How to do a Proof
Using Uno!
What does it mean to
prove something?
PROOF (pruf)
–noun
1.
2.
3.
4.
5.
6.
7.
8.
evidence sufficient to establish a thing as true, or to produce belief in its truth.
anything serving as such evidence: What proof do you have?
the act of testing or making trial of anything; test; trial: to put a thing to the proof.
the establishment of the truth of anything; demonstration.
Law. (in judicial proceedings) evidence having probative weight.
the effect of evidence in convincing the mind.
an arithmetical operation serving to check the correctness of a calculation.
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.
Why do a Proof?
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We will be able to show that ideas
in Geometry will always be true in
any situation.
We can win an argument!
Inductive vs. Deductive
Inductive Reasoning
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–
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.
Deductive Reasoning:
Using Syllogisms
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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.
Finish this Syllogism:
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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.
Finish this Syllogism:
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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.
Finish this Syllogism:
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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.
Finish this Syllogism:
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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.
Building a 2-Column Proof
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We use deductive reasoning to do
proofs.
Ideas must be laid out step by
step using postulates or proven
theorems to build a syllogism.
Postulates and Theorems
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•
Postulates are big ideas that are
accepted as universal truths
without proof.
Theorems are ideas that can be
proven using deductive logic
(through syllogisms).
2-Column Proof Format:
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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.
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.
3 postulates of Uno!
1.
You can play a card of the same color.
2. You can play a card of the same number.
3. You can play a WILD card at any time in
order to change the color.
Sample Proof
• Begin with
• List how to play these cards
• To get to
Given
Same Color
Change Color
You don’t have to use every
postulate you know in every proof.
Same Color
Same Color
Same
Color
Formal 2-Column Proof
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
Given:
Prove:
Using:
Formal 2-Column Proof
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
Given:
Using:
Prove:
Formal 2-Column Proof
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
Given:
Using:
Prove:
Formal 2-Column Proof
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
Given:
Using:
Prove:
Formal 2-Column Proof
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
Given:
Using:
Prove:
Formal 2-Column Proof
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