For the Do Now, students draw a line segment and construct a line parallel to the segment through a point not on the line. This task reviews a skill the students learned in an earlier lesson. I write the Do Now on the board and ask them to perform the construction. After constructing the line, I have the students measure the angles formed by the line that intersects the parallel lines and the parallel lines. This helps students to see the relationships between the angles, which we review further during the Mini-Lesson.
In today's Mini Lesson, students complete a chart about the relationships between angles formed by parallel lines intersected by a transversal. I plan for this activity to begin as a whole class discussion. I find that asking students questions as we complete the chart helps me to assess their prior knowledge.
I then give the students some time to complete the chart as I circulate. Students write a definition of the angle pairs, draw an example, and describe the relationship. They use the construction from the Do Now and information from a previous lesson to complete the chart. We fill in the first two rows of the chart, supplementary angles and interior angles, as an example and then the students complete the rest of the chart independently.
After about 10 minutes, we go over the charts to make sure the information is filled in accurately. In the activity, students use the information from their chart to write proofs. The concepts studied in this lesson will be referred back to in future lessons (MP6).
For the Proofs Involving Parallel Lines Activity, students are given two proofs and a statement to explain. I do not specify how students must write up their proofs. They can use a two-column proof or a paragraph proof (MP3). Both proofs have multiple methods that can be used. However, for Question 2, I tell students to use the information they know about the relationships of the angles formed when two parallel lines are intersected by a transversal.
For students that may have difficulty understanding the postulate in Question 3, the statement can be broken down into smaller parts. Students can look at the first part of the statement, “If a straight line falling on two straight lines,” and write it as “if two lines intersected by a transversal.” Breaking down the statement into chunks helps students interpret the statement. This is an important skill for comprehension throughout all disciplines.
After about 15 minutes, we go over the activity. I call on different students to present their proofs of Questions 1 and 2 on the document projector. If other students have written their proofs differently, I may call on them to explain how they proved the statements. In Question 3, where students are asked to describe Euclid’s fifth postulate in their own words, I call on a few students to read their descriptions and we discuss the validity of their responses.
To summarize the lesson, I ask the students to think about and explain how Euclid’s Fifth Postulate relates to the lesson. The Postulate says:
If two lines intersected by a transversal form interior angles on the same side of the transversal that are not supplementary, then the lines will eventually intersect each other.
I ask students to write the inverse of the Postulate. This connects back to the previous lessons on postulates. The inverse is:
If two lines intersected by a transversal form interior angles on the same side of the transversal that are supplementary, then the two lines will not intersect and are therefore parallel.
Although this is a complex concept for my students, I feel it is important to introduce them to it. The postulate stretches the students' thinking in a way that is relevant to the content learned in the course.