What are Geometric Proofs?
Lesson 4 of 10
Objective: SWBAT explain the term "geometric proof" and define terms associated with writing proofs.
With the implementation of the Common Core standards, students are expected to make sense of problems and apply concepts, which often require students to extract contextual information from verbal problems. I often use passages from Lewis Carroll's books in my lesson to demonstrate applications of mathematical concepts, while helping students improve reading comprehension. Today's lesson begins with a passage from Alice's Adventures in Wonderland.
As my students walk in the room, I hand them a paper with a passage from Alice’s Adventures in Wonderland, in which Alice has a conversation with the Cheshire Cat about madness. I ask students read the passage and use information to prove two statements. They work independently. I want them to write down their interpretation of what is proven true in the statements, to identify the evidence that is presented. I walk around and see what they write, but I keep quiet about whether they have the correct evidence or not, until we go over the Do Now during the Mini-Lesson.
To begin today's Mini-Lesson, we review the do now. We discuss the two statements, “Alice is mad,” and “The Chesire Cat is mad.” In order to prove the two statements, evidence is needed. This evidence can be found directly in the passage. The first statement, “Alice is mad,” can be proved using the Chesire Cat’s quote, “You must be . . .or your wouldn’t have come here.” The second statement, “The Chesire Cat is mad,” can be proved by information based on the Cat’s discussion about dogs and cats.
We then move into a discussion about geometric proofs. I give the students a minute to turn and talk about the following questions:
- What are geometric proofs?
- Why do we need proofs in mathematics?
- What information is needed to write geometric proofs?
After the students have had a minute or two to talk with their partner, I call on a few students to explain their answers to the class.
We then discuss the differences between theorems and postulates using the remaining slides in Intro to Proofs Mini Lesson Presentation. We also look at an example of writing a geometric definition as a bi-conditional statement. This presentation helps my students to appreciate how logical reasoning is used in geometric proof.
In the first part of the activity, students rewrite definitions of geometric terms as conditional statements, converses, and bi-conditional statements. My students have seen these definitions before. Since they are often used in geometric proofs, I want them to take some time to unpack them. After rewriting the definitions in different forms, I find that my students retain the meaning better and can see how definitions can be used to help prove statements in geometry.
Some of my students have difficulty rewording the statements as conditional statements in a logical order. To support their work, I ask, "What they are trying to say?" I help them progress from expressing the definition in their own words towards a more formal mathematical statement. I find that once the conditional is written correctly, my students can successfully write the converse and bi-conditional correctly.
Part B of the worksheet presents geometric statements and examples using the terms defined in Part A. Using the definitions, students write a conclusion based on the given statement. For example, in Question 7, students are given the statement, “Segment KM bisects angle JKL” written in mathematical notation. One conclusion that can be made is “Angle LKM is congruent to angle JKM.” Some of the statements can have more than one conclusion, so I am looking for students to write at least one logically valid conclusion.
After about 20 minutes, we will go over the activity. I call on students to share their conditional statements and biconditionals. We also go over the students’ conclusions in part B. For each question, I call on more than one student, so that we cover multiple statements and students can practice listening and judging the validity of statements.
In order to ensure participation of all students, I call on a student to give his or her answer. If that student doesn't have an answer, I call on another student and then go back to the first student to comment on the second student's answer. This strategy helps my reluctant students gain confidence and experience.
In this unit, students will be using postulates to prove theorems. Understanding the difference between theorems and postulates helps students understand how to write proofs. Today's Exit Ticket helps me to gauge student comprehension. If needed, I can review the vocabulary in the next lesson.
Exit Ticket: Explain the difference between theorems and postulates.