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Questions and Answers
What type of reasoning involves forming a general conclusion based on examining specific instances?
What type of reasoning involves forming a general conclusion based on examining specific instances?
- Deductive reasoning
- Intuitive reasoning
- Inductive reasoning (correct)
- Abductive reasoning
A conclusion reached through inductive reasoning that may or may not be correct is called what?
A conclusion reached through inductive reasoning that may or may not be correct is called what?
- A theorem
- A hypothesis
- An axiom
- A conjecture (correct)
Using inductive reasoning, what number would logically follow the sequence 3, 6, 9, 12, 15?
Using inductive reasoning, what number would logically follow the sequence 3, 6, 9, 12, 15?
- 20
- 18 (correct)
- 16
- 17
Consider the sequence: 1, 3, 6, 10, 15. Using inductive reasoning, what would be the next number?
Consider the sequence: 1, 3, 6, 10, 15. Using inductive reasoning, what would be the next number?
What is the final result if you select 7 as the original number and apply the following procedure: Multiply by 8, add 6, divide by 2, and subtract 3?
What is the final result if you select 7 as the original number and apply the following procedure: Multiply by 8, add 6, divide by 2, and subtract 3?
Given the procedure: Pick a number, multiply by 8, add 6, divide the sum by 2, and subtract 3, what conjecture can be made about the relationship between the original number and the final result?
Given the procedure: Pick a number, multiply by 8, add 6, divide the sum by 2, and subtract 3, what conjecture can be made about the relationship between the original number and the final result?
According to the information provided, what factor primarily determines the period of a pendulum?
According to the information provided, what factor primarily determines the period of a pendulum?
If a pendulum has a length of 49 units, what is its period in heartbeats, based on the observations made?
If a pendulum has a length of 49 units, what is its period in heartbeats, based on the observations made?
What happens to the period of a pendulum if its length is quadrupled?
What happens to the period of a pendulum if its length is quadrupled?
What type of reasoning involves reaching a conclusion by applying general assumptions, procedures, or principles?
What type of reasoning involves reaching a conclusion by applying general assumptions, procedures, or principles?
Using deductive reasoning, demonstrate that the procedure 'Pick a number, multiply it by 8, add 6 to the product, divide the sum by 2, and subtract 3' produces a number that is four times the original number.
Using deductive reasoning, demonstrate that the procedure 'Pick a number, multiply it by 8, add 6 to the product, divide the sum by 2, and subtract 3' produces a number that is four times the original number.
What is the primary function of a chart when solving logic puzzles?
What is the primary function of a chart when solving logic puzzles?
In the logic puzzle involving Sean, Maria, Sarah, and Brian, what conclusion can be directly drawn from the clue that Maria gets home from work after the banker but before the dentist?
In the logic puzzle involving Sean, Maria, Sarah, and Brian, what conclusion can be directly drawn from the clue that Maria gets home from work after the banker but before the dentist?
In the same logic puzzle, if Sarah is the last to get home from work, what can be concluded about her occupation, given that the banker does not get home last?
In the same logic puzzle, if Sarah is the last to get home from work, what can be concluded about her occupation, given that the banker does not get home last?
In the context of the provided logic puzzle, if all possible occupations but one have been ruled out for a person, what does this imply about that person's occupation?
In the context of the provided logic puzzle, if all possible occupations but one have been ruled out for a person, what does this imply about that person's occupation?
What type of reasoning is used when determining whether a tree will produce plums this year based on its production history over the past 10 years?
What type of reasoning is used when determining whether a tree will produce plums this year based on its production history over the past 10 years?
If a contractor estimates a home improvement will cost $35,000, and it is argued that all home improvements cost more than the estimate, what type of reasoning is being used?
If a contractor estimates a home improvement will cost $35,000, and it is argued that all home improvements cost more than the estimate, what type of reasoning is being used?
What is the first step in Polya's four-step problem-solving strategy?
What is the first step in Polya's four-step problem-solving strategy?
Why is it important to have a clear understanding of the problem, according to Polya's strategy?
Why is it important to have a clear understanding of the problem, according to Polya's strategy?
According to Polya, which of the following is part of devising a plan?
According to Polya, which of the following is part of devising a plan?
What should you do if your initial plan does not work while carrying out the problem-solving process?
What should you do if your initial plan does not work while carrying out the problem-solving process?
What is the purpose of reviewing the solution in Polya's problem-solving strategy?
What is the purpose of reviewing the solution in Polya's problem-solving strategy?
Who was the mathematician known for developing a 4-step problem-solving strategy?
Who was the mathematician known for developing a 4-step problem-solving strategy?
In what country was George Polya born?
In what country was George Polya born?
What strategy did Gauss use to quickly solve the problem of summing the first 100 natural numbers?
What strategy did Gauss use to quickly solve the problem of summing the first 100 natural numbers?
How did Gauss simplify the summation of the first 100 natural numbers?
How did Gauss simplify the summation of the first 100 natural numbers?
A baseball team won two out of their last four games. Which listed sequence is NOT a possible order of wins (W) and losses (L)?
A baseball team won two out of their last four games. Which listed sequence is NOT a possible order of wins (W) and losses (L)?
How many different orders are there for a baseball team to win exactly two games out of four?
How many different orders are there for a baseball team to win exactly two games out of four?
In the context of number sequences, what does the subscript notation 'an' typically designate?
In the context of number sequences, what does the subscript notation 'an' typically designate?
What is the purpose of an nth-term formula for a sequence?
What is the purpose of an nth-term formula for a sequence?
For the sequence 1, 4, 7, 10, ..., what is the nth term according to the formula provided?
For the sequence 1, 4, 7, 10, ..., what is the nth term according to the formula provided?
In a difference table, what are the 'first differences'?
In a difference table, what are the 'first differences'?
Given the sequence 2, 5, 8, 11, 14, ..., what is the next term predicted by its first differences?
Given the sequence 2, 5, 8, 11, 14, ..., what is the next term predicted by its first differences?
In a difference table, what are the 'second differences'?
In a difference table, what are the 'second differences'?
Consider the sequence 5, 14, 27, 44, 65, ... If the second differences are constant, what is the next term in the sequence?
Consider the sequence 5, 14, 27, 44, 65, ... If the second differences are constant, what is the next term in the sequence?
Given the sequence: 3, 6, 12, 24, 48,... Identify the type of pattern displayed by the sequence and determine the next number.
Given the sequence: 3, 6, 12, 24, 48,... Identify the type of pattern displayed by the sequence and determine the next number.
Which problem-solving technique suggests trying a similar but simpler problem to gain insight?
Which problem-solving technique suggests trying a similar but simpler problem to gain insight?
Apply deductive reasoning to determine which statement demonstrates valid logic:
I. All squares are rectangles. A is a square. Therefore, A is a rectangle.
II. All rectangles are squares. B is a rectangle. Therefore, B is a square.
III. Some triangles are equilateral. C is a triangle. Therefore, C is equilateral.
Apply deductive reasoning to determine which statement demonstrates valid logic: I. All squares are rectangles. A is a square. Therefore, A is a rectangle. II. All rectangles are squares. B is a rectangle. Therefore, B is a square. III. Some triangles are equilateral. C is a triangle. Therefore, C is equilateral.
Given the following sequence: ACE, BDF, CEG, DFH, ____. Apply inductive reasoning to determine the next term in this sequence.
Given the following sequence: ACE, BDF, CEG, DFH, ____. Apply inductive reasoning to determine the next term in this sequence.
A jar contains 3 red marbles and 5 blue marbles. One marble is drawn, and it is red. Without replacement, what is the probability that the next marble drawn is also red?
Express your answer as a simplified fraction.
A jar contains 3 red marbles and 5 blue marbles. One marble is drawn, and it is red. Without replacement, what is the probability that the next marble drawn is also red? Express your answer as a simplified fraction.
To truly master problem-solving, one must understand nuances and assumptions tied to the problem. You are given a very large number of identical balls, and all of them appear to have exactly the same mass. You are told, however, some of the balls are hollow inside. You can only access a simple balance that tells only if two objects have the same or different masses. What is the smallest number of weighings that will guarantee you can identify a 'fake' ball? Choose the best answer.
To truly master problem-solving, one must understand nuances and assumptions tied to the problem. You are given a very large number of identical balls, and all of them appear to have exactly the same mass. You are told, however, some of the balls are hollow inside. You can only access a simple balance that tells only if two objects have the same or different masses. What is the smallest number of weighings that will guarantee you can identify a 'fake' ball? Choose the best answer.
Flashcards
Inductive Reasoning
Inductive Reasoning
Reaching a general conclusion by examining specific examples.
Conjecture
Conjecture
A conclusion based on examining examples; may or may not be correct.
Deductive Reasoning
Deductive Reasoning
Reaching a conclusion by applying general assumptions, procedures, or principles.
Polya's Four-Step Problem Solving Strategy
Polya's Four-Step Problem Solving Strategy
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Sequence
Sequence
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a_n
a_n
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nth-Term Formula
nth-Term Formula
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Difference Table
Difference Table
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Study Notes
Inductive Reasoning
- Involves reaching a general conclusion by examining specific examples
- It forms a conclusion based on the examination of specific examples
- The conclusion formed is often called a conjecture, it may or may not be correct
Inductive Reasoning Example 1: Predicting a Number
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Problem: Use inductive reasoning to predict the next number in the list 3, 6, 9, 12, 15, ?
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Solution: Each successive number is 3 larger than the preceding number; the next number is 18
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Problem: Use inductive reasoning to predict the next number in the list 1, 3, 6, 10, 15, ?
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Solution: The difference between any two numbers is always 1 more than the preceding difference; the next number in the list will be 21
Inductive Reasoning Example 2: Making a Conjecture
- Procedure: Pick a number, multiply by 8, add 6, divide by 2, subtract 3
- Using the number 5:
- Multiply by 8: 8 x 5 = 40
- Add 6: 40 + 6 = 46
- Divide by 2: 46 / 2 = 23
- Subtract 3: 23 - 3 = 20
- Conjecture: Following the procedure produces a number that is four times the original number
Inductive Reasoning Example 3: Solving an Application
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Galileo Galilei (1564–1642) used inductive reasoning to discover that the time required for a pendulum to complete one swing
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The period of the pendulum, depends on the length of the pendulum
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Pendulum Length & Period Data:
- Length of 1 unit has a period of 1 heart beat
- Length of 4 units has a period of 2 heart beats
- Length of 9 units has a period of 3 heart beats
- Length of 16 units has a period of 4 heart beats
- Length of 25 units has a period of 5 heart beats
- Length of 36 units has a period of 6 heart beats
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If a pendulum has a length of 49 units, its period is 7 heartbeats
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Quadrupling the length of a pendulum doubles its period
Deductive Reasoning
- Involves reaching a conclusion by applying general assumptions, procedures, or principles
Deductive Reasoning Example 1: Establishing a Conjecture
- Show that the following procedure produces a number that is four times the original number using deductive reasoning
- Pick a number
- Multiply it by 8
- Add 6 to the product
- Divide the sum by 2
- Subtract 3
- Solution:
- Let n represent the original number
- Multiply the number by 8: 8n
- Add 6 to the product: 8n + 6
- Divide the sum by 2: (8n+6)/2 = 4n + 3
- Subtract 3: 4n + 3 - 3 = 4n
- This procedure produces a number that is four times the original number
Logic Puzzles
- Can be solved by using deductive reasoning and a chart display the given information in a visual manner
- Example: Four neighbors (Sean, Maria, Sarah, Brian) have different occupations (editor, banker, chef, dentist)
- Clues given to determine each neighbor's occupation:
- Maria gets home after the banker but before the dentist
- Sarah is the last to get home and is not the editor
- The dentist and Sarah leave for work at the same time
- The banker lives next door to Brian
- Solution:
- Maria is not the banker or the dentist
- Sarah is not the editor, and Sarah is not the banker
- Sarah is not the dentist so Sarah is the chef
- Maria is the editor
- Brian is not the banker
- Sean is the banker and Brian is the dentist
Inductive vs Deductive Reasoning
- Argument: Tree produced plums every other year for 10 years, and due to not producing last year, it will produce plums this year.
- Is an example of Inductive Reasoning because the conclusion is based on specific examples
- Argument: All home improvements cost more than the estimate of ₱35,000. Thus, home improvement will cost more than ₱35,000
- Is an example of Deductive Reasoning because the conclusion is a specific case of a general assumption
Polya’s Problem-Solving Strategy
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Devised by mathematician George Polya (1877-1985)
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Four-Step Problem Solving Strategy:
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Understand the problem
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Devise a plan
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Carry out the plan
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Review the solution
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Understanding The Problem: Restate the problem in own words Determine what is known about the types of problems Identify missing information Identify extraneous information Determine the goal
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Devising a Plan: Make a list of the known information Make a list of information thats needed Draw a diagram Make an organized list of all the possibilities Make a table or a chart Work backwards Try to solve a similar but simpler problem Look for a pattern Write an equation Perform an experiment Guess and check a solution
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Carrying Out the Plan: Work carefully Keep an accurate and neat record of all attempts Be prepared to modify or devise another plan
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Reviewing the Solution: Ensure that the solution is consistent with the facts of the problem Interpret the solution in the context of the problem Ask if there are generalizations of the solution to apply to other problems
Polya’s Strategy Example 1: Sum of the First 100 Natural Numbers
- Karl Friedrich Gauss determined the sum of the first 100 natural numbers quickly
- Understand the problem: Sum the numbers 1 to 100
- Devise a plan: Notice 1+100=101, 2+99=101, and there are 50 such pairs
- Carry out the plan: 50 x 101 = 5050
- Review the solution: Addends can be placed in any order without changing the sum
Polya’s Strategy Example 2: Baseball Team Wins
- A baseball team won two of their last four games. In how many orders could they have wins and losses?
- Understand the problem: How many ways can two wins and two losses occur in four games?
- Devise a plan: Make an organized list to ensure each order is listed once
- Carry out the plan:
- WWLL
- WLWL
- WLLW
- LWWL
- LWLW
- LLWW Review the solution: The list has no duplicates, so there are six possible orders
Problem Solving with Patterns: Term of a Sequence
- A sequence is an ordered list of numbers (ex: 5, 14, 27, 44, 65, ...)
- Terms are the numbers in a sequence separated by commas
- 𝑎𝑛 designates the nth term of a sequence
nth-Term Formula for a Sequence
- Formula with "n" that lets you find any term without going up from term to term
- "n" stands for the term number
- Formula: nth term = dn + (a – d), where d is the difference between the terms and a is the first term
- Example: Finding the nth term of 1, 4, 7, 10, ... which has a difference of 3:
- nth term = 3n + (1 - 3) = 3n - 2)
Difference Table
- Shows the differences between successive terms of a sequence
- Numbers in row (1) of the table is the difference between the two closest numbers just above it
- These numbers are called the first differences of the sequence
Predicting Terms with the Difference Table
- To predict the next term of a sequence, look for a pattern in a row of differences
- In cases with no pattern in first differences, compute successive differences of the first differences
- These are called second differences, with the 3rd differences being successive differences of the second differences
Finding Additional Terms with the Difference Table
- In the sequence 5, 14, 27, 44, 65... the second differences are constant at 4
- Add 4 to the first difference (21): 21 + 4 = 25
- Add this difference to the fifth term: 65 + 25 = 90
- This process is repeated to predict other additional terms in the sequence
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