Intro to Proofs
Page 2: Definitions, Postulates, and Theorems
Definitions: Meanings of words used in proofs.
Postulates: Basic ideas accepted as truth without proof.
Theorems: Ideas that have been proven true multiple times.
Page 4: Conjecture and Proof
Conjecture: Propositions that are believed to be true but need verification.
Proof Context: To verify conjectures, one must provide a logical argument.
Page 5: Flowchart of a Proof
A flowchart represents a systematic approach to constructing a proof with:
Conclusions, postulates, definitions, or theorems involved at each step.
Reminder: Counterexamples can disprove statements at any point.
Page 6: Paragraph Proofs
Historical Context: Euclid’s proofs were initially written in paragraph form.
Purpose: To make a precise argument regarding a statement.
Page 7: Structuring 2-Column Proofs
Organization:
Statements: Listed in numbered steps.
Reasons: Use definitions, postulates, and theorems to validate your statements.
Page 8: Importance of Structured Proofs
Reasons for Using 2-Column Proofs:
Enhances academic performance.
An organized method for demonstrating validity or falsehood in claims.
Page 9: Example of a 2-Column Proof
Given: 2x + 5 = 11
Prove: x = 3
Steps:
2x + 5 = 11
2x = 6
x = 3
Reasons:
Given
Subtraction Property
Division Property
Page 10: Conclusions in Proofs
Conclusion: The last statement concludes that the conjecture is true if all prior statements and reasons hold true.
Page 13: UNO Postulates
Postulates/Rules in Playing UNO:
Similar to the proper structuring of proofs, where certain conditions must be adhered to.
Page 15: Subsequent Steps in Proofs
Requirement: Similar conditions to be followed in card play to develop further proof statements.
Page 16: Continued Card Play Dynamics
Ongoing theme of using card game rules as analogies for proof structuring.
Page 17: Preparing to Write Algebra Proofs
Transition: Contextual shift to algebraic proofs and structures.
Page 18: Algebra Proof Method
Example: Solve - Given: 2x + 5 = 11, prove x = 3.
2x + 5 = 11
2x = 6
x = 3
Reasoning includes properties of equality involved.
Page 19: Writing 2-Column Proofs
Emphasizes the necessity to construct true statements and corresponding reasons effectively.
Page 20: Starting Statements
Initialization: The first statement is always the 'Given'.
Page 21: Developing Your Proof
Steps in proof writing must follow a logical flow using appropriate reasoning.
Page 22: Concluding Steps
Final Statement: Always end with the prove statement and its corresponding reason.
Page 23: Commonly Used Properties
Common Proof Reasons: Lists properties crucial for proof development.
Page 24-25: Explanation of Properties
Addition, Subtraction, Multiplication and Division Properties: Explained in terms of their usage in proofs.
Page 26-29: Example Proofs
Example proofs illustrating how to structure statements and reasons effectively.
Page 30-36: More Algebraic Proof Examples
Multiple examples to develop understanding of various algebraic proofs.
Page 37: Introduction to Geometry Proofs
Transition into geometry and proof structures, particularly focusing on transversals.
Page 38-44: Geometry Proof Examples
Detailed geometry proofs demonstrating corresponding angles, alternate interior angles, and their logical structuring using established theorems and postulates.