Parallel Lines and Transversals and Alternate Interior Angles: Oh My!

On the teaching docket this week in my freshman Integrated Algebra and Geometry class: angle relationships formed by parallel lines and transversals!

   
 (Side note: I can mark how accepting my particular community has become of students whose gender identity is fluid or different from that they were assigned at birth entirely through my teaching of the word transversal. Five years ago I had a student who giggled every time I said it, not in a nice way. Thursday when I announced the vocabulary term I had one student yell triumphantly, "that's me!")

Thursday during class the students made predictions about which sets of lines were parallel based on angles they knew, and then we completed guided notes together on what relationships exist between corresponding, alternate interior, alternate exterior, and consecutive interior angles. The students got a bit of practice in with the concepts during the second half of the class, but as they left I realized we needed a bit more time with using these angle relationships to find unknown angles and to identify lines as parallel correctly. Enter Friday, puzzle day!

First, another note: Yesterday morning my entire student body woke up to a dismal snowy forecast and the news that while roughly 400 school districts in Massachusetts had closed school, ours had not. I was truly shocked when 2 of my 3 students for Math 1 arrived at school on time and soaked to the bone, ready to learn despite the worsening road conditions. So, when they whined that it was gross out and could we do something fun, I was all the more inclined to add some goofy elements to my planned lesson. I found a couple of parallel puzzles, drew my parallel line sets on the board, and dug out the Post-Its.

Activity 1: Each student chose a set of parallel lines I had drawn on my whiteboard and was given two small post-it notes. I rapidly called out angle relationships and students had to put the post-its on a set of angles that fulfilled this relationship. The ones I called out were:

• A pair of alternate exterior angles
• A pair of congruent angles
• A pair of corresponding angles
• A different pair of corresponding angles
• A pair of supplementary angles
• A pair of vertical angles 
• A pair of alternate interior angles
• A pair of non-congruent angles

Here are my students working on this activity, though they asked that their faces were not visible. The quote in the middle is thanks to the one on the left, who had the Space Jams theme song stuck in their head and wanted to share. 

My students enjoyed this quick check of different relationships, and liked moving around. And, because they are freshmen, they promptly stuck both post-it notes to their face for the rest of class when we were done.

Activity 2: Both kids were given a puzzle that showed some angle values and were asked to solve for several other variables using the parallel lines shown. I found this puzzle online here. This prompted some good conversations about how many angles you would need to know to completely solve a puzzle. Below is the work my students did:

They both agreed that "q" was the variable that took the most thought to solve

Activity 3: To add to the complexity a bit, I then gave both students the following puzzle, which added in an element of considering triangle angle relationships. I found the puzzle here. The ease with which both students completed the puzzle told me they were ready for more challenging activities with transversals.

Activity 4: The nature of my program is that while my classes are very small, student abilities tend to vary significantly. These two students are both quite capable of the level of the curriculum, but one in particular is capable of working at a higher level through the content. So while one was still completing the two puzzles above, I gave the other a puzzle found at the blog linked above and asked them to identify how many angles they would need to know in order to solve the puzzle. Here's what they did:

Once they were finished, we had a conversation about how we could determine y from knowing angle x and finding the measure of angle acd. Thus, we'd only need angle x! 

Activity 5:  For the remainder of class, which at this point was about 12 minutes, students worked independently to complete problems in the packets I prepare for them for each unit. My more advanced student was writing paragraph proofs for why lines must be parallel given algebraic relationships between pairs of angles, while my other student was practicing their algebra skills to solve for unknown angles and variables in sets of parallel lines. The snow started falling quite heavily at this point, so we got a little caught up in conversation about making snowmen and I forgot to take a picture of this individual work. Oops!

After completing this lesson I feel very confident that my students are ready to move on to proving the triangle angle sum theorem using parallel lines relationships and determining patterns for polygon angle sums next week! While I wish my third student had been present to get some more practice with the parallel puzzles, I will simulate the Post-It activity with him on Monday at the start of class to give him extra practice and perhaps assign him one of the puzzles as homework.