How to do a Proof

1. How to do a Proof Using Uno!

2. What does it mean to prove something? PROOF (pruf) –noun 1. evidence sufficient to establish a thing as true, or to produce belief in its truth. 2. anything serving as such evidence: What proof do you have? 3. the act of testing or making trial of anything; test; trial: to put a thing to the proof. 4. the establishment of the truth of anything; demonstration. 5. Law. (in judicial proceedings) evidence having probative weight. 6. the effect of evidence in convincing the mind. 7. an arithmetical operation serving to check the correctness of a calculation. 8. Mathematics, Logic. a sequence of steps, statements, or demonstrations that leads to a valid conclusion. –adjective 9. able to withstand; successful in not being overcome: proof against temptation. 10. impenetrable, impervious, or invulnerable: proof against outside temperature changes. 11. used for testing or proving; serving as proof. 12. of tested or proven strength or quality: proof armor. –verb (used with object) 13. to test; examine for flaws, errors, etc.; check against a standard or standards.

3. Why do a Proof? • We will be able to show that ideas in Geometry will always be true in any situation. • We can win an argument!

4. Inductive vs. Deductive Inductive Reasoning • Reasoning from detailed facts to general principles. • Any form of reasoning in which the conclusion, though supported by the premises, does not follow from them necessarily. Deductive Reasoning • Reasoning from the general to the particular. • A process of reasoning in which a conclusion follows necessarily from the premises presented, so that the conclusion cannot be false if the premises are true.

5. Deductive Reasoning: Using Syllogisms • A syllogism is like the Transitive Property in Algebra: If a = b, and b = c, then a = c. • If you are accepted to Harvard Medial School, then you will become a doctor. If you are a doctor, then you will be rich. If you go to Harvard Medical School, then you will be rich. • Angle A is 70 degrees. If an angle has a measure less than 90, then it is acute. Angle A is acute. • If JoAnna trick-or-treats, she will get lots of candy. If she get lots of candy, she will eat it. If she eats it, she will get cavities. If JoAnna trick-or-treats, she will get cavities.

6. Finish this Syllogism: • If you live in Manhattan, then you live in New York. • If you live in New York, you live in the United States. • If you live in Manhattan, then you live in the United States.

7. Finish this Syllogism: • If Henry studies his Algebra, then he will pass his test. • If Henry passes his test, then he will get good grades. • If Henry studies his Algebra, then he will get good grades.

8. Finish this Syllogism: • If a number is a whole number, then it is an integer. • If a number is an integer, then it is a rational number. • If a number is a whole number, then it is a rational number.

9. Finish this Syllogism: • If I drive over glass, then I will get a flat tire. • If I get a flat tire, then I have to change it. • If I drive over glass, then I have to change a tire.

10. Building a 2-Column Proof • We use deductive reasoning to do proofs. • Ideas must be laid out step by step using postulates or proven theorems to build a syllogism.

11. Postulates and Theorems • Postulates are big ideas that are accepted as universal truths without proof. • Theorems are ideas that can be proven using deductive logic (through syllogisms).

12. 2-Column Proof Format: • Write the information Given, and what you are trying to Prove. • Draw a T. • The first column is for statements—things that MUST be true. • The second column is for reasons—WHY you know it is true.

13. Writing an Uno Proof • The rules of Uno are our postulates. • Use the first card as the Given. • Use syllogistic logic to list the order in which you would have to play the other cards to finally be able to play the Prove card. • Justify your logic in 2-column format.

14. 3 postulates of Uno! • You can play a card of the same color. • You can play a card of the same number. • You can play a WILD card at any time in order to change the color.

15. Sample Proof • Begin with • List how to play these cards • To get to

16. You don’t have to use every postulate you know in every proof. Given Same Color Change Color Same Color Same Color Same Color

17. Given: Blue 6 Prove: Yellow Reverse Statements(What Card to Play):Reasons(I can play this card because): --------------------------------------------------------------------- 1. Blue 6 ▐ 1. Given 2. Blue Skip ▐ 2. Same Color 3. Wild Draw 4 ▐ 3. Change Color 4. Yellow 5 ▐ 4. Same Color 5. Yellow 1 ▐ 5. Same Color 6. Yellow Reverse ▐ 6. Same Color Formal 2-Column Proof

18. Given: Prove: Using:

19. Given: Blue 5 Prove: Green 6 Statements(What Comes Next):Reasons(I can play this card because): ---------------------------------------------------------- 1. Blue 5 ▐ 1. Given 2. Blue 1 ▐ 2. Same Color 3. Green 1 ▐ 3. Same Number 4. Green 6 ▐ 4. Same Color Formal 2-Column Proof

20. Given: Prove: Using:

21. Given: Blue Seven Prove: Red Nine Statements(What Comes Next):Reasons(I can play this card because ): --------------------------------------------------------------------- 1. Blue 7 ▐ 1. Given 2. Green 7 ▐ 2. Same Number 3. Green 4 ▐ 3. Same Color 4. Yellow 4 ▐ 4. Same Number 5. Yellow 9 ▐ 5. Same Color 6. Red 9 ▐ 6. Same Number Formal 2-Column Proof

22. Given: Prove: Using:

23. Given: Red Reverse Prove: Green Nine Statements(What Card to Play):Reasons(Why I can play the card): --------------------------------------------------------------------- 1. Red Reverse ▐ 1. Given 2. Red 3 ▐ 2. Same Color 3. Blue 3 ▐ 3. Same Number 4. Wild ▐ 4. Change Color 5. Green 5 ▐ 5. Same Color 6. Green 9 ▐ 6. Same Color Formal 2-Column Proof

24. Given: Prove: Using:

25. Given: Yellow Skip Prove: Blue 3 Statements(What Card to Play):Reasons(I can play this card because): --------------------------------------------------------------------- 1. Yellow Skip ▐ 1. Given 2. Yellow 8 ▐ 2. Same Color 3. Red 8 ▐ 3. Same Number 4. Green 8 ▐ 4. Same Number 5. Blue 8 ▐ 5. Same Number 6. Blue 3 ▐ 6. Same Color Formal 2-Column Proof

26. Given: Prove: Using:

27. Given: Yellow 8 Prove: Blue 1 Statements: Reasons: --------------------------------------------------------------------- 1. Yellow 8 ▐ 1. Given 2. Yellow Skip ▐ 2. Same Color 3. Green Skip ▐ 3. Same Number 4. Green Draw 2 ▐ 4. Same Color 5. Red Draw 2 ▐ 5. Same Number 6. Red 5 ▐ 6. Same Color 7. Blue 5 ▐ 7. Same Number 8. Blue 1 ▐ 8. Same Color Formal 2-Column Proof