BIG IDEA: Reasoning and Proof ESSENTIAL UNDERSTANDINGS: Logical reasoning from one step to another...
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Transcript of BIG IDEA: Reasoning and Proof ESSENTIAL UNDERSTANDINGS: Logical reasoning from one step to another...
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Reasoning with Properties from
Algebra
BIG IDEA: Reasoning and ProofESSENTIAL UNDERSTANDINGS:
• Logical reasoning from one step to another is essential in building a proof.
• Reasons in a proof include given information, definitions, properties, postulates, and previously proven theorems.
MATHEMATICAL PRACTICE:Construct viable arguments and critique the reasoning of others
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Follow the steps of the brainteaser using your
age. Then try it using a family member’s age. What do you notice? Explain how the brainteaser works.
Write down your age. Multiply it by 10. Add 8 to the product. Double that answer and then subtract 16. Finally, divide the result by 2.
GETTING READY
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Let be real numbers. Addition Property: Subtraction Property: Multiplication Property: Division Property: Reflexive Property: Symmetric Property: Transitive Property: Substitution Property:
Distributive Property:
Algebraic Properties of Equality
, and a b c
If , then _____+_____=_____+_____a bIf , then _____ _____ _____ _____a b
If , then _____ _____=_____ _____a b If and 0, then _____ _____=_____ _____a b c
For any real number , _____=_____aIf , then _____=_____a b
If and _____=_____, then _____=_____a bIf , then can be ____________ for
in any equation or expression
a b a b
_____ _____a b c
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a)
EX 1: What is the value of x? Justify each step.
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b)
EX 1: What is the value of x? Justify each step.
4 2 3 5 11x x
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The properties of equality reviewed in the
beginning of this lesson add to the students’ “toolbox” for writing proofs using deductive reasoning. It is important to make a connection between the algebraic proofs that you unknowingly completed throughout all of Algebra and the proofs that you will complete in Geometry. You should understand that the process of solving an equation is an algebraic proof, although the justifications for each step are not generally written down. Therefore, proofs are not unique to Geometry.
Math Background
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Reflexive Property:
Symmetric Property:
Transitive Property:
Properties of equality are true for any numbers, while congruence properties are true for geometric figures
Properties of Congruence
AB AB
A A
If , then
If , then
AB CD CD AB
A B B A
If and , then
If and , then
If and , then
AB CD CD EF AB EF
A B B C A C
B A B C A C
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a)
Example 2
Verify that AC BD AB CD
Statements Reasons
1. 1. Given
2. 2.
3. 3.
4. 4. Segment Addition Postulate
5. 5.
6. 6.
BC BC
AC AB BC
AB BC BC CD
AB CD
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b) A baseball diamond is shown below. The
pitcher’s mound is at . Use the information to find the .
Example 2
34m
1 2 3 180
1 2 93
3 4 180
m m m
m m
m m
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Proof: a convincing __________________ that uses
____________________ reasoning. A proof __________________ shows why a conjecture is __________
Two-column proof: lists each __________________ on the __________and the justification or ____________ for each statement is on the __________. Each statement must follow ______________________ from the __________ before it.
Proofs
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Given: Prove:
Writing a Two-Column Proof
Statements Reasons
1. 1. Given
2. 2.
3. 3. Addition Property of Equality
4. 4.
5. 5.
2 2m m
1 3m m
m AEC m DEB
m AEC m DEB
1 2
3 2
m m m AEC
m m m DEB
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Given: Prove:
Writing a Two-Column Proof
5 1 21x
4x
Statements Reasons
1. 1. Given
2. 2.
3. 3.
5 20x
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2.4 p. 9910 – 14 all, 16 – 28 evens, 29 – 32 all, 36 – 45x3
20 questions