control of more routine or fundamental skills" (Goldstein & Levin, 1987). MMW by Joseph G. Taban , UNP  Barriers to Effective Problem  Solving  Failure to recognize the problem  Conceiving the problem too narrowly  Making a hasty choice  Failure to consider all consequences  Failure to consider the feasibility of the solution ***Zaid Ali Alsagoff, Module 7: Problem Solving MMW by Joseph G. Taban , UNP A. Understanding Reasoning   Mathematical reasoning refers to the ability of a person to analyze problem situations and construct logical arguments to justify his process or hypothesis, to create both conceptual foundations and connections, in order for him to be able to process available information.MMW by Joseph G. Taban , UNP NCTM pointed out that….  People who can reason and think analytically tend  To note patterns, structure, or regularities in both real-world situations and symbolic objects;  To ask if those patterns are accidental or if they occur for a reason  To conjecture and prove MMW by Joseph G. Taban , UNP Reasoning  Students are expected to: 1. Define a statement 2. Identify the hypothesis and conclusion in a statement 3. Write conditional statements 4. Write the Converse, Inverse, Contrapositive of a given conditional statement. MMW by Joseph G. Taban , UNP B. Inductive and Deductive Reasoning  What kind of thinking is used when solving problems?  Inductive or deductive? MMW by Joseph G. Taban , UNP  Inductive Reasoning  The type of reasoning that forms a conclusion based on the examination of specific examples is called inductive reasoning. Specific Conclusio Examples n The conclusion formed by using inductive reasoning is often called a conjecture, since it may or may not be correct. MMW by Joseph G. Taban , UNP  Examples Example 1:  A baby cries, then cries, then cries to get a milk. We conclude that if a baby cries, he/she gets a milk. Example 2: Here is a sequence of numbers: 3, 6, 9, 12, ____ What is the 5th number? MMW by Joseph G. Taban , UNP Example 3:  You are asked to find the 6th and 7th term in the sequence: 1, 3, 6, 10, 15, ______ , _____ The first two numbers differ by 2. The 2nd and 3rd numbers differ by 3. The next difference is 4, then 5. So, the next difference will be 6 and Thus the 6th term is 15+ 6 = 21 while the 7 th MMW by Joseph G. Taban , UNP Take note!   Inductive reasoning is not used just to predict the next number in a list.  We use inductive reasoning to make a conjecture about an arithmetic procedure. Make a conjecture about the example 2 and 3 in the previous slide… MMW by Joseph G. Taban , UNP  Exercise Use Inductive Reasoning to Make a Conjecture  A. Consider the following procedure: 1. Pick a number. 2. Multiply the number by 8, 3. Add 6 to the product 4. Divide the sum by 2, and 5. 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. MMW by Joseph G. Taban , UNP  Exercise Use Inductive Reasoning to Make a Conjecture  B. Consider the following procedure: 1. Pick a number. 2. Multiply the number by 9, 3. Add 15 to the product, 4. Divide the sum by 3, and 5. Subtract 5. 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. MMW by Joseph G. Taban , UNP  Exercise Use Inductive Reasoning to Make a Conjecture  C. Consider the following procedure: 1. List 1 as the first odd number 2. Add the next odd number to 1. 3. Add the next odd number to the sum. 4. Repeat adding the next odd number to the previous sum. Construct a table to summarize the result. Use inductive reasoning to make a conjecture about the sum obtained. MMW by Joseph G. Taban , UNP  Exercise Use Inductive Reasoning to Make a Conjecture  D. Observe the two sets of polygons below: What is the name of a polygon that can be used to describe the polygons in column 2? Use inductive reasoning to make a conjecture about the polygons in column 2. MMW by Joseph G. Taban , UNP  Exercise Use Inductive Reasoning to Make a Conjecture  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 “heartbeats.” 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. Use the data in the table and inductive reasoning to answer each of the following questions. 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?  Take note: Conclusions based on  inductive reasoning may be incorrect. As an illustration, consider the circles shown. For each circle, all possible line segments have been drawn to connect each dot on the circle with all the other dots on the circle. For each circle, count the number of regions formed by the line segments that connect the dots on the  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 MMW by Joseph G. Taban , UNP  Exercise 1  Verify that each of the following  statements is a false statement by finding a counterexample. For all numbers x: a. Note: There is no absolute value for 0 because the absolute value changes the sign of the numbers into positive and zero has no sign. b. c. MMW by Joseph G. Taban , UNP  Exercise 2   Verify that each of the following statements is a false statement by finding a counterexample. For all numbers x: MMW by Joseph G. Taban , UNP  DEDUCTIVE REASONING:   Another type of reasoning is called deductive reasoning.  Deductive reasoning is distinguished from inductive reasoning in that it is the process of reaching a conclusion by applying general principles and procedures.    Mathematics is essentially deductive reasoning  Deductive reasoning is always valid  Deductive reasoning makes use of undefined terms, formally defined terms, axioms, theorems, and rules of inference.  Examples of Deductive Reasoning Example 1:  If a number is divisible by 2, then it must be even. 12 is divisible by 2. Therefore, 12 is an even number. Example 2: All math teachers know how to play sudoku. Resty is a math teacher. Therefore, Resty knows how to play  Examples of Deductive Reasoning Example 3:  If a student is diligent in class, he gets a high grade. If a student gets a high grade, he gets a reward. Therefore, if a student is diligent in class, then he gets a reward. Example 4: If ∠A and ∠B are supplementary angles. If m∠A = 100º, then m∠B = 80º Take note:   The essence of deductive reasoning is drawing a conclusion from a given statement.  The deductive reasoning works best when the statements used in the argument are true and the statements in the argument clearly follow from one another.  Logic Puzzles   Logic Puzzles can be solved by deductive reasoning and a chart that enables us to display the given information in a visual manner. Example 1: Each of four 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.  SOLUTION CLUES: 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. Edito Bank Chef  3. The dentist and Sarah leave for work at the same time. 4. The banker lives next door to Brian. Denti  We know from clue 1 that the r er st banker is not the last to get Sean home, and we know from clue X X X 2 that Sarah is the last to get Maria home; therefore, Sarah is not X X X  the Frombanker. clue 3, Sarah is not the Sarah X X X dentist. Brian X X X  As a result, Sarah is the Chef.  Maria is the Editor.  From clue 1: Maria is not  From clue 4, Brian is not the the banker or the dentist. banker.  Brian is the Dentist.  From clue 2, Sarah is not the editor.  Sean is the Banker.  EXAMPLE   Brianna, Ryan, Tyler, and Ashley were recently elected as the new class officers (president, vice president, secretary, treasurer) of the sophomore class at Summit College. From the following clues, determine which position each holds: 1. Ashley is younger than the president but older than the treasurer. 2. Brianna and the secretary are both the same age, and they are the youngest members of the group. 3. Tyler and the secretary are next-door neighbors.  SOLUTION Pres. V. P. Sec.  Treas. Briann a X X X Ryan X X X Tyler X X X Ashle y X X X President -- Tyler Vice –President –Ashley Secretary— Ryan Treasurer -- Brianna