Logical Reasoning (An Introduction to Geometry) MATHEMATICS Grade 8
If a number is even, then it is divisible by 2 The statement above is written in conditional form, or in if-then form. A conditional statement has 2 parts: A hypothesis, denoted by p, and A conclusion, denoted by q. In symbols, If p, then q. is written as p => q.
Remember! A conditional statement may be true or false. To show that a conditional statement is false, you need to find one example (called a counterexample) in which the hypothesis is fulfilled and the conclusion is not fulfilled.
Remember! To show that a conditional statement is true, you must construct a logical argument using reasons. The reasons can be a definition, an axiom, a property, a postulate, or a theorem.
Converse The converse of the conditional statement is formed by interchanging the hypothesis and conclusion. For instance, the converse of p => q is q => p. The converse may also be true or false.
Examples: If m A = 45, then A is acute. This statement is true because 45 <90. Converse: If A is acute, then m A = 45. The converse is false, because some acute angles do not measure 45.
Examples: If m B = 90, then B is right angle. This statement is true because the measure of the right angle is exactly 90. Converse: If B is right angle, then m B = 90. The converse is true. (Explanation same as above)
Examples: If today is Sunday, then it is a weekend day. This statement is true because Sunday is a weekend day. Converse: If today is a weekend day, then it is Sunday. The converse is false. Saturday (a counterexample) is also a weekend day.
Other statements Conditional: p => q If p, then q Inverse: ~p => ~q If not p, then not q. Contrapositive: ~q => ~p If not q, then not p. The symbol (~) shows the negative of the hypothesis and conclusion.
Remember: To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion. To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement.
Examples: Conditional: If m A = 45, then A is acute. Converse: If A is acute, then m A = 45. Inverse: If m A is not 45, then A is not acute. Contrapositive: If A is not acute, then m A is not 45.
Examples: Conditional: If m B = 90, then B is right angle. Converse: If B is right angle, then m B = 90. Inverse: If m B is not 90, then B is not a right angle. Contrapositive: If B is not a right angle, then m B is not 90.
Examples: Conditional: If today is Sunday, then it is a weekend day. Converse: If today is a weekend day, then it is Sunday. Inverse: If today is not Sunday, then it is not a weekend day. Contrapositive: If today is not a weekend day, then it is not Sunday.
For NOW: Conditional: p => q If p, then q Converse: q => p If q, then p Inverse: ~p => ~q If not p, then not q. Contrapositive: ~q => ~p If not q, then not p.
Remember: If the statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true. In determining if the inverse, converse, and contrapositive of the statement is true or false, assume that the given statement is true.
Seatwork (1 whole int. pad) Write the converse, inverse, and contrapositive of each conditional statement. Determine the truth value of each statement. If the statement is false, give a counterexample.
Seatwork (1 whole int. pad) 1. If the degree measure of an angle is between 90 and 180, then the angle is obtuse. 2. If a quadrilateral has four congruent sides, then it is a square. 3. If a bird is an ostrich, then it cannot fly. 4. If today is Friday, then tomorrow is Saturday. 5. If there is no struggle, then there is no progress.
3. Converse: If an angle is obtuse, then its degree measure is between 90 and 180. Inverse: If the degree measure of an angle is not between 90 and 180, then the angle is not obtuse. Contrapositive: If an angle is not obtuse, then its degree measure is not between 90 and 180. All statements are TRUE.
4. Converse: If a quadrilateral is a square, then it has four congruent sides. Inverse: If a quadrilateral has no congruent sides, then it is not a square. Contrapositive: If a quadrilateral is not a square, then it has no congruent sides. All statements are TRUE.
8. Converse: If a bird cannot fly, then it is an ostrich. Inverse: If a bird is not an ostrich, then it can fly. Contrapositive: If a bird can fly, then it is not an ostrich. Converse and Inverse statements are FALSE. Counterexample: Penguin Contrapositive is TRUE.
1. Converse: If tomorrow is Saturday, then today is Friday. Inverse: If today is not Friday, then tomorrow is not Saturday. Contrapositive: If tomorrow is not Saturday, then today is not Friday.
Deductive Reasoning Joash Caleb Z. Palivino MATHEMATICS Grade 8
Deductive Reasoning To deduce means to reason from the known facts. Deductive Reasoning is the process of using facts, rules, definitions, or properties to reach logical conclusions from given statements.
Deductive Reasoning In deductive reasoning, assume that the hypothesis is true, and then write a series of statements that lead to the conclusion. Each statement is supported by a reason that justifies it.
Deductive Reasoning Law of Detachment Draws conclusion from a true conditional statement p q and a true statement p. If p q is a true statement and p is true, then q is true.
Deductive Reasoning Law of Detachment (Example) Given: If a car is out of gas, then it will not start. Sarah s car is out of gas. Valid Conclusion: Sarah s car will not start.
Law of Detachment Given: If two numbers are odd, then their sum is even. The numbers 3 and 5 are odd numbers. Conclusion: The sum of 3 and 5 is even. Given: If you want good health, then you should get 8 hours of sleep each day. Aaron wants good health. Conclusion: Aaron should get 8 hours of sleep each day.
Law of Detachment Given: If you are a good citizen, then you obey traffic rules. Aaron is a good citizen. Conclusion: Aaron obeys traffic rules. VALID CONCLUSION.
Law of Detachment Given: If a pet is a rabbit, then it eats carrots. Jennie s pet eats carrots. Conclusion: Jennie s pet is a rabbit. INVALID CONCLUSION. There are other animals that eat carrots besides rabbit, like hamster.
Seatwork (Math NB Ans. only) Determine if the conclusion is valid or invalid. If invalid, explain your reasoning by giving a counterexample.
1. Given: If students pass an entrance exam, then they will be accepted into college. Latisha passed the entrance exam. Conclusion: Latisha will be accepted to college.
2. Given: Right angles are congruent. 1 and 2 are right angles. Conclusion: 1 and 2 are congruent. 3. Given: An angle bisector divides an angle into two congruent angles. Ray KM is an angle bisector of JKL Conclusion: JKM and MKL are congruent.
4. Given: Rating Age EC 3 and older E 6 and older E10+ 10 and older T 13 and older M 17 and older If a game is rated E, then it has content that may be suitable for ages 6 and older. Cesar buys a computer game that he believes is suitable for his little sister who is 7. Conclusion: The game Cesar purchased has a rating of E.
5. Given: All vegetarians do not eat meat. Theo is a vegetarian. Conclusion: Theo does not eat meat. 6. Given: If a figure is a square, then it has four right angles. Figure ABCD has four right angles. Conclusion: Figure ABCD is a square.
7. Given: If you leave your lights on while your car is off, your battery will die. Your battery is dead. Conclusion: You left your lights on while the car was off.
8. Given: If Dante obtains a part-time job, he can afford a car payment. Dante can afford a car payment. Conclusion: Dante obtained a part-time job.
9. Given: If the temperature drops below 32 degrees Fahrenheit, it may snow. The temperature did not drop below 32 degrees Fahrenheit on Monday. Conclusion: It did not snow on Monday.
10. Given: Some nurses wear blue uniforms. Sabrina is a nurse. Conclusion: Sabrina wears blue uniform.
Answers 1) The conclusion is valid. 2) The conclusion is valid. 3) The conclusion is valid. 4) The conclusion is invalid. The rating can also be EC. 5) The conclusion is valid.
Answers 6) The conclusion is invalid. The figure could be a rectangle. 7) The conclusion is invalid. The battery could be dead for another reason. 8) The conclusion is invalid. Dante could afford a car payment for another reason. 9) The conclusion is valid. 10) The conclusion is invalid. Not all nurses wear blue uniform.
Deductive Reasoning Law of Syllogism Draw conclusions from two true statements when the conclusion of one statement is the hypothesis of another. If p q is true and q r is true, then p r is also true.
Deductive Reasoning Law of Syllogism (Example) Given: If two angles of a triangle are congruent, then the sides opposite these angles are also congruent. If two sides of triangle are congruent, then the triangle is isosceles. Valid Conclusion: If two angles of a triangle are congruent, then the triangle is isosceles.
Law of Syllogism Given: If a number is a whole number, then the number is an integer. If a number is an integer, then it is a rational number. Conclusion: If a number is a whole number, then it is a rational number.
Determine if a valid conclusion can be reached from the given statements Given: If an angle is supplementary to an obtuse angle, then it is acute. If an angle is acute, then its measure is less than 90. Conclusion: If an angle is supplementary to an obtuse angle, then its measure is less than 90.
Determine if a valid conclusion can be reached from the given statements Given: If a parallelogram has a right angle, then it is a rectangle. If a parallelogram has a right angle, then it is a square. Conclusion: NO VALID CONCLUSION.
Determine if a valid conclusion can be reached from the given statements Given: If an angle is a right angle, then the measure of the angle is 90. If two lines are perpendicular, then they form a right angle. Conclusion: If two lines are perpendicular, then the measure of the angle formed is 90.
Determine if a valid conclusion can be reached from the given statements Given: If you are a good citizen, then you pay your taxes. If you are a good citizen, then you obey traffic rules. Conclusion: NO VALID CONCLUSION.
Seatwork (Math NB Ans. only) Use the Law of Syllogism to draw a valid conclusion from each set of statements, if possible. If no valid conclusion is possible, write no valid conclusion.
1. If Tina has a grade of 90% or greater, she will be on the honor roll. If Tina is on the honor roll, then she will have her name in the school paper. 2. If the measure of an angle is between 90 and 180, then the angle is obtuse. If an angle is obtuse, then it is not acute. 3. If a number ends in 0, then it is divisible by 2. If a number ends in 4, then it is divisible by 2.
4. If a triangle is a right triangle, then it has an angle that measures 90. If a triangle has an angle that measures 90, then its acute angles are complementary. 5. If you interview for a job, then you wear a suit. If you interview for a job, then you will update your resume.
6. If two lines in a plane are not parallel, then they intersect. If two lines intersect, then they intersect in a point. 7. If it continues to rain, then the soccer field will become wet and muddy. If the soccer field becomes wet and muddy, then the game will be canceled.
8. If the bank robber steals the money, then the sheriff will track him down. If the bank robber steals the money, then the bank robber will be rich. 9. If the truck runs over some nails, then a tire will go flat. If a tire goes flat, then the deliveries will not be made on time. 10. If Jane encounters a traffic jam today, she reports to work late. If Jane reports to work late, her boss penalizes her.
Inductive Reasoning Joash Caleb Z. Palivino MATHEMATICS Grade 8
Identifying a Pattern Monday, Wednesday, Friday, Alternating days of the week make up the pattern. The next day is Sunday. 3, 6, 9, 12, 15, Multiples of 3 make up the pattern. The next multiple is 18.
Inductive Reasoning Inductive reasoning is a process of observing data, recognizing patterns, and making generalizations from observations. Inductive reasoning is reasoning from specific to general. In using inductive reasoning to make a generalization, the generalization is called a conjecture.
More on Identifying a Pattern 1, 2, 4, 8, 16, Each term is 2 times the previous term. The next two terms are 32 and 64. 1, 4, 9, 16, 25, Each term is a square number. The next two terms are 36 and 49.
Making a Conjecture The product of an even number and an odd number is. List some examples and look for a pattern. (2)(3) = 6 (2)(5) = 10 (4)(3) = 12 (4)(5) = 20 The product of an even number and an odd number is even.
Making a Conjecture Study each number patterns: 12 + 28 = 40-14 + 6 = -8-10 + 30 = 20 0 + 22 = 22 18 + 16 = 34 8 + 38 = 46 Conjecture: The sum of two even numbers is an even number.
Making a Conjecture Study each number patterns: 4 (5) = 20 9 (8) = 72 11 (6) = 66-12 (-3) = 36-5 (8) = -40-41(4) = -164 Conjecture: The product of an odd number and an even number is an even number.
Remember! Inductive reasoning may not always lead to the right conclusion. To show that a conjecture is always true, you must prove it. To show that a conjecture is false, you have to find only one example in which the conjecture is not true. This case is called a counterexample.
Seatwork (Math NB Ans. only) Use inductive reasoning to find the next two terms of each sequence. Justify your answer.
Use inductive reasoning to find the next two terms of each sequence. Justify your answer. 1. 1, 10, 100, 1000,, 2. 1, 3, 9, 27, 81,, 3. 1, 1, 2, 3, 5, 8, 13,, 4. 0, 2, 6, 12, 20, 30, 42,, 5. O, T, T, F, F, S, S, E, N,, 6. J, F, M, A, M, J, J,, 7. ½, ¼, 1 / 8, 1 / 16,, 8. ½, 9, 2 / 3, 10, ¾, 11,, 9. S, M, T, W, T,, 10. A, C, E, G,,
Logic Puzzle Alice met a lion and a unicorn. Suppose that the lion lies on Monday, Tuesday, and Wednesday and the unicorn lies on Thursday, Friday, and Saturday. At all other times both animals tell the truth. Alice has forgotten the day of the week during her travels through the Forest of Forgetfulness. Lion:Yesterday was one of my lying days. Unicorn: Yesterday was one of my lying days, too! Alice, who was very smart, was able to deduce the day. What day of the week was it? Explain.
Logic Puzzle Tweedledum and Tweedledee are identical twins who decided to entertain themselves by confusing Alice. One of the brothers of course, we don t know which says, In this puzzle, each of us will pick one of two cards, either an orange one or a blue one. The one with the orange card will always tell the truth. The one with the blue card will always lie.
Logic Puzzle I have the blue card, and I am Tweedledee! You are not! I am Tweedledee. Alice picks out Tweedledee immediately! Which one is it, and how did she figure it out?
Logic Puzzle Tweedledum is now carrying a blue card! Alice looked confused for a moment, then thought as logically as she could and solved the puzzle. Who is Tweedledum? How can you tell?
Logic Puzzle Three sisters are identical triplets. The oldest by minutes is Sarah, and Sarah always tells anyone the truth. The next oldest is Sue, and Sue always will tell anyone a lie. Sally is the youngest of the three. She sometimes lies and sometimes tells the truth. Victor, an old friend of the family's, came over one day and as usual he didn't know who was who, so he asked each of them one question.
Logic Puzzle Victor asked the sister that was sitting on the left, "Which sister is in the middle of you three?" and the answer he received was, "Oh, that's Sarah." Victor then asked the sister in the middle, "What is your name?" The response given was, "I'm Sally." Victor turned to the sister on the right, then asked, "Who is that in the middle?" The sister then replied, "She is Sue." This confused Victor; he had asked the same question three times and received three different answers. Who was who?
Performance Task # 2: Logic Puzzle Task: You work for a company that publishes logic puzzle booklets. Your task is to create an original logic puzzle that requires the use of inductive and/or deductive reasoning to determine the solution.
Performance Task # 2: Logic Puzzle Mechanics: You must use at least 3 people/objects for your puzzle. You must provide list of statements (clues) that will help solve the puzzle. Test your puzzle on at least 2 people (not your groupmates).
Performance Task # 2: Logic Puzzle Mechanics: Submit the following on March 15, 2017: One (1) blank puzzle sheet + One (1) puzzle sheet solution key. [FINAL] (Format: Short bond paper (8.5 x 11 ), computerized, font style and size of your choice (but should be legible and understandable)) All copies of DRAFT and TESTED puzzle sheets.
Performance Task # 2: Logic Puzzle Mechanics: Reflection Journal (individual). (Format: Computerized on short bond paper, Arial, 12, 1.5 spacing, 1 margin all sides). The reflection paper must address the following: What geometry skills are used for the project? Can I use these skills outside of class? How? How did we get started? What were my first thoughts? How does our team work? How do each member contribute to the group s success?
Performance Task # 2: Logic Puzzle Mechanics: Late submission of Performance Task will have a demerit of 2 points each day. Rubric: GROUP (80%) Content (25%) Clues (25%) Solution (20%) Mechanics (10%) INDIVIDUAL Reflection Journal (20%)