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Learning Outcomes:
At the end of this lesson, the students will be able to:
1. Analyze problems using different types of reasoning.
2. Apply different types of reasoning to justify
statements and arguments made about mathematics
and mathematical concepts.
INDUCTIVE REASONING
It is the process of reaching a
general conclusion by
examining specific examples.
INDUCTIVE REASONING
The conclusion formed by using
inductive reasoning is often
called a conjecture, since it may
or may not be correct.
Example 1:
Use inductive reasoning to predict the next number in
each item.
1. 2,8,14,20,26, _____
2. 1,2,5,10,17,26, _____
Example 2:
A. Every sports car I have ever seen is red. Thus, all
sports cars are red.
B. The coin I pulled from the bag is a 5-peso coin.
Another 5-peso coin is drawn from the bag. A third
coin from the bag is again a 5-peso coin. Therefore,
all the coins in the bag are 5-peso coins.
Example 3:
Consider the following. Pick a number. Multiply the
number by 4, add 8 to the product, divide the sum by 2,
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.
Solution:
Suppose we start with seven as the original number. Then
repeat the process for different numbers. The procedure yields
the following:
We conjecture that the given procedure produces a number
that is one less than twice the original number.
Remarks:
When we use inductive reasoning, we have no
guarantee that our conclusion is correct. Just because
a pattern is true for a few cases, it does not mean the
pattern will continue. A statement is a true statement
provided that it is valid in all cases. If we can find one
case for which a statement is not valid, called a
counterexample, then it is a false statement.
DEDUCTIVE REASONING
It is the process of reaching a
conclusion by applying
general assumptions, procedures,
or principles.
DEDUCTIVE REASONING
Deduction starts out with a general
statement, or hypothesis, and
examines the possibilities to reach a
specific, logical conclusion.
Example 1:
1. All men are mortal. Kahwi is a man. Therefore,
Kahwi is mortal.
2. Corresponding parts of congruent triangles are
congruent. Triangle ABC is congruent to triangle
DEF. Angle B and angle E are corresponding angles.
Thus, angle B is congruent to angle E.
GENERALITIES
SPECIFIC
CASES
DEDUCTIVE INDUCTIVE
Determine if each of the following statement
uses inductive or deductive reasoning.
1. Teacher Erica is an enthusiastic and passionate teacher.
Therefore, all teachers are enthusiastic and passionate.
2. All dogs are animals. Dhai is a dog. Thus, Dhai is an
animal.
3. I got low score on the first long exam. I just recently took
the second long exam and I got low score. Therefore, I will
also get a low score on the third long exam.
IR
DR
IR
Determine if each of the following statement
uses inductive or deductive reasoning.
4. My classmates are disrespectful toward our instructor.
Hence, all students are disrespectful.
5. Last Wednesday it was raining. Today is Wednesday and it
is raining. Therefore, on the next Wednesday, it will also rain.
6. For any right triangle, the Pythagorean Theorem holds.
ABC is a right triangle, therefore for ABC the Pythagorean
Theorem holds.
7. All basketball players in your school are tall, so all
basketball players must be tall.
IR
IR
DR
IR
A logic puzzle is a puzzle deriving from the
mathematics field of deduction.
Logic puzzles can be solved by using deductive
reasoning and by organizing the data in a
given situation.
A logic puzzle is basically a description of an
event or any situation. Using the clues
provided, one has to piece together what
actually happened. This involves clear and
logical thinking, hence the term “logic” puzzles.
Example 1
Three musicians appeared at a concert. Their last names were Benton, Lanier, and
Rosario. Each plays only one of the following instruments: guitar, piano, or saxophone.
1. Benton and the guitar player arrived at the concert together.
2. The saxophone player performed before Benton.
3. Rosario wished the guitar player good luck.
Who played each instrument?
Guitar Piano Saxophone
Benton
Lanier
Rosario
Example 2
You have a basket containing ten apples. You have 10 friends,
who each desire an apple. You give each of your friends, one
apple. Now all your friends have one apple each, yet there is an
apple remaining in the basket. How?
Solution:
✎You give an apple to your first nine friends, and a basket with an apple
to your tenth friend. Each friend has an apple, and one of them has it
in a basket.
✎ Alternative answer: one friend already had an apple and put it in the
basket.
Example 3
A census-taker knocks on a door, and asks the woman inside
how many children she has and how old they are. “I have three
daughters, their ages are whole numbers, and the product of
their ages is 36,” says the mother. “That’s not enough
information”, responds the census-taker. “I’d tell you the sum of
their ages, but you’d still be stumped.” “I wish you’d tell me
something more.” “Okay, my oldest daughter Annie likes dogs.”
What are the ages of the three daughters (Zeitz, 2007)?
Solution:
After the first reading, it seems impossible- there isn’t enough information to
determine the ages. The product of the ages is 36, so there are only a few possible
triples of ages. Here is a table of all the possibilities
Age 1, 1,36 1, 2,18 1, 3, 12 1, 4, 9 1, 6, 6 2, 2, 9 2, 3, 6 3, 3, 4
Sum 38 21 16 14 13 13 11 10
Now we see what is going on. The mother’s second statement (“I’d tell you the
sum of their ages, but you’d still be stumped.) gives valuable information. It tells
that the ages are either 1,6,6 or 2,2,9, for in all other cases, knowledge of the sum
would tell unambiguously what the ages are. The final clue now makes sense, it
tells that there is an oldest daughter, eliminating 1, 6,6. The daughters are thus 2,
2, and 9 years old.

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Lesson 2.3 Mathematical Reasoning and Logic

  • 1.
  • 2. Learning Outcomes: At the end of this lesson, the students will be able to: 1. Analyze problems using different types of reasoning. 2. Apply different types of reasoning to justify statements and arguments made about mathematics and mathematical concepts.
  • 3.
  • 4. INDUCTIVE REASONING It is the process of reaching a general conclusion by examining specific examples.
  • 5. INDUCTIVE REASONING The conclusion formed by using inductive reasoning is often called a conjecture, since it may or may not be correct.
  • 6. Example 1: Use inductive reasoning to predict the next number in each item. 1. 2,8,14,20,26, _____ 2. 1,2,5,10,17,26, _____
  • 7. Example 2: A. Every sports car I have ever seen is red. Thus, all sports cars are red. B. The coin I pulled from the bag is a 5-peso coin. Another 5-peso coin is drawn from the bag. A third coin from the bag is again a 5-peso coin. Therefore, all the coins in the bag are 5-peso coins.
  • 8. Example 3: Consider the following. Pick a number. Multiply the number by 4, add 8 to the product, divide the sum by 2, 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.
  • 9. Solution: Suppose we start with seven as the original number. Then repeat the process for different numbers. The procedure yields the following: We conjecture that the given procedure produces a number that is one less than twice the original number.
  • 10. Remarks: When we use inductive reasoning, we have no guarantee that our conclusion is correct. Just because a pattern is true for a few cases, it does not mean the pattern will continue. A statement is a true statement provided that it is valid in all cases. If we can find one case for which a statement is not valid, called a counterexample, then it is a false statement.
  • 11. DEDUCTIVE REASONING It is the process of reaching a conclusion by applying general assumptions, procedures, or principles.
  • 12. DEDUCTIVE REASONING Deduction starts out with a general statement, or hypothesis, and examines the possibilities to reach a specific, logical conclusion.
  • 13. Example 1: 1. All men are mortal. Kahwi is a man. Therefore, Kahwi is mortal. 2. Corresponding parts of congruent triangles are congruent. Triangle ABC is congruent to triangle DEF. Angle B and angle E are corresponding angles. Thus, angle B is congruent to angle E.
  • 15. Determine if each of the following statement uses inductive or deductive reasoning. 1. Teacher Erica is an enthusiastic and passionate teacher. Therefore, all teachers are enthusiastic and passionate. 2. All dogs are animals. Dhai is a dog. Thus, Dhai is an animal. 3. I got low score on the first long exam. I just recently took the second long exam and I got low score. Therefore, I will also get a low score on the third long exam. IR DR IR
  • 16. Determine if each of the following statement uses inductive or deductive reasoning. 4. My classmates are disrespectful toward our instructor. Hence, all students are disrespectful. 5. Last Wednesday it was raining. Today is Wednesday and it is raining. Therefore, on the next Wednesday, it will also rain. 6. For any right triangle, the Pythagorean Theorem holds. ABC is a right triangle, therefore for ABC the Pythagorean Theorem holds. 7. All basketball players in your school are tall, so all basketball players must be tall. IR IR DR IR
  • 17. A logic puzzle is a puzzle deriving from the mathematics field of deduction. Logic puzzles can be solved by using deductive reasoning and by organizing the data in a given situation.
  • 18. A logic puzzle is basically a description of an event or any situation. Using the clues provided, one has to piece together what actually happened. This involves clear and logical thinking, hence the term “logic” puzzles.
  • 19. Example 1 Three musicians appeared at a concert. Their last names were Benton, Lanier, and Rosario. Each plays only one of the following instruments: guitar, piano, or saxophone. 1. Benton and the guitar player arrived at the concert together. 2. The saxophone player performed before Benton. 3. Rosario wished the guitar player good luck. Who played each instrument? Guitar Piano Saxophone Benton Lanier Rosario
  • 20. Example 2 You have a basket containing ten apples. You have 10 friends, who each desire an apple. You give each of your friends, one apple. Now all your friends have one apple each, yet there is an apple remaining in the basket. How? Solution: ✎You give an apple to your first nine friends, and a basket with an apple to your tenth friend. Each friend has an apple, and one of them has it in a basket. ✎ Alternative answer: one friend already had an apple and put it in the basket.
  • 21. Example 3 A census-taker knocks on a door, and asks the woman inside how many children she has and how old they are. “I have three daughters, their ages are whole numbers, and the product of their ages is 36,” says the mother. “That’s not enough information”, responds the census-taker. “I’d tell you the sum of their ages, but you’d still be stumped.” “I wish you’d tell me something more.” “Okay, my oldest daughter Annie likes dogs.” What are the ages of the three daughters (Zeitz, 2007)?
  • 22. Solution: After the first reading, it seems impossible- there isn’t enough information to determine the ages. The product of the ages is 36, so there are only a few possible triples of ages. Here is a table of all the possibilities Age 1, 1,36 1, 2,18 1, 3, 12 1, 4, 9 1, 6, 6 2, 2, 9 2, 3, 6 3, 3, 4 Sum 38 21 16 14 13 13 11 10 Now we see what is going on. The mother’s second statement (“I’d tell you the sum of their ages, but you’d still be stumped.) gives valuable information. It tells that the ages are either 1,6,6 or 2,2,9, for in all other cases, knowledge of the sum would tell unambiguously what the ages are. The final clue now makes sense, it tells that there is an oldest daughter, eliminating 1, 6,6. The daughters are thus 2, 2, and 9 years old.

Editor's Notes

  1. Reasoning is a process based on experience and principles that allow one to arrive at a conclusion.
  2. Reasoning is a process based on experience and principles that allow one to arrive at a conclusion.
  3. We make generalizations from the part to the whole. If we are not careful, it can lead to erroneous or mistake conclusions:
  4. Conjecture- estimation or guess
  5. Each successive number is 6 larger than the preceding number.
  6. It is not enough that the deduction is logically sound; the assumption (1) must also be true. Consider the following: "1) All cats are red. 2) Kitty is a cat, therefore Kitty is red." It is logically valid but leads to a non-valid conclusion because not 'all cats are red'.
  7. In mathematics, deductive reasoning makes use of definitions, axioms, theorems and rules and inference.