Graphic Organizer: Reasoning and Proof Unit Essential Quetions

Transcription

1 Start Here Page 1 Graphic Organizer: Reasoning and Proof Unit Essential Quetions Right Side of Page

2 Right Sight Page 2 Unit 2 Reasoning and Proof (handout: orange sheet) Right side

3 Right Sight Page 3 Proof Book (back side) Sunday, October 09, :55 AM

4 Left Side Page 4 Homework: 4 Types of Conditional Statements (left side) Type 1: Symbol: Words Type 4: Symbol: Words 4 Types of Conditional Statements Type 2: Symbol: Words Type 3: Symbol: Words

5 Conditional Statement Notes Conditional Statement Notes Page 5

6 Conditional Statement Notes Page 6 Converse Guided Notes Converse Statement: If you see lightning, then you hear thunder. Converse: If you hear thunder, then you see lightening. The converse is written by switching the hypothesis and the conclusion.

7 Conditional Statement Notes Page 7 Inverse, Contrapositive, Biconditional Statements Notes (Right side) Sunday, October 09, :22 AM Symbolic Notation p : represents the hypothesis q : represents the conclusion : is read as implies If the sun is out, then the weather is good. p q The conditional statement can be written as: If p, then q or p q Symbolic Notation The converse of If p, then q is If the weather is good, then the sun is out. q p The converse can be written as: If q, then p or q p Symbolic Notation A biconditional statement can be written as: If p, then q and if q, then p OR p q OR p if and only if q Using Symbolic Notation Let p be the value of x is - 5 and let q be the absolute value of x is 5 A) Write p q If the value of x is - 5, then the absolute value of x is 5. B) Write q p If the absolute value of x is 5, then the value of x is - 5. C) Decide whether p q is true. A) True. B) False, x can be 5. C) p q is False. Inverse Statement: If you see lightning, then you hear thunder. Inverse: If you do not see lightening, then you do not hear thunder. The inverse is written by making the hypothesis and the conclusion negative. Symbolic Notation The inverse of If p, then q is If, the sun is not out, then the weather is not good. ~ p ~ q The inverse can be written as: If ~ p, then ~q or ~ p ~ q

8 Conditional Statement Notes Page 8 Contrapositive Statement: If you see lightning, then you hear thunder. Inverse: If you do not see lightening, then you do not hear thunder. Contrapositive: If you do not hear thunder, then you do not see lightening. The contrapositive is written by switching the hypothesis and the conclusion and making them negative. Symbolic Notation The contrapositive of If p, then q is If the weather is not good, then the sun is not out. ~ q ~ p The contrapositive can be written as: If ~q, then ~ p or ~ q ~ p 8 9 Using Symbolic Notation Let p be it is raining and let q be the soccer game is canceled A) Write ~ p ~ q If it is not raining, then the soccer game is not canceled. B) Write ~ q ~ p If the soccer game is not canceled, then it is not raining.

9 Warmups Page 9 Warm-Up 9/26 (left side) Warm-up Conditional Statements Complete the If-Then statement. Identify the hypothesis in purple and conclusion in pink. Then write the converse, inverse, and contrapositive. 1) If I run the red light, then.. Example: If I run the red light, then I would get a ticket. Converse: If I get a ticket, then I ran the red light. Inverse: If I do not run the red light, then I would not get a ticket. Contrapositive: If I did not get a ticket, then I did not run the red light.

10 Warmups Page 10 Warm-Up 9/27 Statement Example Truth Value Symbols Conditional If the measure of angle A is 95 o, then angle A is obtuse. T p q Converse Inverse If angle A is obtuse, then the measure of angle A is 95 o. If the measure of angle A is not 95 o, then angle A is not obtuse. Contrapositive If angle A is not obtuse, then the measure of angle A is not 95 o. F q p F ~p ~q T ~q ~p

11 Warmups Page 11 Warm-Up 10/5 Sunday, October 09, :57 AM Read the following statements: a) I scored below 60% on the past 3 tests. The next test I take will also be below a 60%. b) If I score below a 60% on a test, then I earned an F. I earned an F on my last test therefore, I scored below 60%. Are both statements always true? Explain your thinking in at least 2 sentences.

12 Warmups Page 12 Warm-Up 10/10 Sunday, October 09, :17 AM Warm-up (Add to end of Notes on Deductive Reasoning) Is each conclusion the result of inductive or deductive reasoning? 1) There is a myth that you can balance an egg on its end only on the spring equinox. A person was able to balance an egg on July 8, September 21, and December 19. Therefore, this myth is false. Inductive Reasoning 2) There is a myth that the Great Wall of China is the only man-made object visible from the Moon. The Great Wall is barely visible in photographs taken from 180 miles above Earth. The Moon is about 237,000 miles from Earth. Therefore, the myth cannot be true. Deductive Reasoning

13 Homework Page 13 9/27 Explanation of 4 types of Conditional Statements(left side) Sunday, October 09, :21 PM On a loose leaf of paper 1) Write one conditional statement. 2) Write its converse, inverse, and contrapositive. 3) Determine whether each is true or false. If either is false, state the counterexample.

14 Reflection Page 14 Reflection 10/3 (left side) Reflection Date 1) Highlight/Lowlight of the class 2) Now that you ve taken a test and quizzes, how do you think you will do in this class? 3) What grade would you like to achieve? 4) What do you need to do to get there? 5) What do you need from Ms. Washington to help you get there?

15 Classifying Inductive vs Deductive Page 15 Classifying Inductive vs Deductive (Right side) NHS

16 Inductive vs Deductive Reasoning Notes Page 16 Inductive vs Deductive Notes (Right side) Sunday, October 09, :19 AM Deductive Reasoning Inductive vs. Deductive Reasoning Inductive: previous examples and patterns are used to form a conjecture. Andy knows that Kd is a sophomore and Maria is a junior. All the other juniors that Andy knows are older than Kd. Therefore, Andy reasons inductively that Maria is older than Kd based on past observations. Inductive vs. Deductive Reasoning Deductive: uses facts, definitions, and accepted properties in a logical order to write a logical argument. Andy knows that Maria is older than Jason. She also knows that Jason is older than Kd. Andy reasons deductively that Maria is older than Kd based on accepted statements. Laws of Deductive Reasoning Law of Detachment: If p q is true conditional statement and p is true, then q is true. Example: Devin knows that if he misses the practice the day before a game, then he will not be a starting player in the game. Devin misses practice on Tuesday so he concludes that he will not be able to start in the game on Wednesday. Using the Law of Detachment State whether the argument is valid. If two angles form a linear pair, then they are supplementary ; A and B are supplementary, so A and B form a linear pair. Not valid: p q and q (the conclusion) are not true. The argument implies that all supplementary angles form a linear pair. Laws of Deductive Reasoning Law of Syllogism: If p q and q r are true conditional statements, then p r is true. p : Rebecka visits California. q : Rececka spends New Year s Day in Pasadena. r : Rebecka goes to watch the Rose Parade. If Rebecka visits California, then she will go to watch the Rose Parade.

17 Inductive vs Deductive Reasoning Notes Page 17 Using the Law of Syllogism Write some conditional statements that can be made from the following true statements using the Law of Syllogism. 1) If a bird is the fastest bird on land, then it is the largest of all birds. 2) If a bird is the largest of all birds, then it is an ostrich. Possible Conditional Statements If a bird is the fastest bird on land, then it is an ostrich (1 & 2) If a bird is the fastest bird on land, then it is flightless. (1 & 3) 3) If a bird is the largest of all birds, then it is flightless.

18 Types of Reasoning Page 18 Graphic Organizer: Inductive vs Deductive (Right side) Sunday, October 09, :53 PM

19 Table 1.13 Page 19 Inductive vs Deductive (left side) Sunday, October 09, :30 PM

20 Algebraic Proof Notes Page 20 Algebraic Proof Notes(Right side) Tuesday, October 18, :46 PM Algebraic Properties of Equality Reasoning with Properties of Equality Let a, b, and c be real numbers. Addition Property of Equality: If a = b, then a + c = b + c Abbr. Add. Prop. of = Subtraction Property of Equality: If a = b, then a - c = b - c Abbr. Subt. Prop. of = Multiplication Property of Equality: If a = b, then ac = bc Abbr. Mult. Prop. of = Division Property of Equality: If a = b and c 0, then a/c = b/c Abbr. Div. Prop. of = Algebraic Properties of Equality continued Reflexive Property: For any real number a, a = a Abbr. Refl. Prop. of = Symmetric Property: If a = b, then b = a Abbr. Sym. Prop. of = Transitive Property: If a = b and b = c, then a = c Abbr. Trans. Prop. of = Substitution Property: If a = b, then a can be substituted for b in any equation or expression. Abbr. Subs. Prop. of = Use properties to support your argument. All the algebraic properties of equality can be used to solve equation. Other properties, such as the distributive property can also be used. a( b + c ) = ab + ac Abbr. Distrib. Prop. Two-Column Proof Given p 1 = 6, prove that p = 7. Statements Reasons Statements Reasons Usually written in symbols and mathematical notation. A definition, property, postulate or theorem that proves or explains why the statement is true. Usually written in symbols and mathematical notation. A definition, property, postulate or theorem that proves or explains why the statement is true. 1) Statement 1 2) Statement 2 3) Last statement is what you wanted to prove. 1) Reason 1 is always Given 2) Property,etc. that explains the change from the previous step to this one. 3) Same as Reason 2. 1) p - 1 =6 2) p = 7 1) Given 2) Addition Prop. of Equality

21 Algebraic Proof Notes Page 21 Given 2r 7 = 9, prove that r = 8. Statements Usually written in symbols and mathematical notation. 1) 2r 7 = 9 2) 2r = 16 3) r = 8 Reasons A definition, property, postulate or theorem that proves or explains why the statement is true. 1) Given 2) Addition Prop. of Equality 3) Division Prop. of Equality Solve 3(2t + 9) = 30 and state the reason for each step. Statements 1) 3(2t + 9) = 30 2) 6t + 27 = 30 3) 6t = 3 4) t = 0.5 Reasons 1) Given 2) Distrib. Prop. 3) Subt. Prop. of = 4) Div. Prop of =

22 Warm-up Page 22 Warm-Up week of 10/10 (left side) Tuesday, October 18, :01 PM Warm-up (Add to end of Notes on Deductive Reasoning) Is each conclusion the result of inductive or deductive reasoning? 1) There is a myth that you can balance an egg on its end only on the spring equinox. A person was able to balance an egg on July 8, September 21, and December 19. Therefore, this myth is false. Inductive Reasoning 2) There is a myth that the Great Wall of China is the only man-made object visible from the Moon. The Great Wall is barely visible in photographs taken from 180 miles above Earth. The Moon is about 237,000 miles from Earth. Therefore, the myth cannot be true. Deductive Reasoning Warm-up Solve the following equations and show each step. 1) 5x 18 = 3x + 2 2) 55z 3(9z+12)= 64 2x 18 = 2 2x = 20 x = 10 55z 27z 36 = 64 28z 36 = 64 28z = 28 z = 1 Info Info Info Info Clue Clue Clue Clue Conclusion Conclusion Conclusion Conclusion Determine which triangle best represent "Inductive Reasoning" and which best represents "Deductive Reasoning" and explain why.

23 Algebraic Proof H.W Page 23 Algebraic Proof Homework (left side) Tuesday, October 18, :04 PM

24 H.W. Faulty Logic Page 24 Faulty Logic (left side) Tuesday, October 18, :31 PM

25 Proof Sheet Page 25 Proof Sheet 1 (Right side) Tuesday, October 18, :56 PM

26 Proof Shee 2 Page 26 Proof Sheet 2 Tuesday, October 18, :16 PM

27 Matching Proof Sheet Page 27 Proof Sheet Classwork 10/18 Tuesday, October 18, :18 PM

28 Warm Up Oct 17 Page 28 WEEK 10/17/2011 WARMUP (left side) Tuesday, October 18, :29 PM Warm-up Identify each as a statement or a reason. 10/17 1) Combine Like Terms 2) 12 = 4( 6x 3) 3) m = 11 4) Distrib. Prop. 5) Add. Prop of = 6) m A = m A 7) Subs. Prop of = 8) AB = CD 9) m HAG = m GAK 10) Trans. Prop of = 1) Reason 2) Statement 3) Statement 4) Reason 5) Reason 6) Statement 7) Reason 8) Statement 9) Statement 10)Reason Rearrange the statements and reasons to write a two-column proof. Solve for x, given that 12 = 4( 6x 3). Rearrange the statements and reasons to write a two-column proof. Solve for x, given that 12 = 4( 6x 3). 0 = 24x Distrib. Prop. 12 = 4( 6x 3) Given Subt. Prop. of = 0 = x Div. Prop. of = 12 = 24x ) 0 = 24x 4) 0 = x 1) 12 = 4( 6x 3) 2) 12 = 24x + 12 Subt. Prop. of = Distrib. Prop. Div. Prop. of = Given 10/19

29 DEDUCT. REASONING REFLECTION Page 29 Deductive Reasoning Reflection (left side) Tuesday, October 18, :37 PM Deductive Reasoning Story Project SELF-EVALUATION AND REFELCTION Name: Period: 1. Refer to the Knowledge of Topic portion of your rubric. How do you feel you scored? Why? Explain your reasoning. 2. Refer to the Organization portion of your rubric. How do you feel you scored? Why? Explain your reasoning. 3. Refer to the Presentation portion of your rubric. How do you feel you scored? Why? Explain your reasoning. 4. What went well in this project? What did you learn or what was refreshed or reinforced for you? 5. If you could do it all over again, what would you change? How would you improve your project