Problem Solving and Reasoning

Problem solving is the process of finding solutions to difficult or complex issues ,w hile reasoning is the action of thinking about something in a logical, sensible way.

1. By Prof. Liwayway Memije-Cruz Problem Solving and Reasoning

2. Mathematical Logic Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. A statement or proposition, is a declarative statement that is either true or false, but not both.

3. Solve:

4. 1. When asked how old she was, Mara replied “In two years I will be twice as old as I was five years ago”. How old is she ? 2 . If you have two coins totaling 11p, and one of the coins is not a penny, what are the two coins ? 3 . Divide 40 by half and add ten. What is the answer? 4. To the nearest cubic centimeter, how much soil is there in a 3m x 2m x 2m hole ? 5. A farmer has 15 cows, all but 8 die. How many does he have left ? 6 . The ages of a mother and her graduate son add up to 66. The mother’s age is the son’s age reversed. How old are they ? 7. The amount of water flowing into a tank doubles every minute. The tank is full in an hour. When is the tank half full? Mathematical Logic Test

5. Inductive Reasoning Uses specific examples to reach a general conclusion called conjecture. a logical process in which multiple premises, all believed true or found true most of the time, are combined to obtain a specific conclusion . often used in applications that involve prediction, forecasting, or behavior. not logically rigorous. Imperfection can exist and inaccurate conclusions can occur. moves from the particular to the general. It gathers together particular observations in the form of premises, then it reasons from these particular premises to a general conclusion.

6. Mara leaves for school at 7:00 a.m. Mara is always on time. Mara assumes, then, that she will always be on time if she leaves at 7:00 a.m . Two-thirds of the students at BSU receive student aid. Therefore, two-thirds of all college students receive student aid . All children in the day care center like to play with Legos. All children, therefore, enjoy playing with Legos. There are varying degrees of strength and weakness in inductive reasoning, and various types including statistical syllogism, arguments from example, causal inferences, simple inductions, and inductive generalizations. They can have part to whole relations, extrapolations, or predictions.

7. Read and analyze Examples 1 – 4, pp. 43-46 of Mathematics in the Modern World. https :// www.jobtestprep.co.uk/images/free-pdf/free-logical-reasoning-questions-answers.pd f Exercises on Inductive Reasoning

8. Deductive Reasoning process of reaching a general conclusion by applying general assumptions, procedures and principles. is a logical process in which a conclusion is based on the concordance of multiple premises that are generally assumed to be true. syllogism The Greek philosopher Aristotle, who is considered the father of deductive reasoning, wrote the following classic example : All men are mortal. Socrates is a man. Therefore, Socrates is mortal.

9. or deduction, is one of the two basic types of logical inference. A logical inference is a connection from a first statement (a “premise”) to a second statement (“the conclusion”) for which the rules of logic show that if the first statement is true, the second statement should be true . Examples : Premise: Socrates is a man, and all men are mortal. Conclusion: Socrates is mortal . Premise: This dog always barks when someone is at the door, and the dog didn’t bark. Conclusion: There’s no one at the door . Premise: Sam goes wherever Ben goes, and Ben went to the library. Conclusion: Sam also went to the library. Deductive Reasoning

10. Read, study and solve examples 5 and 6 in your book Exercises:

11. Problem Solving with Patterns

12. Fibonacci sequence A sequence is an ordered list of numbers. Numbers separated by commas are called terms.

13. Fibonacci Numbers

14. Jacques Philippe Marie Binet a French mathematician, physicist and astronomer born in Rennes; he died in Paris, France, in 1856. made significant contributions to number theory, and the mathematical foundations of matrix algebra. Binet's Formula expressing Fibonacci numbers in closed form is named in his honour, although the same result was known to Abraham de Moivre a century earlier.

15. Binet’s Formula

16. Blaise Pascal a French mathematician, physicist, inventor, writer and Catholic theologian . Known for his Pascal triangle

17. Pascal's Triangle One of the most interesting Number Patterns is Pascal's Triangle (named after Blaise Pascal, a famous French Mathematician and Philosopher). To build the triangle, start with "1" at the top, then continue placing numbers below it in a triangular pattern.

18. History of Pascal's Triangle Properties of Pascal's Triangle

19. Primes When you look at Pascal's Triangle, find the prime numbers that are the first number in the row. That prime number is a divisor of every number in that row.

20. Powers of 2 Now let's take a look at powers of 2. If you notice, the sum of the numbers is Row 0 is 1 or 2^0. Similiarly, in Row 1, the sum of the numbers is 1+1 = 2 = 2^1. If you will look at each row down to row 15, you will see that this is true. In fact, if Pascal's triangle was expanded further past Row 15, you would see that the sum of the numbers of any nth row would equal to 2^n

21. Fibonacci's Sequence If you take the sum of the shallow diagonal, you will get the Fibonacci numbers.

22. Fractal If you shade all the even numbers, you will get a fractal. This is also the recursive of Sierpinski's Triangle.

23. Polya’s Problem Solving Strategy George Pólya was a Hungarian mathematician. In 1945 George Polya published the book How To Solve It which quickly became his most prized publication. It sold over one million copies and has been translated into 17 languages

24. Polya’s Problem Solving Strategy

25. Polya’s First Principle: Understand the problem Polya taught teachers to ask students questions such as: • Do you understand all the words used in stating the problem? • What are you asked to find or show? • Can you restate the problem in your own words? • Can you think of a picture or diagram that might help you understand the problem ? • Is there enough information to enable you to find a solution?

26. Polya’s Second Principle: Devise a plan Polya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is included: Guess and check Look for a pattern Make an orderly list Draw a picture Eliminate possibilities Solve a simpler problem Use symmetry Use a model Consider special cases Work backwards Use direct reasoning Use a formula Solve an equation Be ingenious

27. This step is usually easier than devising the plan. In general, all you need is care and patience, given that you have the necessary skills. Persist with the plan that you have chosen. If it continues not to work discard it and choose another. Don’t be misled , this is how mathematics is done, even by professional Polya’s Third Principle: Carry out the plan

28. Polya’s Fourth Principle: Look back Polya mentions that much can be gained by taking the time to reflect 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.

29. Read and study the examples on Polya’s strategy from pp. 55 – 60 Read and answer Exercises Set 3 from pp. 60-62 Exercises

30. https:// whatis.techtarget.com/definition/inductive-reasoning https:// www.jobtestprep.co.uk/free-inductive-reasoning-examples https:// www.jobtestprep.co.uk/images/free-pdf/free-logical-reasoning-questions-answers.pdf https:// whatis.techtarget.com/definition/deductive-reasoning https:// artofproblemsolving.com/wiki/index.php?title=Binet%27s_Formula http:// www.milefoot.com/math/discrete/sequences/binetformula.htm http:// jwilson.coe.uga.edu/EMAT6680Su12/Berryman/6690/BerrymanK-Pascals/BerrymanK-Pascals.html https://math.berkeley.edu/~ gmelvin/polya.pdf References: