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.

The questions they've always needed to ask

I promise I'll get to questioning in my classroom, but please bear with me for some brief* self-questioning first!

This past summer I had the privilege of attending Alan November's Building Learning Communities (BLC) conference where I stuffed my brain past capacity for three full days thinking about teaching, learning, equity, improving all student outcomes, and how to use technology effectively to improve the collaboration and the scope of the work we do in classrooms. I left with about 15 pages of notes, a list of roughly 100 education folks I wanted to start following on Twitter, and so many ideas that I spent a good part of last August prepping new tasks and ideas to incorporate into my classes.

One of the ideas that really stuck with me from the conference was hardly a new one - homework - but my thinking about it changed significantly. As I heard teacher after teacher talking about how homework has been shown to serve those who will succeed no matter what the circumstances are while it may harm those who struggle, I really began to question my own thoughts on assigning homework most nights. My particular student population is overwhelmingly biased on the "those who struggle" side of learners, and I've spent most days of my first five years teaching lamenting how few complete homework assignments were turned in. Particularly as the curriculum grows harder in my 11th and 12th grade classes, fewer and fewer kids do their homework and I see huge gaps start to form in understanding core content between my (admittedly few) students based on their homework decisions. So, throwing caution to the wind, I decided to go all in on a homework-free pilot of my polynomials unit with my juniors this year. The unit goes as follows:

determining end behavior from standard form of polynomials --> graphing polynomials in factored form --> factoring and solving (factor-able) polynomials --> long and synthetic division of polynomials --> rational roots theorem --> Fundamental Theorem of Algebra --> combining everything we know about polynomials to solve any polynomial

Anybody who has taught this material before can understand why kids could fall very behind if they weren't completing homework! In past years by the time we got to the rational roots theorem I usually spent the 20-25 minutes of independent working time at the end of class running around answering questions for kids varying from "What specific error did I make here?" to "What's a root, again?" And when at the end of class I told students to go home and do more similar problems for homework to independently practice, I knew that my strugglers had little chance of completing the assignment alone and that they would more likely never bother to start rather than face the frustration. Hoping to help all of my kids learn this material better, I made a deal with them. For the entirety of our polynomials unit, they won't have any homework in exchange for us working productively together through every minute of class. When class ends, we stop where we are. We pick back up the next day, and kids work at their own pace with peers as they want when we get to independent practice times after collaborative instruction.

*sorry that was in no way brief.

OK, so questioning. Since every day in class we work together through structured notes/exploration/independent practice with the material, there is ample room for kids to ask questions about the content. We've had some really great moments. They asked terrific questions as we learned about standard form, like "do we really need to bother fully expanding a polynomial in factored form if we can just find the first term?" They also made great connections to multiplicity patterns of roots, like "Hey! The shape the graph makes at the root is like the shape of the parent graph for that exponent! If it's squared, it looks like a parabola!" One student has made it a habit every day of asking students who ask questions and receive answers from myself or peers to repeat what they just asked and learned, explaining that she "learns the most when she hears what connections or questions other students have."

We've also had some not great moments, where students ask basic algebra questions that initially make me concerned for their previously learned skills. When determining the degree, students asked often about how to combine exponents: "do we add them or multiply them?" When factoring polynomials, "Can you remind me how we know which method to use?" Or a rather concerning one, "What's a perfect square?"

I'm standing firm so far that while these questions give me doubt to students' past conceptual understanding, they would have these questions anyway. Since we are learning together in class, these questions come out naturally and everybody can benefit from a miniature review. Previously, when those questions would come up on a homework assignment at 10 p.m. when my students were alone, they proved to be the perfect excuse to give up before really digging into current material and simply pronouncing the next day in class that they "didn't get" the homework. My most struggling student, who has had a very spotty math skills background for the past few years due to major emerging mental illness and a handful of different housing situations, has asked more questions in the three weeks we've spent learning about polynomials so far than he has asked in two years. And learning from those answers, coupled with knowing that homework assignments where he'll feel completely, hopelessly lost and lose credit for them in class are off the table, means that he recently earned one of the highest scores on the Polynomials section of my midterm exam.

While we have spent more time on the first part of our Polynomials unit than I have yet in my teaching career since there are no homework assignments, average student understanding is vastly improved. I'll keep with my no-homework pilot for this unit (which at this point might well take us until March, but damnit they'll be Polynomials masters) and continue to evaluate student understanding. From all of the great learning I've seen in this short pilot so far, homework may well be on its way out in my classroom...

Scavenging for My Favorite

One of my absolute favorite ways to have students review material before an assessment or practice skills once I've taught them a few ways to do so (a la solving quadratic equations or solving systems of linear equations) is to have them do a scavenger hunt around our school. Because we're a small program and the entire building isn't that large, I often put clues all over the school to encourage the kids to walk around and actually have to hunt for things, which adds an element of fun. During our most recent scavenger hunt practicing operations with polynomials one student commented that he had walked more during class than he does typically in a day. I reminded him that walking is good for you, put on my best Calvin's-dad face and voice, and pointed out that it builds character. 

From the great Calvin and Hobbes

Later that day when our program director notified students that defacing school property can get you in loads of trouble (aimed at a few kids doing art on the bathroom walls) my students were happy to suggest that the scavenger hunt pieces I'd taped all over the school technically fell under "defacing property." I assured them I would be happy to end future scavenger hunts and only do boring work in the classroom day after day and the mutiny was quickly squelched. 

A typical scavenger hunt will start with each student being given a problem to solve. Depending on the kind of skill we are practicing (and, realistically, how much prep time I've had with that particular hunt) once they've solved it, they either 

• Find a clue with the correct answer at the top and a new problem at the bottom of the card and continue this process
• Find a clue with the correct answer and pull a new problem from an envelope attached to the clue and continue to combine parts of previous answers to form future clue solutions
• Go to the coordinate location of their solution to find the next clue and continue that way (see below*)

I always make my scavenger hunt pieces with a unique sticker on each clue, and give students organizers to complete their work that calls for them to identify the sticker that was on each subsequent clue/problem card. I do that for four reasons:

1. I can very quickly check students' work when they've completed the activity. I can have each student start at a different clue so they're not tripping over each other, and since my scavenger hunts always end where they began I just check their circle of problem stickers against my key. Here's what a typical key looks like: 

   

   My stickers are animal, vehicle, and "teacher rewards" themed at the moment. 

2. I can easily differentiate for my different students. I can make sure that students start on a clue that proves to be an appropriate level of challenge for them and cycle from there, and for a student who is struggling I can set a number of clues that I would like them to get through in the allotted time period (i.e. "You'll have completed the task when you get to the Giraffe clue."
3. If a student gets partway through and can't find a matching solution, I can easily check that their clue progression has been correct or redirect them if they went awry somewhere along the way.
4. If we get interrupted during the scavenger hunt (fire drills, students having to leave early, etc.) or we run out of time they can finish the next day by simply finding the sticker they left off on and picking up from there. 

*If the skills we're practicing happen to relate to graphing on the coordinate plane I pull out the big guns (read: my floorplan of the school with a coordinate plane overlay) and their solutions take them to the exact location of the next clue instead of having to hunt around. This serves to both check their solving skills and help refresh their memory on how we work with coordinate points and how we read a map. 

A Day in the Life...a few days late

When I saw last week's "A Day in the Life" blogging challenge, I was prepared. I took careful notes of my day last Monday, then got so busy with the "days in the life" of the rest of the week that I never got around to blogging. Oops. Here you are, a little late:

A Day in the Life of Sarah on a Monday

         5 a.m. Alarm goes off. I get up, bundle up, and squeeze in a quick 5k length run.

         7:10 a.m. I should be heading out the front door, but I took too long picking out an outfit. In the words of the great Liz Lemon, blurgh.

         7:17 a.m. I actually am heading out the front door.

         7:44 a.m. I arrive to school. On my drive in I've been thinking about the quiz I need to create for beginning polynomials for my sophomores, so I'm going to try to start that before our morning staff meeting.

         7:52 a.m. I comb through my binders of old curriculum to help inform the quiz. While combing I also realize that my binders are a bit of a mess and add that project to my to-do list.

         8:00-8:20 a.m. Our morning staff meeting. We plan for schedules, talk about kids who are struggling, learn of a potential new student, and I successfully convince our program director to order a BreakoutEDU kit with the program budget.

         8:21 a.m. I order the BreakoutEDU kit. Yay!

         8:27 a.m. I give up on finding a good quiz that I created in the past. Make new quiz until 8:51.

         8:55-9:11 a.m. Make all of the photocopies.

         9:15-9:40 a.m. My statistics students are doing a variation experiment on the Stroop Test looking at if conflicting stimuli cause people's response times to increase, and whether it increases more based on whether the response is oral, written, or physical clicking online. I organize and gather information from various students so that we can start conducting the experiment today. 

         9:45 - 10:15 a.m. Create alumni survey for dispersion online to inform our large-scale program planning.

         10:16 a.m. Refill water bottle for the second time. Hooray hydration!

         10:40 a.m. My freshmen arrive for class. Two-thirds are present. Sounds dramatic, but I have three students, so my two most consistent are here. We explore what other information we would need given two pieces (sides or angles) in order to perfectly match a given triangle. They independently determine the four triangle congruence postulates by the end of class. We have a conversation about why ASS doesn't always work, and my boss walks by my classroom while ASS is written very large on my board. Good thing he trusts me!

          11:30 a.m. My juniors come in whining about how tired they are and how they hate Mondays. We start our exploration into graphing polynomials from factored form by looking at multiplicities of roots patterns and what they look like on a graph.

          11:42 a.m. I regret my second water bottle. It's a loooooong time until I can pee.

          12:20 p.m. Sweet relief! It's lunchtime, so of course no students are in the bathroom. After relieving myself I eat my same lunch I eat every day: turkey sandwich, apple, water. 

          1:04 p.m. While continuing our multiplying polynomials scavenger hunt from the previous week, one student asks "Is this going to be an IEPeagle claw?" This is her own term for the visual pattern created by distributing between a binomial and a trinomial when multiplying them together. Something like this: 

Note: I teach in a Special Ed program, so my kids all have IEPs. They know they're documents that help kids like them learn the same as students without similar emotional and/or learning issues. Her comment, including IEP in her "eagle claw" distribution pattern, identifies the problem as unique, and is not meant to be disparaging at all.

           1:32 p.m. The same student comes across a trinomial multiplied by another trinomial and decides that she'll call this a "Super IEPeagle claw." 

           1:59 p.m. Kids in study skills ask about the Powerball Odds. I briefly explain how combinations work.

           2:15 p.m. Walk between study skills rooms looking for kids who are "working" but actually flipping screens and help students having a hard time getting started pick something to work on productively. 

           2:20 - 2:50 p.m. Boring administrative stuff. Do my attendance, grades, points.

           3 - 3:30 p.m. Host "homework cafe" for students who need a quiet space still at school to get their work done. We eat, we work, I pester my most faithful attendee to actually read the articles he claims to be using for research instead of just gathering them.

          3:45 p.m. Head for home. Relax, eat, prepare for tomorrow!

A great debate: Polynomials

My senior year of college I ran into my former Linear Algebra professor leaving the bathroom, who was at the time dusted head to toe with chalk. Middlebury's math professors were amongst those who loved the "classic" feel of chalk, so most things in the math building had a fine layer of grit if you stuck around long enough, but she was COVERED. When she saw me she acknowledged how ridiculous she looked, then explained her appearance with an exuberant "I was teaching the Fundamental Theorem of Calculus!"

That memory sprang to mind vividly today during my junior Algebra 2 advanced class, when I was getting excited about the discussion my students were having to develop rules for the end behavior of polynomials. After discovering and agreeing on how the degree and leading coefficient can help you figure out the very basic shape of a graph, they moved on to how we could use factored form to graph polynomials more specifically. How we could work backwards from a solution followed, and as we went along I could feel my voice growing louder with excitement. I prefer modern technologies, so in lieu of chalk dust by the end of class I found my hands covered in whiteboard marker flecks, but in that moment I was very much channeling Professor Proctor. I at one point exclaimed "...and all of that is so important that we call the idea the Fundamental Theorem of Algebra!" so red-faced and loudly that my students asked if I was alright, while one kid (lovingly) pointed and shouted "Nerd!"

In one poignant moment of our discussion a student, M, raised her hand and said "So to solve a polynomial you just need to know its x-intercepts?" We talked about this for awhile and my excitement ballooned as everybody realized that the x-intercepts also tell us the solutions of a polynomial and that now we can put our previously learned quadratics skills around graphing and factoring and solving to work to get results from the standard form of a polynomial. M raised her hand again and asked in a somewhat annoyed tone "Why didn't you tell us all this months ago? It would have made things so much easier!" Smugly, I responded with "Because all that work we did months ago led to all of you coming to the conclusions we just reached. You guys just discovered everything on your own." And as she slowly released her breath in a damn-it-you're-right-but-I-still-want-to-be-mad-at-you-for-keeping-this-from-me way, I considered dropping the metaphorical mic and walking out of my classroom, my work complete.

The more I teach and read about teaching, the more I question the practice of assigning nightly homework. My students and I have agreed that all of our polynomials work will happen in the classroom, with all class minutes focused on productive work and discussion. If a student feels the need to work independently at home (or needs to catch up after absences), they may, but I'm not assigning homework. We'll see how it goes as the weeks progress, but since these moments of great conversation happened 3 days into the pilot, I'm feeling pretty good about the decision so far!

Getting ready!

I'm so excited to begin blogging as part of the Explore #MTBos Blogging Initiative!

This year I want to incorporate even more new ideas into my teaching, practice mindfulness daily in any way that I can, and begin blogging about my work to share my ideas and get input from others. So far 2016 has brought me strep throat and sick days, so I'm looking forward to being well enough to fully commit to my resolutions!