Angles and Parallel Lines (Day 1 of 2)
Lesson 11 of 16
Objective: SWBAT understand angle measure relationships associated with two lines cut by a transversal.
To begin the lesson I choose pairs of students at random and hand each pair a copy of the The I Like Billy Parallel Line task. I also project the image on the board for all to see. Before allowing students to begin the task, I have a volunteer read the text below the image out loud, and when everyone understands what it is they are to do, I tell the students that they have a few minutes to discuss their ideas with their partner and write their response in the space below the image. Students usually ask what tools they are allowed to use to accomplish their goal, I tell the class that they can use any geometry tool, but must state so in their written response. We won't actually perform the tasks yet.
As students write responses I walk around listening in and asking that they not "leak" the answers to nearby groups. I want to make sure that everyone has time to figure out what to do. Some students may say that they would simply paint a white line parallel to the other lines that are still visible. I ask these students how they can be 100 percent certain that the lines are parallel. I make sure that my students understand that there is more work to be done.
The answer I'm looking for of course is that in order to draw a line parallel to the others in the parking area, they would use angle measurements formed by the intersecting lines and then paint the hidden line using these measurements. Students usually come up with this after a few minutes of thought and discussion, and no direct lead on my part.
Afterwards, I call on student volunteers to share there ideas. I ask that they explain themselves and go up to the projected image if necessary. After listening to a few students, I open a Geometer's Sketchpad document and show students the relationships formed when parallel lines are crossed by a transversal.
Video Source: http://screencast.com/t/DQ8MQrkOq (Accessed May 14, 2014)
Before flowing into the New Info section, it should be clear to students that in order to paint the hidden parking line perfectly parallel to the others, the indicated (corresponding) angles must be congruent (or, the same side interior angles must be supplementary). In this section, the students learn the names of these angle pairs.
I begin this section of the lesson by projecting the New Info image on the smart-board and ask the class the following questions:
- How many angles are formed by the two lines and the transversal?
- Are the two lines, cut by the transversal parallel?
- How many intersections do you see?
Teacher's Note: When discussing these questions it's important to use a pointer to indicate angles and lines as the questions are being answered, so all students know exactly what is being addressed.
I proceed to state that pairs of angles, one angle from each intersection, have different names depending on their positions in the figure. Many of the angles are named in relation to the transversal, line t. Some of these angle pairs are quite obvious. I then ask students to observe the diagram and raise their hands to volunteer answers to the following questions.
- Who can give me the numbers of one pair of exterior angles on the same side of the transversal, one from each intersection
- Who can give me a pair of interior angles on the same side of the transversal, again, one from each intersection?
I state that the word "alternate" means on different sides of the transversal. Then, I ask two more questions:
- Who can identify a pair of Alternate exterior angles?
- Who can name a pair of Alternate interior angles?
If students struggle to come up with the angles, I may specify a pair, so they can come up with another pair. This always works.
Finally, I tell the class that other pairs of angles are named by their relation to each other. The word "corresponding" means that the angles are in the same relative position in the intersections and there are 4 pairs. I provide the first pair, 1 and 5, and ask the students to provide the other three pairs of corresponding angles. This approach usually works out fine, but like I mention in the reflection (see Developing a Conceptual Understanding: A note on vocabulary) students learn these after repeated exposure and providing the meanings of the terms, when asked.
Before finishing this presentation, I make sure that I make the written specifications of the angle pairs on the projected image so that the class can reference it during the next activity. In the HW assignment, students will organize angle pairs and reinforce the naming of these.
Our work today is laying the foundation for students to master these measurement questions in Day 2 of our work with Angles and Parallel Lines.
With the annotated projected image displayed on the board, I hand each student a copy of the Map of the Village in NYC. I ask my students to answer the questions and label the diagram as specified below the map. I allow students to work in pairs, but I provide each student with their own handout. Students will practice the names of angle pairs in a way that reinforces the fact that angles do not have to be formed by parallel lines to be corresponding, alternate interior, same side interior, and so forth.
When students get to Question 8, I expect them to make the connection to the I Like Billy task from the Launch. Students should answer that the intersection of West St. and 20th street (far west side) also makes a 53 degree angle because 18th and 20th streets are parallel.
To conclude Day 1 of this lesson, I ask student pairs to come up with their handouts and project their map on the board using the document camera. I encourage students to tell the class their answers and to carefully indicate where their angle pairs are positioned in the diagram.
If mistakes are made, I ask the class to specify the mistake and suggest a correction. I call up as many groups as time permits, but I make sure that before the period ends, I question the class to make sure they understand that the names of angle pairs have nothing to do with the lines being parallel or not. And, I tell them that when the lines are parallel, the corresponding angles are congruent. The other angle measure relationships will be seen in tomorrow's lesson.
In this lesson's homework assignment, HW Angles and Parallel Lines Day 1, the first 5 questions are true or false response items. When I distribute the task, I emphasize to the students that they must provide an explanation and the correct answer to any false statements.
In questions 4 and 5, I included a triangle in the image because many students have trouble seeing certain special angle pairs when they are "hidden" in triangle images. I make sure I go over these in class together.
I added a table so that students can reinforce their understanding of the angle pairs, by explaining them in their own words and drawing a sketch. I encourage students not to simply copy an image covered in class, but to imagine a figure that contains the angle pair.