MMW Chapter 3 LOGIC PDF

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MATHEMATICS IN THE MODERN WORLD PR PROB OB OBLEM LEM SOL SOLVING VING

GENARO B. ABREU

Inductive and Deductive Reasoning

01

PROBLEM SOLVING Logic Puzzles

02

Polya‘s Four Steps in Problem Solving

03

Recreational Problems using Mathematics

04

Problem Solving

Definition

Agenda Style

❑

Refers to mathematical tasks that have the potential to provide intellectual challenges in order to enhance a person’s mathematical understanding and development.

❑

It pointed out that people who can reason and think analytically tend to: (a) note patterns, structure, or regularities in both real-world situations and symbolic objects; (b) ask if those patterns are accidental or if they occur for a reason; (c) conjecture and prove.

❑

An important skill not only in dealing with Mathematics, but also in making decisions in life. Decision-making is a significant part of problem-solving.

Inductive and Deductive Reasoning

Inductive Reasoning

Definition

Agenda Style

❑ The process of reaching a general conclusion by examining specific examples. ❑ It involves looking for patterns and making generalizations. ❑ The conclusion formed by using inductive reasoning is often called a conjecture, since it may or may not be correct. .

Illustrative Examples: 1. Write a conjecture that describe the pattern 2, 4, 12, 48, 240. Then use the conjecture to find the next item in the sequence.

Step 1. Look for a pattern. 2, 4, 12, 48, 240, … ? Step 2. Analyze what is happening in the given pattern. The numbers are multiplied by 2, then 3, then 4, then 5. The next number will be the product of 240 times 6 or 1,440. Step 3: Make a conjecture Now, the answer is 1,440

Example: 2. Write a segments

ow many

Step 1. Look for a pattern. 3-segments

9-segments

18-segments

Step 2. Analyze what is happening in the given pattern.

This could be written in a form of:

(3)(2)

(3)(3)

(3)(4)

(3)(5)

The figure will increase by the next multiple of 3. If we add 15, the next or the fifth figure is made of 45 segments.

Step 3. Make a conjecture. Hence the fifth figure will have 45 segments.

Exercises: Complete the following 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. 1. Consider the following procedure: a. Pick a number. b. Multiply the number by 8, c. Add 6 to the product d. Divide the sum by 2, and e. Subtract 3.

2. Consider the following procedure: a. Pick a number. b. Multiply the number by 9, c. Add 15 to the product, d. Divide the sum by 3, and e. Subtract 5. 3. Consider the following procedure: f. List 1 as the first odd number g. Add the next odd number to 1. h. Add the next odd number to the sum. i. Repeat adding the next odd number to the previous sum.

Deductive Reasoning

Definition

Agenda Style

❑ The process of reaching a conclusion by applying general principles and procedures. ❑ It involves making a logical argument, drawing conclusions, and applying generalizations to specific situations.

Illustrative Examples: 1.

If a number is divisible by 2, then it must be even. 12 is divisible by 2. Therefore, 12 is an even number.

2. All Engineers are very good at Math. Quirina is an Engineer. Therefore, Quirina is very good at Math. 3. If a student is a DOST scholar, he receives a monthly allowance. If a student receives a monthly allowance, his parents will be happy. Therefore, if a student is a DOST scholar, his parents will be happy. 4. If ∠A and ∠B are supplementary angles, their sum is 180º. If m∠A = 100º, then m∠B = 80º

Exercise 3.1. Classify the reasoning employed in the following arguments as INDUCTIVE or DEDUCTIVE.

________1. All even numbers are divisible by 2. Twenty-eight is even. Therefore, 28 is divisible by 2. ________2. 3,6,9,12,15,

. The next term is going to be 18.

________3. Since all squares are rectangles, and all rectangles have four sides, all squares have four sides. ________4. For any right triangle, the Pythagorean Theorem holds. ABC is a right triangle, therefore for ABC the Pythagorean Theorem holds. ________5. The population of Baguio City has risen steadily for the past 40 years. It is logical to predict that the population of Baguio City will also rise next year.

___________6. Two figures are said to be congruent if they have the same shape and size or if one has the same shape and size as the mirror image of the other. My figure is the mirror image of my mirror, therefore my figure and my mirror image are congruent. ___________7. If x = 4 and if y = 1, Then 2x + y = 9 __________8. Based on a survey of 3300 randomly selected registered voters, 56.2% indicate that they will vote for the incumbent officials in the upcoming election. Therefore, approximately 56% of the votes in the upcoming election will be for the incumbent.

__________9. Jack is taller than Jill. Jill is taller than Joey. Therefore, Jack is taller than Joey. _________10. It usually takes 2–3 days for a delivery to ship from the warehouse to your door via most major shipping services. You ordered on Tuesday morning, so it‘s safe to assume your package will arrive Thursday or Friday.

Logic Puzzles

Logic Puzzles

Illustrative Examples: 1.

Each of four neighbors, Mark, Zen, Linda, and Roy, has a different occupation (teacher, banker, chef, or broker). From the following clues, determine the occupation of each neighbor. CLUES 1. 2. 3. 4.

Zen gets home from work after the banker but before the broker. Linda, who is the last to get home from work, is not the teacher. The dentist and Linda leave for work at the same time. The banker lives next door to Roy. Solution: From clue 1, Zen is neither the banker nor the broker. From clue 2, Linda is not the teacher.

฀

We know from clue 1 that the banker is not the last to get home, and we know from clue 2 that Linda is the last to get home; therefore, Linda is not the banker

฀

From clue 3, Linda is not the broker. As a result, Linda is the Chef.

฀

Since Linda is the Chef, it could not be Zen. Zen, therefore, is the Teacher.

฀

From clue 4, Roy is not the banker.

฀

And since Linda is the Chef and Zen is the Teacher, Roy must be the Broker.

฀

Mark is the Banker, the only occupation not filled up. Teacher

Banker

Chef

Broker

Mark

x

/

x

x

Zen

/

x

x

x

Linda

x

x

/

x

Roy

x

x

x

/

Exercise 3.2. Satisfy the following Logic Puzzles. 1. Quirina, Rey, Bibo and Venus were recently elected as the new class officers (president, vice president, secretary, treasurer) of the sophomore class at Basilan State College. o

From the following clues, determine which position each holds: Quirina is younger than the president but older than the treasurer.

o

Rey and the secretary are both the same age, and they are the youngest members of the group.

o

Bibo and the secretary are next-door neighbors. Vice President Quirina Rey Bibo Venus

President

Secretary

Treasurer

2. There are 5 foreign vessels in Batangas port: 1. The Estonian ship leaves at four and carries bananas. 2. The Ship in the middle has a brown exterior. 3. The Panamanian ship leaves at six. 4. The Dutch ship with blue exterior is to the left of a ship that carries bananas. 5. To the right of the ship carrying cocoa is a ship going to Bohol. 6. The Costa Rican ship is heading for Marinduque. 7. Next to the ship carrying rice is a ship with a orange exterior. 8. A ship going to Zamboanga City leaves at five. 9. The Nigerian ship leaves at seven and is to the right of the ship going to Bohol. 10. The ship with a red exterior goes to Palawan. 11. Next to the ship leaving at seven is a ship with a black exterior. 12. The ship on the border carries sugarcane. 13. The ship with a brown exterior leaves at eight. 14. The ship carrying sugarcane is anchored next to the ship carrying rice. 15. The ship to Palawan leaves at six. Which ship goes to Surigao? Which ship carries corn?

Table Ship

Estonian Dutch Costa Rican

Nigerian Panamanian

Departure

Product

Exterior

Destination

Einstein Puzzles

3. Five travelers are standing in a queue for plane tickets in Manila. Names: Ping, Leni, Bongbong, Bato and Isko Social Media Platform: You Tube, Facebook, Selfie, Tiktok and Telegram. Destinations: Bukidnon, Tacloban, Catanduanes, Dumaguete and Batanes Ages: 47, 50, 52, 55 and 58 Hairstyle: Wavy, long, straight, curly and bald Where they live: Malabon, Pasig, Quezon City, Valenzuela and Taguig 1. The person in the middle watches Tiktok. 2. Ping is the first in the queue 3. The person who watches the You Tube is next to the person who lives in Valenzuala City 4. The person going to Dumaguete City is behind Isko 5. The person who lives in Quezon City is 52 6. The person who is going to Tacloban City has straight hair 7. The person travelling to Dumaguete City watches Tiktok 8. The 47-year-old is at the end of the queue 9. Bongbong is busy doing a Selfie. 10. The person heading to Batanes has long hair

Einstein Puzzles

11. Leni lives in Quezon City 12. The 50-year-old is bald 13. The fourth in the queue is going to Catanduanes 14. The people who are in Tiktok and Telegram are standing next to each other 15. The person who watches Facebook stands next to the person with a wavy hair 16. A person next to Isko has a wavy hair 17. The 55-year-old lives in Valenzuala City. 18. The person who watches Facebook has long hair 19. The 58-year-old lives in Taguig 20. The person who is travelling to Bukidnon lives in Malabon 21. Bato is not next to the person with straight hair Who hails from Pasig City? Who watching on Telegram?

Table Traveler Bato Ping Leni Bongbong Isko

Age

Destination

Address

SocMed Interest

Hairstyle

Polya’s Four Steps in Problem Solving

Polya‘s four steps in Problem-Solving ฀

Understanding 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. These are some questions that you may be asked to yourself before you solve the problem. ✔ Are all words in a problem really understand and clear by the reader?

✔ Do the reader really know what is being asked in a problem on how to find the exact answer? ✔ Can a reader rephrase the problem by their own without deviating to its meaning? ✔ If necessary, do the reader can really visualize the real picture of the problem by drawing the diagram? ✔ Are the information in the problem complete or is there any missing information in a problem that could impossible to solve the problem?

฀

Devise a Plan – strategize. Pólya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is bes learned by solving various problems. Applying strategies to devise a plan requires skill and own judgment. Some strategies are as follows: ✔ As much as possible, list down or identify all important information in the problem. ✔ Sometimes, to be able to solve problem easily, you need to draw figures or diagram and tables or charts. ✔ Organized all information that are very essential to solve a problem. ✔ You could work backwards so that you could get the main idea of the problem ✔ Look for a pattern and try to solve a similar but simpler problem. ✔ Create a working equation that determines the given (constant) and variable. ✔ You could use the experiment method and sometimes guessing is okay.

฀

Carry out the plan After devising a plan, the next logical step is to carry out that plan. ✔ Implement the strategy in Step 2 and perform any necessary actions or computations. ✔ Check each step of the plan as you proceed; this may be intuitive checking or a formal proof of each step. ✔ Keep an accurate record of your steps as you implement your devised plan. ✔ Persist with the plan that you have chosen, and if it continues not to work, discard it and choose another.

฀

❑ ❑

Look back/Review the solution Pólya mentions that much can be gained by taking the time to reflect, examine, and look back at what you have done - what worked and what didn't; doing this will enable you to predict what strategy to use to solve future problems, if these relate to the original problem.

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. ✔ Examine the solution obtained. Check the results in the original problem (in some cases, this will require a proof).

✔ Interpret the solution in terms of the original problem. Find out if your answer makes sense or is reasonable. Ensure that the solution is consistent with the facts of the problem. ✔ Determine whether there is another method of finding the solution. ✔ If possible, determine other related or more general problems for which the techniques will work; find out if there are generalizations of the solution that could apply to other problems.

Illustrative Examples: 1. One number is 7 more than another. Twice the larger is equal to four times the smaller decreased by 2. Find the numbers. Step 1: Understand the Problem We are looking for two numbers wherein one is 7 more than another, and twice the larger is equal to four times the smaller decreased by 2. Step 2: Devise a plan. We can use the ―Formulate an equationǁ strategy x = smaller number x + 7 = larger number Equation: 2(x + 7) = 4x – 2

Step 3: Carry out the plan 2(x + 7) = 4x – 2

2x + 14 = 4x – 2

Subtract 4x and subtract 14 from the both sides of the equation 2x – 4x + 14 – 14 = 4x – 4x – 2 – 14 – 2x = – 16 then divide both sides by – 2 x = 8 and x + 7 = 8 + 7 = 15 Step 4: Look back 15 is 7 more than 8; twice 15, which is 30, is four times 8 less 2. Thus: The smaller number is 8 and the larger number is 15.

2. Anne is 2 years older than Betty. Last year Anne was 2 times as old as Betty. How old is Anne? Step 1: Understand the Problem We are looking for the age of Anne at present. She is now 2 years older than Betty, and last year she was twice as old as Betty. Step 2: Devise a plan We can also use a table such as this: Age now

Age last year

Betty

x

x-1

Anne

x+2

(x + 2) - 1

Last year, Anne was twice as old as Betty Thus, the equation is: 2(x – 1) = x + 2 -1

Step 3: Carry out the plan 2(x – 1) = x + 2 -1 2x – 2 = x + 1 2x – x = 1 + 2 x = 3 and x + 2 = 5 Step 4: Look back/Check If Anne is 5 years old now, Betty is 3 years old. Last year, Anne was 4 and Betty was 2, that is, Anne was twice as old as Betty.

Mathematical Problems Involving Patterns Andres cashes a PhP1,800 check and wants the money in PhP100 and PhP200 bills. The bank teller gives him 12 bills. How many of each kind of bill does Andres receive? Solution Method 1: Making a Table Understand Andres gives the bank teller a PhP1800 check The bank teller gives Andres 12 bills. These bills are a mix of PhP100 and PhP200 bills. We want to know how many of each kind of bill Andres receives. Strategy Let’s start by making a table of the different ways Andres can have twelve bills in 100s and 200s. Andres could have twelve PhP100 and zero PhP200 bills, or eleven PhP100 and one PhP200 bills, and so on. We can calculate the total amount of money for each case.

Apply strategy/solve

PhP100 12 11 10 9 8 7 6 5 4 3 2 1 0

1 2 3 4 5 6 7 8 9 10 11 12

PhP200

Total amount

0 100(12) + 200(0) = PhP1200 100(11) + 200(1) = PhP1300 100(10) + 200(2) = PhP1400 100(9) + 200(3) = PhP1500 100(8) + 200(4) = PhP1600 100(7) + 200(5) = PhP1700 100(6) + 200(6) = PhP1800 100(5) + 200(7) = PhP1900 100(4) + 200(8) = PhP2000 100(3) + 200(9) = PhP2100 100(2) + 200(10) = PhP2200 100(1) + 200(11) = PhP2300 100(0) + 200(12 )= PhP2400

In the table, we listed all the possible ways you can get twelve PhP100 and PhP200 bills and the total amount of money for each possibility. The correct amount is given when Andrew has six PhP100 bills and six PhP200 bills. Answer: Andres gets six PhP100 bills and six PhP200 bills Check Six PhP100 bills and six PhP200 bills→6 (PhP100)+6 (PhP200) = PhP600+PhP1200 = PhP1,800

Method 2: Looking for a Pattern Understand Andres gives the bank teller a PhP1800 check. The bank teller gives Andrew 12 bills. These bills are a mix of PhP100 bills and PhP200 bills. We want to know how many of each kind of bill Andres receives. Strategy Let’s start by making a table just as we did above. However, this time we will look for patterns in the table that can be used to find the solution. Apply strategy/solve Let’s fill in the rows of the table until we see a pattern.

PhP100

PhP200

12 11 10

1 2

Total amount

0 100(12) + 200(0) = PhP1200 100(11) + 200(1) = PhP1300 100(10) + 200(2) = PhP1400

We see that every time we reduce the number of PhP100 bills by one and increase the number of PhP200 bills by one, the total amount increases by PhP100. The last entry in the table gives a total amount of PhP1400, so we have PhP400 to go until we reach our goal. This means that we should: Reduce the number of PhP100 bills by four and Increase the number of PhP200 bills by four. That would give us six PhP100 bills and six PhP200 bills.

6 (PhP100)+6 (PhP200) = PhP600+PhP1200 = PhP1,800 Answer: Andres six PhP100 bills and six PhP200 bills.

Exercise 3.2. Solve the following Mathematical Problems.

❑ Jerry is 7 years older than Jan. In three years Jerry will be twice as old as Jan. Find their present ages. ❑ Malik and Marites left at 8A.M. from the same point. Malik is traveling east at an average speed of 50 mph and Marites travelling south at an average speed at 60 mph. At what time to the nearest minute will they be 300 miles apart? ❑ The sum of three consecutive positive integers is 165. What are these three numbers? ❑ There are ten students in a room. If they give a handshake for his classmate once and only once, how many handshakes can be made?

❑ Kiko is trying to cut down on drinking coffee. His goal is to cut down to 6 cups per week. If he starts with 24 cups the first week, then cuts down to 21 cups the second week and 18 cups the third week, how many weeks will it take him to reach his goal? ❑ A new big resort opens in San Luis, Batangas. On opening day, the resort has 120 visitors; on each of the next three days, the resort has 10 more visitors than the day before; and on each of the three days after that, the resort has 20 more vis...