Reasoning in Reasoning in GeometryGeometry

Will JaramilloWill Jaramillo

““Logical Reasoning in Geometry” ProjectLogical Reasoning in Geometry” Project Mr. JaramilloMr. Jaramillo Objectives:Objectives: Students will use technology to create a presentation on Geometric Reasoning. Students will discuss the Students will use technology to create a presentation on Geometric Reasoning. Students will discuss the

significance and difference between inductive and deductive reasoning. In addition, students will explore the four significance and difference between inductive and deductive reasoning. In addition, students will explore the four statements (conditional, converse, inverse, and biconditional) to determine the truth value of each statement. Lastly, statements (conditional, converse, inverse, and biconditional) to determine the truth value of each statement. Lastly, students must determine whether the conditional statement meets the conditions of a biconditional statement.students must determine whether the conditional statement meets the conditions of a biconditional statement.

TEKS:TEKS: G.1.A: Develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical G.1.A: Develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical

reasoning, and theorems.reasoning, and theorems. G.2.B: Make conjectures and determine the validity of the conjectures.G.2.B: Make conjectures and determine the validity of the conjectures. G.3.B: Construct and justify statements about geometric figures, statements, and their properties.G.3.B: Construct and justify statements about geometric figures, statements, and their properties. G.3.D: Use inductive reasoning to formulate a conjecture.G.3.D: Use inductive reasoning to formulate a conjecture. G.3.E: Use deductive reasoning to prove a statement.G.3.E: Use deductive reasoning to prove a statement. Criteria:Criteria: * Groups of 2* Groups of 2 * Construct a PowerPoint Presentation* Construct a PowerPoint Presentation * Use voice recorder* Use voice recorder * Use webcam* Use webcam *Animations/Slide designs*Animations/Slide designs Slides Must Include, But Not Limited To:Slides Must Include, But Not Limited To: Title slide with names.Title slide with names. Deductive and Inductive LogicDeductive and Inductive Logic

An example, justifying if it is deductive or inductive.An example, justifying if it is deductive or inductive. Conditional statement with its converse, inverse, and Contrapositive.Conditional statement with its converse, inverse, and Contrapositive. Truth table with descriptions of statement validities.Truth table with descriptions of statement validities. Webcam, at least for the conclusionWebcam, at least for the conclusion Questions to Consider:Questions to Consider: What is the difference between deductive and inductive reasoning?What is the difference between deductive and inductive reasoning? What are the four types of conditional statements, and how do they relate?What are the four types of conditional statements, and how do they relate? When can a conditional statement also be written as a biconditional statement?When can a conditional statement also be written as a biconditional statement? What did you learn from this project in terms of curriculum and technology?What did you learn from this project in terms of curriculum and technology? What did you like most about the project? Least?What did you like most about the project? Least? Grading Outline:Grading Outline: ______ 5 pts. ______ 5 pts. At least 6 slides (MUST USE 6 x 6 RULE)!At least 6 slides (MUST USE 6 x 6 RULE)! ______ 4 pts. ______ 4 pts. Use of voice recorder for at least one minute.Use of voice recorder for at least one minute. ______ 4 pts. ______ 4 pts. Use of webcam to conclude the presentation.Use of webcam to conclude the presentation. ______ 5 pts. ______ 5 pts. Conclusion answers questions 4 and 5.Conclusion answers questions 4 and 5. ______ 5 pts. ______ 5 pts. Creative show design.Creative show design. ______ 8 pts. ______ 8 pts. Deductive and Inductive Reasoning with original example and justifications.Deductive and Inductive Reasoning with original example and justifications. ______ 10 pts. ______ 10 pts. Four conditional statements with truth table and justifications.Four conditional statements with truth table and justifications.

Inductive vs. Deductive Inductive vs. Deductive ReasoningReasoning

Inductive Reasoning:Inductive Reasoning: Patterns of observationPatterns of observation

Deductive Reasoning:Deductive Reasoning: Logic with facts and properties.Logic with facts and properties.

Inductive or Deductive?Inductive or Deductive? There is a myth that bumblebees should There is a myth that bumblebees should

not fly because their weight is more than not fly because their weight is more than their wings can support. However, if you their wings can support. However, if you were to observe bumblebees, you would were to observe bumblebees, you would see that they fly. see that they fly.

Response?Response?(Student’s use record narration, as on this (Student’s use record narration, as on this

slide)slide)-Deductive Reasoning: Based on fact that -Deductive Reasoning: Based on fact that

bumblebees do fly.bumblebees do fly.

Conditional StatementsConditional Statements

Conditional Statement means Conditional Statement means if if p, p, then then qq

Converse Statement: Converse Statement: if if q, q, thenthen p p Inverse Statement: Inverse Statement: if not if not p, p, then not then not qq Contrapositive Statement: Contrapositive Statement: if not if not q, q, then then

notnot p p Biconditional Statement: pBiconditional Statement: p if and only if if and only if

qq Can reverse Can reverse pp and and qq as conditionals as conditionals

Writing StatementsWriting Statements

If it is a tiger, then it has four legs.If it is a tiger, then it has four legs. Converse Statement:Converse Statement:

If it has four legs, then it is a tiger.If it has four legs, then it is a tiger. Inverse Statement:Inverse Statement:

If it is not a tiger, then it does not have If it is not a tiger, then it does not have four legs.four legs.

Contrapositive Statement:Contrapositive Statement: If it does not have four legs, then it is not If it does not have four legs, then it is not

a tiger. a tiger.

Verifying ValidityVerifying Validity

p qp q TrueTrue

q p q p FalseFalse

~~p ~qp ~q FalseFalse

~~q ~pq ~p TrueTrue

Since not all statements are true, this is not a tautology. Also, since all statements are not false, this is not a fallacy.

This cannot be a biconditional statement (Converse Statement is false).

p: It is a tiger

q: It has four legs

Project EvaluationProject Evaluation