Through the Looking Glass or Why is a Raven Like a Writing-Desk? One of Lewis Carroll's unsolved riddles was "Why is a Raven Like a Writing Desk? We will never know his answer but we can make our own conjectures based on our collected knowledge. We will compare this inductive reasoning to the deductive reasoning that Euclidean Geometry is built upon. A proof is a convincing argument that a statement is true. This unit is about reasoning and proof. We will discuss what makes a good definition and use our ideas to create definitions for funny shapes as well as geometric terms. We will read about Lewis Carroll's works and how they are relevant to the field of Logic. We will analyze patterns, solve logic puzzles and make conjectures. The difference between deductive and inductive reasoning will be explored and formal proofs will be introduced. Topics: Proofs and Logical Arguments Resources: Geometry, Holt, Rinehart, Winston, 2001 Web Links: What is a Proof?- a little help from Lewis Carroll Standards MA Frameworks : Write simple proofs of theorems in geometric situations, such as theorems about congruent and similar figures, parallel or perpendicular lines. Distinguish between postulates and theorems. Use inductive and deductive reasoning, as well as proof by contradiction. Given a conditional statement write it's inverse, converse, and contrapositive. GG1 Enduring Understandings Students will understand that they can use logical arguments to make a definite point. Students will understand that a proof shows that a statement must hold true for every case. Students will understand that a single counterexample will disprove a statement. Students will understand that if-then statements are one-way statements unless it is a definition. Students will understand that geometry is logically built upon a few basic axioms and undefined terms. Students will understand that in geometry every new statement follows from a previously proved or accepted statement or definition. Essential Questions: What is proof? Why is it important to prove things? How do you construct a proof? What is a logical argument? What makes a definition good? Could we build a geometry using fewer [ "ttp://www.sparknotes.com/math/geometry3/axiomsandpostulates/section1.html ]axioms than Euclid did? When would you use inductive reasoning and when would you use deductive reasoning? Is one better than the other? Which one is Euclidean geometry built upon? Knowledge and Skills Students will be able to write good definitions. Students will write conditional statements and model them with an Euler diagram. Students will form the converse of a conditional. Students will know that if the converse of a conditional is true it is a definition. Students will use conditionals in logic chains and to form logical arguments. Students will use inductive reasoning to form conjectures. Students will link the steps of a proof using algebraic properties and the overlapping segment and angle theorems. Students will write two column and paragraph proofs. Day 15 An Introduction to Proof-  Step 1,   2.1 Oct. 23/24 Daily Objective: Use inductive reasoning to make generalizations from patterns. Homework: page 85/ 33-37 Day 16 Conditional Statements, 2.2 Oct 25/26 Daily Objectives: Write conditional statements and their converes.                        Understand the significance of  a counter example. Introduction to conditional statements and Euler Diagrams . Practice making converse statements and discussion on why converse maybe false. Creation of logic chains. Make logic chains by passing around a sheet of paper where each student writes a conditional statement linking to the previous one. ( Deductive Reasoning) Notes on If -then Transitive Property. Homework: page 95,  9-14, 17-25,  33-35 Day 17 Definitions 2.3, Oct 27/30 Daily Objectives: Use distinguishing characteristics to write a good definition.                       Write biconditional statements. Transparency 16,  Defining a “Flopper” Write a biconditional statement by combining the conditional statement and its converse. Activity 1 page 100 Capturing the "Essence" of a Thing. Transparency 17 -Activity2 page 101 Write a definition for adjacent angles based on observation and pattern. (Inductive reasoning) Practice 2.3 Performance Task - "What is a Dog?" due Nov 3 Homework: page 102, 8-28 even Day 18 Step 2 Drawing a Diagram Step 2 and Step 3  Oct 31/Nov 1 Daily Objectives: Draw a diagram from written instructions.                       State the given information and the information to be proven.                       Use the equivalent properties to prove an algebraic solution. Regions in a Circle- If you connect points on a circle what is the maximum number of regions you would create. Create a table and look for a pattern. Write a conjecture based on the pattern. Algebraic Properties of Equality p108 Introduction to theorems and two column proofs  with algebra. Equivalent Properties p110 Midchapter quiz Homework: page 112, 9-12 Day 19 Conjectures that Lead to Theorems 2.5 Nov 2/3 Lines and Angles Activity The Vertical Angles Conjecture (Inductive Reasoning) 5 Steps of a Proof 5 Step 2 column proof of the Vertical Angles Theorem (Deductive Reasoning) Homework: Collect 3 Ads for project and write conditional statements. Day 20 Conjectures that Lead to Theorems 2.5 Nov. 6/7 Daily objective: Prove overlapping segments theorem and overlapping angles theorem                    Solve angle problems with algebra. Practice solving vertical angle and linear pair problems involving algebra. Discussion of difference between conjectures and theorems p120, -communication questions Practice 2.5 & Writing Activities Homework: page 122, 10-20, 39 &40 Review:   Inductive Reasoning & Deductive Reasoning Day 21 What is the truth value of a biconditional statement?  Nov. 8/9 Creation of Conditionals in Advertising Poster Homework: Finish poster due by next class,  study for Test on Chapter 2. Day 22  Assessment Chapter 2   Nov. 13/14 Test Chapter 2 & Regions in a Plane Homework:  Additional Exploration -Logic Chart