MMWModule- Chapter 3 Reasoning and Problem Solving PDF

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FM-AA-CIA-15 Rev. 0 10-July-2020

Study Guide in Mathematics in the Modern World

GE7 Mathematics in the Modern World

Module 3 : Reasoning and Problem Solving

MODULE 3 REASONING AND PROBLEM SOLVING MODULE OVERVIEW

This module consist of three lessons: Inductive and Deductive Reasoning, Polya’s Problem Solving Strategy, Recreational Problems using Mathematics. Each lesson was designed as a self-teaching guide. Definitions of terms and examples had been incorporated. Answering the problems in “your turn” will check your progress. You may compare your answers to the solutions provided at the later part of this module in that way you will be able to measure your achievement and as well as the effectiveness of the module. Exercises were prepared as your assignment to deepen your understanding about the topics. MODULE LEARNING OBJECTIVES

At the end of the module, you should be able to:  Use different types of reasoning to justify statements and arguments made about mathematics and mathematical concepts.  Solve problems involving problems and recreational problems following Polya’s four steps.  Organize one’s methods and approaches to proving and solving problems. LEARNING CONTENTS (INDUCTIVE AND DEDUCTIVE REASONING)

Introduction Mathematics has always been seen as a tool for problem solving. Math by nature is based on logical and valid reasoning so that it used for decision – making. A good decision maker is one who can find resolution using his/ her reasoning ability and mathematical strategy. In this chapter, you will learn to organize your own methods and approaches to solve mathematical problems.

Discussion Lesson 1. Inductive and Deductive Reasoning Inductive Reasoning The type of reasoning that uses specific examples to reach a general conclusion is called inductive reasoning. The conclusion formed by using inductive reasoning is called conjecture which may or may not correct. For instance, the following are examples of inductive reasoning: - Jenny leaves for school at 7:00 am. Jenny is always on time. Therefore, Jenny assumes then that if she leaves for school at 7:00 am., she will always be on time. The conclusion, however, may not be accurate because Jenny would have still be late even she leaves early due to she might encounter some unexpected circumstances causing her to be late. - The chair in the living room is red . The chair in the dining room is red . The chair in the bedroom is red. Therefore, all chairs in the house are red. PANGASINAN STATE UNIVERSITY

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Study Guide in Mathematics in the Modern World

GE7 Mathematics in the Modern World

in

Module 3 : Reasoning and Problem Solving

The conclusion, however, may not be correct. There might some other chair/s that house that is/are not red.

Even when you examine a list of numbers and predict the next number in the list according to some pattern you have observed, you are also using inductive reasoning.

Example

Use Inductive Reasoning to Predict a Number

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 in the list is 3 larger than 15, which is 18. 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. . Your turn

Use inductive reasoning to predict the next number in each of the following lists.

a. 5, 10, 15, 20, 25, ?

b. 2, 5, 10, 17, 26, ?

Example 2

Use inductive reasoning to make a conjecture out of 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 5. 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 8 ×5=40 Multiply by 8 : 40 + 6= 46 Add 6: PANGASINAN STATE UNIVERSITY

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Study Guide in Mathematics in the Modern World

GE7 Mathematics in the Modern World

Divide by 2: Subtract 3:

Module 3 : Reasoning and Problem Solving

46 ÷ 2=23 23−3=20

We started with 5 and followed the procedure to produce 20. Starting with 6 as our original number produces a fi nal 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 times the original number. We conjecture that following the given procedure produces a number that is four times the original number.

Consider the following procedure: Pick a number. Multiply the number by 9, add 15 to the product, divide the sum by 3, and 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. Your turn

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.

Example 3

Length of pendulum, in units 1

Period of pendulum, in heartbeats

4

2

9

3

16

4

25

5

36

6

1

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? Solution PANGASINAN STATE UNIVERSITY

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Study Guide in Mathematics in the Modern World

GE7 Mathematics in the Modern World

FM-AA-CIA-15 Rev. 0 10-July-2020 Module 3 : Reasoning and Problem Solving

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. Your turn

A tsunami is a sea wave produced by an underwater earthquake. The height of a tsunami as it approaches land depends on the velocity of the tsunami. Use the table at the left and inductive reasoning to answer each of the following questions. a. What happens to the height of a tsunami when its velocity is doubled? b. What should be the height of a tsunami if its velocity is 30 feet per second?

Conclusions based on inductive reasoning may not always be true. In other words, a conjecture formed by using inductive reasoning may be incorrect. To illustrate this , consider the results below, 11 ×(1) ( 101 )=1111 11 × ( 2 ) (101 ) =2222 11 × ( 3 ) (101 ) =3333 11 ×(4) ( 101 )=4444 11 × ( 5 ) (101 ) =5555 11 × ( 6 )( 101 ) =? Simple arithmetic shows that the answer is 6666; hence it is conjectured that the product of 11 and a multiple of 101 is number where all digits are equal. But is 11 ×n (101 )=nnn true for all n , ( n , a natural number)? Suppose n=10 . Then 11 ×10 ( 101 ) =11,110 which obviously does not satisfy the previous conclusion. This method of disproving a statement is to give a counterexample .

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 4 PANGASINAN STATE UNIVERSITY

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Study Guide in Mathematics in the Modern World

GE7 Mathematics in the Modern World

FM-AA-CIA-15 Rev. 0 10-July-2020 Module 3 : Reasoning and Problem Solving

Find a counterexample. Verify that each of the following statements is a false statement by finding counterexample. For all number x : a. |x| > 0 b. x 2> x ¿x c. √ x2 Solution

A statement may have many counterexamples, but we need only find one counterexample to verify that the statement is false. a. Let x=0 . Then |0|=0 . Because 0 is not greater than 0, we have found a counterexample. Thus “for all numbers x, |x| > 0 ” is a false statement. b. For x=1 we have 12=1 . Since 1 is not greater than 1, we have found a counterexample. Thus “for all numbers x, x 2> x ” is a false statement. c. Consider x=−3 . Then √ (−3 )2=√ 9=3 . Since 3 is not equal to −3 , we have found a ¿ x ” is a false statement. counterexample. Thus “for all numbers √ x2

Your turn

Verify that each of the following statements is a false statement by finding a counterexample for each.

For all numbers x: x =1 a. x b.

x +3 =x +1 3

c.

√ x2 +16=x+ 4

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 . For instance, the following are examples of deductive reasoning: -

All squares are rectangle. All rectangles have four angles. Therefore, logic tells us that all squares have four right angles.

-

All dogs have good sense of smell. Blackeye is a dog. Therefore, using deductive reasoning tells us that Blackeye has a good sense of smell.

Example 5

Use deductive reasoning to show that the following procedure produces a number that is four times the original number.

Procedure: 1. Pick a number. 2. Multiply the number by 8, 3. add 6 to the product, PANGASINAN STATE UNIVERSITY

Note that Example 5 is the same as Example 2 except that in this example deductive reasoning is used. 5

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Study Guide in Mathematics in the Modern World

GE7 Mathematics in the Modern World

Module 3 : Reasoning and Problem Solving

4. divide the sum by 2, 5.. and subtract 3. Solution Let n represent the original number. Multiply the number by 8: Add 6 to the product: Divide the sum by 2: Subtract 3:

8n 8 n+6 8 n+ 6 =4 n+3 2 4 n+3−3=4 n

We started with n and ended with 4n. The procedure given in this example produces a number that is four times the original number.

Your turn

Use deductive reasoning to show that the following procedure produces a number that is three times the original number.

Procedure: 1. Pick a number. 2. Multiply the number by 6, 3. add 10 to the product, 4. divide the sum by 2, 5.and subtract 5. Hint: Let n represent the original number.

Logic Puzzles 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 6

Solve a Logic Puzzle

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. 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 (which stands for “ruled out by clue 1”) in the Banker and the Dentist columns of Maria’s row. Editor Sean Maria Sarah

Banker X1

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Chef

Dentist X1

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Study Guide in Mathematics in the Modern World

GE7 Mathematics in the Modern World

Module 3 : Reasoning and Problem Solving

Brian 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.

Sean Maria Sarah Brian

Editor

Banker

X2

X1 X2

Chef

Dentist X1

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. Editor X3 √ X2 X3

Sean Maria Sarah Brian

Banker X1 X2

Chef X3 X3 √ X3

Dentist X1 X3

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. Editor Banker Chef Dentist √ Sean X3 X3 X4 √ Maria X1 X3 X1 √ X3 Sarah X2 X2 Brian X3 X4 X3 √ Sean is the banker, Maria is the editor, Sarah is the chef, and Brian is the dentist

Your turn

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. PANGASINAN STATE UNIVERSITY

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Study Guide in Mathematics in the Modern World

GE7 Mathematics in the Modern World

Module 3 : Reasoning and Problem Solving

3. Tyler and the secretary are next-door neighbors. LEARNING POINTS The type of reasoning that uses specific examples to reach a general conclusion is called inductive reasoning . The conclusion formed by using inductive reasoning is called conjecture which may or may not correct. Deductive reasoning is distinguished from inductive reasoning in that it is the process of reaching a conclusion by applying general principles and procedures LEARNING ACTIVITY 1

In exercises 1 to 4. Use inductive reasoning to predict the next number in each of the following lists. 1. 5, 10, 15, 20, 25, ___ 3 5 7 9 11 13 2. , , , , , 5 7 9 11 13 15 3. 2, 5, 10, 17, 26, ____ 4. 2,7, −3,2, −8,−3, −13,−8, −18 , __ ¿ 5. 1, 8,27, 64,125, ¿ 6. 80, 70,61, 53, 46, 40, _____ 7. 1,5, 12, 22,35, ______

, ______

In exercises 8 to 11. Determine whether the argument is an example of inductive reasoning or deductive reasoning. 6. Emma enjoyed reading the novel Under the Dome by Stephen King, so she will enjoy reading his next novel. 8. All pentagons have exactly five sides. Figure A is a pentagon. Therefore, Figure A has exactly fi ve sides. 9. Every English setter likes to hunt. Duke is an English setter, so Duke likes to hunt. 10. Cats don’t eat tomatoes. Tigger is a cat. Therefore, Tigger does not eat tomatoes. 11. Two computer programs, a bubble sort and a shell sort, are used to sort data. In each of 50 experiments, the shell sort program took less time to sort the data than did the bubble sort program. Thus the shell sort program is the faster of the two sorting programs. In exercises 12 to 16. Verify that each of the following statements is a false statement by finding a counterexample for each. 12. (x+ y)2=x 2 + y 2 13. For all x , |x +3 |=| x|+3 14. For all x , x+x >x (x +1)( x−1) =x +1 15. For all number x , (x−1) 16. For all numbers x , −x< x .

In exercises 17 to 18. Use the data in the table and by inductive reasoning , answer the following question below.

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Study Guide in Mathematics in the Modern World

GE7 Mathematics in the Modern World

Earthquake Magnitude 7.5 7.6 7.7 7.8 7.9 8.0 8.1 8.2 8.3

Module 3 : Reasoning and Problem Solving

Max. Tsunami Height(meters) 5 9 13 17 21 25 29 33 37

17. if the earthquake magnitude is 8.5, how high (in meters) can the tsunami be? 18. Can a tsunami occur when the earthquake magnitude is less than 7? Explain you answer. 19. Use deductive reasoning to show that the following procedure always produces the number 5. Procedure: Pick a number. Add 4 to the number and multiply the sum by 3. Subtract 7 and then decrease this difference by the triple of the original number. 20. Solve a logic puzzle . Each of the four friends Donna, Sarah, Nickkie, and Xhanelle , has a different pet(fish, cat, dog, and snake). From the following clues, determine the pet of each individual. 1. Sarah is older than her friend who owns the cat and younger than her friend who owns the dog. 2. Nikkie and her friend who owns the snake are both of the same age and are the youngest members of their group. 3. Donna is older than her friend who owns the fish. LEARNING CONTENTS (POLYA’S PROBLEM SOLVING STRATEGY)

Lesson 2. Polya’s Problem Solving Strategy One of the foremost recent mathematicians to make a study of problem solving was George Polya (1887–1985). He was born in Hungary and moved to the United States in 1940.In his book How to Solve It”, George Polya enumerates the four steps of problem –solving : 1. 2. 3. 4.

Understand the problem. Devise a plan. Carry out the plan. 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? PANGASINAN STATE UNIVERSITY

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Study Guide in Mathematics in the Modern World

GE7 Mathematics in the Modern World

FM-AA-CIA-15 Rev. 0 10-July-2020 Module 3 : Reasoning and Problem Solving

■ 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? 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 i...