Proving that Lines are Parallel
Lesson 2 of 5
Objective: SWBAT use angle pairs to prove that lines are parallel, and construct a line parallel to a given line.
When the students arrive, there is a sentence displayed on the board. I collect their homework and ask that they direct their attention to the sentence: If I leave the house, then I must have my cell phone on me. I ask the students to identify the hypothesis and the conclusion of this conditional, as well as the truth value, and then ask the students to provide a few more examples of conditional statements, for which we again identify the hypotheses and conclusions, and discuss the truth values (MP2).
Then I introduce the concepts of inverse, converse, and contrapositive using the statement If p, then q. I ask that the students create the inverse, converse and the contrapositive of the cell phone statement I originally gave them, and I raise the issue of the truth values of these statements. If the original statement is true, do the other statements have to be true?
In order to help my students understand the possibilities with regard to truth values, I use a Venn diagram, as I demonstrate in this video. Our ultimate conclusion is that the original statement and the contrapositive will always have the same truth value; the inverse and the converse only sometimes share that truth value.
[In New York State, we used to teach an entire unit on logic, including logic proofs. I really miss this unit (my students always seemed to enjoy it!) and think it's important to include at least a little logic in the course. When talking about the isosceles triangle theorems, for example, the use of the word "converse" is really helpful, and serves, I think, to solidify the students' understanding of these theorems.]
Proving Lines are Parallel
I remind the students that we began our study of angles and parallel lines with this postulate: If two parallel lines are cut by a transversal, the corresponding angles are congruent. I ask them to tell me what the converse of this statement might be, and explain that we will postulate the converse as well. From this postulate, we then prove the theorem "If two parallel lines are cut by a transversal, the alternate interior angles are congruent."
I provide the students with the hand out, Proving Lines are Parallel, and work through these proofs with the class. In the second of the proofs, the students can choose which angles pairs they use in their proof, so that some can use alternate interior angles, while others can use corresponding angles. We discuss both options. (MP3)
In this lesson, I have chosen to do two-column proofs. In future lessons, we will also use flow chart proofs and paragraph proofs. I think it's important for students to be exposed to the different approaches to proof, and believe they should be able to choose the format that is most natural for them.
Constructing a Parallel Line
Next I pose this question to the class: Suppose you were in charge of a parking lot, and you needed to create one more parking space by painting one more white line. This line needs to be parallel to the one next to it. How can we use our knowledge of parallel lines to ensure that the two lines will be parallel?
This question leads us into the construction of a line parallel to a given line through a given point. (MP5)
I hand out the sheet on which the students will do the construction. On the back of the sheet (the plain side) I introduce the students to the construction for copying an angle. Then we turn to the front of the sheet and I instruct the students on the parallel line construction.
When we have completed copying the angle and drawing in the parallel line, I ask the students to tell me what type of angle pair we just constructed and the theorem that we used to guarantee that our lines are parallel.
(See here for instructions to this particular construction.)
We now change gears entirely and focus on algebra. With a 90 minute class, I think it's crucial to change activities and focus often. Otherwise the class could get awfully boring!
I announce that we are going to refresh our memories of algebra, and write two equations with two unknowns on the board that can be easily solved using substitution. I ask the students to solve for x and y, and we discuss their answers and their approach. I provide a few more examples for them to try, and then I give the students a similar problem, but one that is more easily solved using the elimination (addition/subtraction) method. We discuss this method, and I give the students several problems to work through.
Now it's time to apply our algebra skills to parallel lines. I hand out the Parallel Line Problems and the students work together in their groups .
These problems require that the students closely examine the structure of the diagrams in order to identify angle pairs created by parallel lines (MP7), establish the relationships of these angles (congruent or supplementary), and use a variety of algebra skills to solve the problems (MP1). By working together and discussing the problems, the students become comfortable using the vocabulary contained in this unit and play an active role in the problem solving (MP3).
I tell the class that whatever they didn't finish on the Parallel Lines Problems is their homework.
Then we review the logic concepts (if-then statements, inverse, converse, and contrapositive) with which we began the class. I distribute the Ticket out the Door, a quote from a popular book in which the author mislables one of these terms. On it, I ask the students to correctly identify the concept being used.
This exit ticket will allow me to assess my students' understanding of the logic concepts, while also allowing them to feel slightly superior to Dan Brown! I also think it's important for my students to see that these concepts are not just something that they will see only in math class - that references to them can actually be found in the real world, too!