Where We've Been: We've just finished a cycle of reasoning about vertical angles and linear pairs. Students used inductive reasoning to make conjectures about vertical angles and linear pairs. Next we worked with the linear pair postulate, and then used that postulate to prove the vertical angles theorem.
Where We're Going: We are now beginning an analogous cycle of reasoning about parallel lines cut by a transversal. We'll start with inductive reasoning: making conjectures. Then we'll move on to writing rigorous two-column proofs about parallel lines cut by a transversal, theorems that rely on the vertical angle theorem and linear pair postulate. I'm trying to give students the feeling that we are gradually building an axiomatic system.
So in this section, I'm making sure that students still know the vertical angles theorem and the linear pair postulate. I also make sure that students know the meaning of congruent angles and supplementary angles.
To achieve these goals, I do some basic check for understanding using whiteboard strategies.
I use the following prompts:
1. Draw a diagram of angle 1 vertical to angle 2
2. What does the vertical angle theorem say about angle 1 and angle 2?
3. Draw a diagram of angle 1 and angle 2 forming a linear pair.
4. What does the linear pair postulate say about angle 1 and angle 2?
5. Measure of Angle A is 42 degrees; Angle B is supplementary to angle A. What is the measure of angle B?
6. Measure of Angle A is 42 degrees; Angle C is congruent to angle A. What is the measure of angle C?
I love the vocabulary in this lesson because it actually makes sense. It's fun to see the students understanding the words, as opposed to merely memorizing them. I start by defining a transversal as a line that intersects two or more lines. Next, I draw two non-parallel lines cut by a transversal. I number the angles 1 through 8 in random order. This prevents students from using number patterns to classify angle pairs. Warning, if you like to read, please continue, but if you'd rather get a sense of how I teach this section check out the [educreation link].
First, to make sure students are seeing things clearly, I ask them to consider which of the angles in the diagram form vertical angles. Same for linear pairs. Since we've hit this already in the previous section, this is a little less formal. I call out prompts and let students respond as they wish. In any case, the students should be rock solid on vertical angles and linear pairs at this point.
Once we've exhausted the vertical angles and linear pairs angle, I move on to giving students notes for today's lesson.
Next I give the students their bearings, so to speak. I ask them what the word exterior means. Exterior in this context, I tell them, means outside of the two lines that are being cut by the transversal. Then as a class, we identify all four exterior angles. Then, similarly, we talk about the general meaning of interior and the meaning in this context. Next we talk about the word alternate as implying some type of "otherness".
I define corresponding to mean "in the same relative position". As we know, there are four pairs of corresponding angles formed on a transversal when it cuts two lines. I start out writing on the board (for example), Corresponding Angles: Angle 1 and Angle 7 (as I explain why). Then I give another pair of corresponding angles with an explanation. Next, i ask for a volunteer to identify another pair. Finally, I have every student decide on the fourth pair and write it in their notes (this ensures that students are thinking and not just copying notes mindlessly).
Next, I write on the board, Alternate Interior Angles. As the name implies, I tell the students, these are interior angles that are on opposite sides of the transversal (and on different lines). This parenthetical note is presented more as a matter of practicality. I point out that the "alternate interior" angles that are on the same line, already have a name (linear pair) and we don't want to confuse things. So I write Alternate Interior Angles and give students the first of the two pairs. Then they must come up with the second pair on their own.
We do a similar treatment of alternate exterior, same-side interior, and same-side exterior angles.
Once the students have all of the notes, it's now time to check for (and develop) understanding. Using whiteboard strategies, I give them prompts to be sure that they can identify all types of angle pairs. Students should walk away from this section of the lesson feeling confident that they know all the different types of angle pairs formed when lines are cut by a transversal.
The discovery exercise continues the inductive reasoning process that we've been conducting in various contexts. This time students start by creating parallel lines cut by a transversal on the coordinate plane and making observations as they find the angle measures. For this first part of the exercise, we use the Parallel Lines Cut By Transversal on Coordinate Plane resource.
This next activity can be done after the first activity is completed, or possibly in lieu of the first activity. In any case, we head over to the computer lab to do some investigation using dynamic geometry software (Geometer's Sketchpad in my case). I am careful to hand out the directions for the activity (Inductive Reasoning about Parallel Lines Cut by a Transversal) before leaving my classroom. It's important for the students to understand their mission before we leave the classroom so that when we get to the lab, we hit the ground running.
Once in the lab, students will need to create parallel lines cut by a transversal and measure all eight angles that are formed. Beforehand, I've instructed students to call me over for an inspection when they have reached this critical milestone. I realistically do not have time to help 33 students at once, so as students emerge as frontrunners for this activity, I activate them as resources for other students.
As I walk around helping students, one thing I focus on is making their process as systematic as possible. For example, I might walk over to a student and ask, "Which type of angle pair are you currently investigating?" [Let's say they are looking at alternate interior angles] "Ok then, show me a pair of alternate interior angles...Good....What are their measures?" [Typically students have the angle measures for all eight angles clustered together in the top left corner of the screen where Sketchpad dumps them by default.] "I want to make your process more systematic. Let's drag the angle measures into their correct positions on the diagram so that you can see them more clearly...Now, how can you come up with a conjecture for the relationship between these angle? Are there any other angle pairs you need to consider before making your conjecture?"
Another thing that happens, typically, is students over-generalize when writing their conjectures. For example, I see things like "Alternate Interior are always the same" First, I'd have to urge the student to use academic vocabulary. I'd also ask them if there are any conditions that need to be satisfied in order for alternate interior angles to be congruent. I'd instruct them to do some experimenting to see what happens if the lines are not parallel and see if that helps them to refine their conjecture.
To close the lesson, I model how to use SketchPad to create the diagram. Then I go through each angle pair type and demonstrate how I go about looking for a relationship and making a conjecture. By the end of this section, each student should have notes that include conjectures about the angle relationships when parallel lines are cut by a transversal.