Inductive and Deductive Reasoning

Legally Blonde

Pass out Work Sheet Always, Sometimes or Never Activity   Determine if each statement is always true (A), sometimes true (S), or never true (N). Circle your choice. If the statement is always true provide an example. If the statement is never true, provide a counterexample. If the statement is sometimes true, provide an example of when it is true and an example of when it is false.

1. Two lines that intersect form right angles. Determine if each statement is always true (A), sometimes true (S), or never true (N). 1. Two lines that intersect form right angles. Sometimes Example Counter Example

2. Two angles that are congruent, are vertical angles. Determine if each statement is always true (A), sometimes true (S), or never true (N). 2. Two angles that are congruent, are vertical angles. Sometimes Example Counter Example

3. An equilateral triangle is congruent to a right triangle. Determine if each statement is always true (A), sometimes true (S), or never true (N). 3. An equilateral triangle is congruent to a right triangle. Never Counter Example

4. If two angles are complimentary, then they form a right angle. Determine if each statement is always true (A), sometimes true (S), or never true (N). 4. If two angles are complimentary, then they form a right angle. Sometimes Example Counter Example 45 45

Determine if each statement is always true (A), sometimes true (S), or never true (N). 5. For any two points 𝐴 and 𝐵, there is exactly one line that contains them. Always Example A B B A A B

Inductive vs. Deductive Reasoning Reasoning Inductive Reasoning- uses a number of specific examples to come up with a generalization or prediction Deductive Reasoning- uses facts, rules, definitions, or properties to reach logical conclusions.

Inductive Reasoning Inductive Reasoning- uses a number of specific examples to come up with a generalization, prediction, or what is called a conjecture. A conjecture is an educated guess based on known information.

Inductive Reasoning Examples 1. 4, 8, 12, 16, ____ 20 2. 20, 15, 11, 8, ____ 6 3. If 10 people walk into a room and they all have cell phones. Conjecture: ______________________________ Counterexample:__________________________ All people have cell phones. Find someone without a cell phone

Inductive Reasoning Examples Given: B is between A and C, and A, B, and C are collinear Conjecture:______________________ DEF of Segment Addition A B C AB + BC = AC EX 2) Given: If B is the midpoint of 𝐴𝐶 Conjecture:______________________ Def of MIDPOINT 𝐴𝐵  𝐵𝐶

Inductive Reasoning Examples Given: M is between ARC Conjecture:______________________ DEF of Angle Addition A M R C mARM + mCRM = mARC EX 4) Given: 𝑀𝑅 bisects angle ARC Conjecture:______________________ Def of Angle Bisector ARM  CRM

Inductive Reasoning Examples B C D EX 5) Given: 𝐴𝐵  𝐶𝐷 Conjecture:______________________ DEF of congruent segments AB = CD F E A B C D EX 6) Given: ABC  DEF Conjecture:______________________ Def of congruent angles mABC = mDEF

Inductive Reasoning Examples Given: 𝐷𝐵 and 𝐴𝐶 are perpendicular Conjecture:______________________ DEF of perpendicular lines A B C D E DBC is a right angle EX 8) Given: DBC is a right angle Conjecture:______________________ Def of Right Angle mDBC = 90

Inductive Reasoning Examples Given: 1 and 2 are supplementary Conjecture:______________________ DEF of Supplementary Angles 1 2 m1 + m2 = 180 EX 10) Given: ARM and CRM are complementary angles Conjecture:______________________ Def of Complementary Angles A M R C mARM + mCRM = 90

Deductive Reasoning Deductive Reasoning- uses facts, rules, definitions, or properties to reach logical conclusions. Two types of Deductive Reasoning are: 1. Law of Detachment 2. Law of Syllogism

Deductive Reasoning Law of Detachment (STAAR) If pq is true, and p is true, then q is also true. EX) (1) If it’s raining, then it’s wet. (2) On Thursday it was raining. (3) Therefore, it was wet on Thursday.

Deductive Reasoning Law of Detachment (STAAR) If pq is true, and p is true, then q is also true. EX) (1) If the measure of an angle is greater than 90, then it is obtuse. (2) mABC > 90 (3) ABC is obtuse

Deductive Reasoning Law of Detachment (STAAR) If pq is true, and p is true, then q is also true. EX) (1) If the measure of an angle is less than 90, then it is acute. You Try It! (2) mABC < 90 (3) ABC is acute

Deductive Reasoning Law of Syllogism (STAAR) If p  q and q  r, then p  r Fancy way to say transitive EX) (1) If you are a ballet dancer, you like classical music p q q r (2) If you like classical music, then you enjoy the opera. r p (3) If you are a ballet dancer, then you enjoy the opera.

Deductive Reasoning Law of Syllogism (STAAR) If p  q and q  r, then p  r Fancy way to say transitive EX) (1) If 1 and 2 form a linear pair, then they are supplementary. p q q r (2) If 1 and 2 are supplementary, then m1 + m2 = 180. p r (3) If 1 and 2 form a linear pair, then m1 + m2 = 180.

Deductive Reasoning Law of Syllogism (STAAR) If p  q and q  r, then p  r Fancy way to say transitive EX) (1) If two lines are perpendicular, then they form a right angle. p q q r (2) If a right angle is formed, then the measure is 90. r p (3) If two lines are perpendicular, then the angle measure is 90.

Pass out Case Sheets Your job is to prove whether each case is guilty or not guilty based on the evidence that was collected. Be able to defend the victim with conditional. What conjectures can you make? Explain and justify your statement without a reasonable doubt.

Case #1 Guilty or Not Guilty? Facts: John pleads innocent from being accused of breaking and entering. John was at his mom’s house at 4:00 PM wearing a black hoodie. The burglary happened around 6:00 PM. At 5:30 PM John made a call to his friend. A crowbar was found inside the trunk of John’s car. The door was forced open showed signs of breaking and entry at the scene of the crime. John’s mom said John left around 5:35 PM. Neighbors around the victim’s house said they saw two suspects leave the house with personal belongings. The neighbor was unable to recognized clearly who they were. Your job is to prove John is guilty based on the evidence that the police reported and collected. Defend the victim with conditional statements to prove John guilty. What conjectures can you make? Explain and justify your statement.

Case #2 Guilty or Not Guilty? Daisy has been accused of vandalizing school property. Prove that Daisy is innocent of the crime she was convicted for. With the following facts what conjecture can be made? Explain and justify your statement. Facts: Spray cans were found in Daisy’s locker. Cameras show a female spray painting the wall with graffiti. Further inspection of the locker showed signs of break in attempts. Inside the locker, there was hair that did not match that of Daisy’s. At the scene of the crime, there was also hair that matched the one at the locker.

Case #3 Cat Thief or Not? Help determine if the police can make a conclusion from the following scenario. Cat thief or Not? With the following facts what conjecture can be made? Explain and justify your statement.   Facts: Meow was last seen outside sleeping on her bed. Rain began to fall while Meow was sleeping. Fresh paw prints were found on the ground. The next door neighbor complains about Meow sitting on the porch. Upon making more observations, another set of paw prints were next to Meow. The day after Meow disappearance the neighbor was seen wearing new fur earmuffs. When the paw prints were followed, they led to a tree nearby.