logic
science of correct reasoning
reasoning
The drawing of inferences or conclusions
inductive or deductive
provlem solivng and reasoning reasoning can either be...
not the study of persuasion and manipulative manipulative rhetorical devices
successful argumentβ does not mean persuasive argument
provlem solivng and reasoning logic is not...
Human fallibility and manipulative rhetoric
____ and ____ led people to accept poor reasoning and reject good reasoning
T
T OR F: In a successful argument if the premises are true, then the conclusion is either guaranteed to be true or likely to be true.
1. Inductive reasoning
2. Deductive reasoning
1. ______ is the process of reaching a general conclusion by examining specific examples.
2. ______ is the process of reaching a conclusion by applying general assumptions, procedures, or principles.
arrangement of inductive reasoning vs deductive reasoning (pyramid)
inductive vs deductive reasoning formula
inductive
INDUCTIVE OR DEDUCTIVE REASONING?You are a good student.
You get all 90+.
Therefore, your friends must get all 90+ too.
deductive
INDUCTIVE OR DEDUCTIVE REASONING?Ninety percent of humans are right-handed.
Edryshe is human, therefore Edryshe is right-handed
inductive reasoning uses patterns to arrive at a conclusion (conjecture)
deductive reasoning uses facts, rules, definitions or properties to arrive at a conclusion
what inductive reasoning use vs deductive reasoning use
30 degrees, deductive
INDUCTIVE OR DEDUCTIVE REASONING?identify angle x as well
shade at lower left, inductive
INDUCTIVE OR DEDUCTIVE REASONING?"
deductive
INDUCTIVE OR DEDUCTIVE REASONING?
All oranges are fruits.
All fruits grow on trees.
Therefore, all oranges grow on trees.
inductive
INDUCTIVE OR DEDUCTIVE REASONING?
Diding hails from Visayas and Visayans are accented in their mother tongue when they speak English.
Therefore, Diding is accented.
inductive
INDUCTIVE OR DEDUCTIVE REASONING?
1, 1, 2, 3, 5, 8 . . .
inductive
INDUCTIVE OR DEDUCTIVE REASONING?
6, 13, 20, 27, ...
deductive
INDUCTIVE OR DEDUCTIVE REASONING?The sum of two odd integers is an even
m and n are odd integers
Thus, m + n is even.
inductive
INDUCTIVE OR DEDUCTIVE REASONING?
3 + 5 = 8, 7 + 11 = 18, and 9 + 21 = 30
Therefore, the sum of two odd integers is even.
1. 452. no, the tsunami height would be negative and height can never be negative3. minimum is 7.4 wherein the max tsunami height is 14. inductive
INDUCTIVE OR DEDUCTIVE REASONING?
1. If the earthquake magnitude is 8.5, how high (in meters) can the tsunami be?
2. Can a tsunami occur when the earthquake magnitude is less than 7? Why?
3. What is the earthquake magnitude where the start of the max tsunami height can be determined?
4. inductive or deductive reasoning?
Not always true. If x = 0, the result is not equal to 1.
ANSWER THE PROBLEM BY INDUCTIVE REASONING
Prove that the following is not true or not always true.
not true
ANSWER THE PROBLEM BY INDUCTIVE REASONING
Prove that the following is not true or not always true.
the number is a multiple of 5
ANSWER THE PROBLEM BY DEDUCTIVE REASONING
Consider the following procedure:
Pick a number.
Multiply the number by 10, add 8 to the product, divide the number by 2, and subtract by 4.
1. sequence
2. terms of sequence
1. _____ an ordered list of numbers
2. the numbers seperated by commas are called the _____
πΉ8 = 21
πΉ10 = 55
PROBLEM SOLVING WITH PATTERNSFind the πΉ8 πππ πΉ10
Binet's formula
what formula is used to identify large fibonacci numbers.
state the formula
provlem solivng and reasoning USING BINET'S FORMULAFind the:1. 20th fibonacci number2. 50th fibonacci number
Pascal's triangle
what is the triangular arrangement of numbers that gives the coefficients in the expansion of any binomial expression.
1. ones2. counting numbers3. triangular numbers4. marbles
where the arrows are pointed
1. 12. 33. 64. 105. 15
identify the number of dots for the to create ff triangle pattern1. 12. 23. 34. 45. 5
1. 12. 43. 104. 205. 35
identify the number of marbles are needed for a stack of a certain height 1. height=12. height=23. height=34. height=45. height=5
2n
210 = 1024
formula to get the sum of the row of pascal's triangle
210 = ?
use pascal's triangle as a guide to get the expansion of the ff algebraic expressions:
1. (a+b)02. (a+b)13. (a+b)2
1. (a+b)32. (a+b)43. (a+b)5
1. = π₯6 + 6π₯5π¦ + 15π₯4π¦2 + 20π₯3π¦3 + 15π₯2π¦ 4 + 6π₯π¦5 + π¦6
2. = π₯7 - 7π₯6π¦ + 21π₯5π¦2 - 35π₯4π¦3 + 35π₯3π¦4 - 21π₯2π¦ 5 + 7π₯π¦6 - π¦7
3. = 1(π₯) 4(β2)0 + 4(π₯)3(β2) + 6(π₯)2(β2)2 + 4(x)(β2)3 + 1(π₯)0(β2)4
= (π₯)4 β 8(π₯)3 + 24(π₯)2 β 32π₯ + 16
= π₯4 β 8π₯3 + 24π₯2 β 32π₯ + 16
USING PASCAL'S TRIANGLE AS A GUIDE, EXPAND THE FF ALGEBRAIC EXPRESSIONS:1. (x+y)62. (x-y)73. (x-2)4
TRUE
T OR F: Can fibonacci numbers can be found in pascal's triangle?
George Polya
father of problem-solving
He is one of the recent mathematicians who outlined a strategy for solving problems from virtually any discipline
What is he also known for?
How to Solve It
In his book, β____β, he wrote: βA great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and triumph of discovery.
1. Understand the problem
2. Devise a plan
3. Carry out the plan
4. Review the solution
Steps in polya's problem-strategy
b
This part of the problem-solving is sometimes (if not always) neglecteda. Carry out the planb. Understand the problemc. Review the solutiond. Devise a plan
b
In order to solve a problem, one must first know what is being asked, and what information or data can be extracted from what is given.a. Carry out the planb. Understand the problemc. Review the solutiond. Devise a plan
b
You can state the problem in your own words.a. Carry out the planb. Understand the problemc. Review the solutiond. Devise a plan
b
problem solving and reasoning You can state the problem in your own wordsa. Carry out the planb. Understand the problemc. Review the solutiond. Devise a plan
d
Listening, tabulating are some of the strategies to solve a problema. Carry out the planb. Understand the problemc. Review the solutiond. Devise a plan
- listing, tabulating- drawing a diagram- guessing then proving if your answers (guesses)
types of strategies to solve the problem
a
provlem solivng and reasoning "Implementing the strategy chosen (in the second step) until the problem is solved.
a. Carry out the planb. Understand the problemc. Review the solutiond. Devise a plan
c
Counter checking if your answers are correcta. Carry out the planb. Understand the problemc. Review the solutiond. Devise a plan
1024 ways
210 = 1024
APPLICATION OF POLYA'S STRATEGYA quiz consists of ten TRUE or FALSE questions. How many possible ways can a student answer the quiz?
Thus, there are tourists who made a side trip to HK
APPLICATION OF POLYA'S STRATEGYAn agency charged P15,000 for a 3-day and 2-night tour in Macau and P20,000 for the same tour with a side trip to Hong Kong. Ten persons joined the trip, which enable them to collect P170,000.
How many tourists made a side trip to Hong Kong?
Let x = tourist visiting Macao y = tourist visiting Macao with a side trip to Hong Kong
1. Modulo
2. congruent
_____ is when two integers a and b are said to be ______modulo n, within being a natural number, if πβπ π is an integer. In this case, we write π β‘ ππππ π.
The number n is called the modulus. The statement π β‘ ππππ π is called a congruence, (Aufmann, 2015).
1. modulus
2. congruence
3. Aufmann
Two integers a and b are said to be congruent modulo n, within being a natural number, if πβπ π is an integer. In this case, we write π β‘ ππππ π.
The number n is called the _____. The statement π β‘ ππππ π is called a _____, (_____, 2015).
Wednesday
MODULOFinding a Day of the Week.
In 2017, Venusβ birthday fell on a Saturday, June 3. On what day of the week does her birthday fall in 2020? (Note that 2020 is a leap year).
Saturday
MODULOFinding a Day of the Week.
In 2020, Venusβ birthday fell on a Wednesday, June 3. On what day of the week does her birthday fall in 2017? (Note that 2020 is a leap year).