Opening Day Activity – Evolving

I am not 100% sold on how I have divided the data, but I want to think out loud about how my opening day activity for Precalculus Honors is developing.

Last year, I took the advice of a number of online colleagues and divided my classroom into table sets of 3 for student groups. I anticipate that I will have five of these student groups in Precalculus Honors (PCH from now on in the post) so I set up five subdivisions of the data that I previously shared (in this post) and I will start off with the cleanest of the data sets. At the bottom of the post I will attach GeoGebra file links as well as the current status of my handout. I will present screen shots of the data subsets and discuss my hopes and dreams for how this activity will unfold.

Group #1 will have an image similar to this when they graph their data. I will share this exact one with the whole class. I set up five GeoGebra files with the same screen dimensions (at least I think the dimensions are exactly the same, hope so!) My hope is that this will feel largely quadratic to them. I do not think that they have previous experience with regression equations on their TI or with any online tool. I am suggesting that they use Desmos in class. I will present GeoGebra with the goals of exposing them right away to two of my favorite tools. We’ll discuss which one might feel better for different situations. In the past few years I have had a number of students adopt both programs as a natural part of their problem solving. On more than one occasion a student has written a note on a problem set along the lines of ‘Desmos agrees with my conclusion!’ A natural direction for them to go is to use point H as a vertex and develop a quadratic that fits this reasonably well. Hopefully, we can incorporate a discussion throughout the year of the messiness of real world data compared to mathematical models. I love quoting George Box here – ‘All models are wrong, but some are useful’

Group two will generate something similar to this image. Again, I am anticipating a quadratic guess. This time point E should be seen as the vertex. Playing a bit with my TI and anchoring my guess at point E yields the following promising picture. 

Now, we are getting somewhere!

Group three sees this data and has some decisions to make. This is where I am really questioning myself. I worry that there is too much here for a first day activity. I think I may tweak this data set so that it is also more deceptively quadratic in nature.

Group 4 picture above.

Group five data picture.

And, finally, the whole set together here –

As I write this and think about my goals for that first day, I am sure that I want to modify the data that group 3 gets. I don’t want this play to become frustrating on day one. Let’s save frustration for later on!

I am hoping to plant seeds for a number of interesting conversations to have over the course of the year. I want the students to think about decisions based on small sets of data versus larger sets. I want them to think about periodicity and where/when/why to expect that behavior. I want to present them with an intentionally open question on our first day together to set a tone for open questions together throughout the year. I want them to remember something fundamental about quadratics and I expect to present two forms (standard and vertex) on the board after a little nudging from the groups.

I worry, as I often do, that I am being too ambitious. We have about 30 minutes together on day one due to a whole school convocation that day. I am really debating whether to make this the activity and conversation for our first full class day together. In that case, I would pull out an old favorite problem to have as the conversation seed for day one and mix in a bit of the boring old syllabus, etc on day one. Any advice there?

Here is a link to a dropbox folder with the GeoGebra files as they currently are arranged as well as my Word handout.

As always, advice/comments/questions are welcome here or over on twitter where I am @mrdardy

Preparing for Precalculus

For the first time since the 2010 – 2011 school year, I will be teaching our Precalculus Honors course. Since I left the course we changed texts and five teachers in my department have arrived since then, including the colleague who I will be working with on this course. Now that it’s August, I sat down this morning with our text (we are using the Demana, Waits, Kennedy, etc) book and I have some thoughts/questions that I want to air out. I will understand my own thoughts better after writing and I anticipate some helpful wisdom coming my way through this site or over on twitter. We start off in Chapter Four of this text, doing right into trigonometry.

Things I know I don’t like

  • Any formula to convert angles to arc length. Just emphasize part/whole relationships!
  • Language of vertical or horizontal shrinks or stretches. I just want to talk about amplitude and period. It feels like this extra language just clutters things up.
  • Inverse trig function using the odd negative 1 power. I want to write arccos x or arcsin x. Pointing out where it is and what it looks like on their calculator is a necessary nuisance.

 

Things I think I don’t like

  • The text has an odd emphasis on the word sinusoid. I don’t know why I would want to use that word, not clear on any benefit.
  • DMS notation. Why? Not sure, other than in surveying, when they will encounter this.
  • Introduction of cosine as x / r and sine as y / r – I kind of want to talk about the fact that all triangles are similar and simply scale down to ‘unit right triangles’ with a hypotenuse of length 1. This feels like a natural lead into the unit circle.

 

Things I know I like

  • An activity my colleague shared with paper plates, strings, and discovery that an arc that is equal to the radius will be subtended by the same central angle no matter the size of the plate.
  • The opening day activity I am tweaking that involves the length of daylight hours as a function of days after January 1.
  • Conversations I am planning on having about why it is usually sufficient to solve a triangle by knowing three facts out of six (three side lengths and three angle measures) and when it would not be sufficient.

 

I have taught Precalc at each of the four schools where i have worked and I always enjoy the course despite its weird, buffet style curriculum. The kids are fun to work with, sophisticated enough to have serious math conversations. We do not have the AP calendar breathing down our necks and our new schedule that includes a 90 minute class once every seven school days really lends itself to some meaningful play time in this course. I’m excited.

As always, please share any opinions/advice/questions here in the comments or over on twitter where I remain @mrdardy

 

Hijacking My Own Blog

I have used this space for years as a platform to talk about math teaching. Today I am going to use it for another passion of mine. Today I am thinking about music. Many of you (most? ALL?) follow me on twitter and you know that I have been fortunate enough in the past year to score a DJ gig at a local college. I have been posting all of my playlists on Spotify where I can be found (as on twitter) as mrdardy. I have been a bit obsessive about music since my middle school years. In college I worked in a record store. After college, in my early teaching years, I wrote for the local newspaper and then for a free news monthly. Later, when I moved to south Florida I wrote about music for another magazine there. I am pretty deeply obsessed.

As evidence – here are my CD and LP racks in my basement.

 

For reasons I do not understand, each image is upside down. Oh well, the point is made.

Part of what has come along with the music obsession is a good bit of snobbery. Long ago I stopped listening to radio to find music and dove deeply into music criticism and word of mouth. I can recognize labels and producers and often purchase something simply because of some arcane connection to something I already love. The I had a child. Then, six years later, another one. I stopped going to shows. I moved. I let myself wallow in my collection and got swept up a bit in the huge world of streaming music. First Pandora, then Spotify. I probably could have gone for the rest of my life without really digging in and learning new music again but the DJ gig saved me from that. Somewhat at the same time, my son (now 15) started really getting interested in music. In the past year I have taken him to see Gorillaz (I quite like them and was happy to go), Tyler the Creator (not quite as happy), and Kendrick Lamar (glad I went but much of the show was not my style). I have been really delighted to see him so animated by something important to me and it gives us a safe venue to talk which is not always easy with 15 year olds in the house. I don’t love the music he loves. I feel life an old fart being bothered by the language of much of the music he listens to, but I remind myself of generations of parents complaining about ‘that noise’ that delights their children.

All of this concert business with my son has made my soon to be 9 year old daughter (birthday Wednesday!) jealous. She has started listening to pop radio and has her own pretty extensive playlist on Spotify. Listening to this station of hers is trying for me. I am not a fan of the style and they seem to play only about 8 songs over and over again. On the other hand, it brings her joy and it is music. So, last Friday I took her to Philadelphia to see Charli XCX, Camila Cabello, and Taylor Swift. I think that all of you reading this know Taylor Swift even if you don’t know much of her music. The other artists, probably not so much. I had not knowingly ever heard Charli XCX but I discovered that I did, in fact, know a few of her songs. So, when she asked me to go to see her #2 favorite singer (Camilla) and her #3 favorite singer (Taylor) I kind of had to say yes.

The show was at Lincoln Financial Field – home of the super bowl champion Philadelphia Eagles. I am pretty sure that the last time I went to see a stadium show was when I saw The Who as a freshman in college (1982 or 1983) with Joan Jett and the B-52s as opening acts. I am used to seeing shows in clubs where I can be 50 feet from the performer or in nice old theaters where the acoustics are great and the seats are pretty comfy. We were FAR from 50 feet away from Taylor, Camilla, and Charli but that did not phase my lil one bit. From the time that Charli XCX came on at about 7:10 PM to a crown only about 1/3 of what it would soon be, my daughter was engaged. Charli XCX was energetic and passionate. She worked what crowd was there and she kind of won me over. Camilla Cabello had a tight, synchronized, and choreographed set. She played most of the CD my lil has and she mixed in bits of Frank Sinatra and Prince in the middle of songs. She is still pretty new to this business (she was previously in the girl group 5th Harmony) but she will likely have a long career ahead of her. Then Taylor Swift came on.

What a spectacle. There were probably 40 semi trailers in the parking lot that had been carrying the three stages for her performance in addition to the fire, the 40 to 50 foot video screen, the fireworks, the cables that transported her from one stage to another while she sang, the cameras, etc. etc. etc. I have to say I was there out of a sense of duty and a sense of wanting my daughter to have a meaningful memory of an adventure with dad. She knows I love music and I think that she wants to connect with me on this field (as does my son – I am flattered in both cases) but I was fighting my snobbery and my cynicism. Let me tell you, it melted away quickly. Her opening numbers were from newest album and I am not terribly familiar with them. I bought my daughter the 1989 album and know all those songs. I know some of the older country-ish songs and I know a couple of the newest ones. It didn’t matter. The show was so energetic and spectacular, the songs are so carefully crafted, and the JOY of over 50,000 people cheering and singing along is simply transformative. I found myself so swept up by the whole thing. My daughter was taken by it, the crowd near us, the whole damned stadium was in the palm of her hand.

Now, I do not imagine that I will be dialing up the music of any of these artists during my free listening time. I will hear them because I live in this world and because I have a young child in my home who is taken by this music. I will, however, hold on to the memory of this show. Not just because I hope my daughter will treasure this but because it cracked through my grumpy music snob exterior. It made me smile and sing along to songs that don’t particularly matter to me because I was in a crowd of about 50,000 people to whom these songs really mattered. Because this talented artist who was playing in her home town really cared about the fact that she was coming home.  Because she has crafted a show (hell, a career) designed to bring joy to large groups of people together. Because it was a summer night and a beautiful one at that. Because I was proud of myself for having made arrangements for this event. Because I just love music. I love being in its presence, even when it is music I do not inherently love.

I have seen hundreds of nights of live music in my life. I am totally lucky in that respect. I have distinct memories of dozens of those shows/nights. This show will definitely live in the small set of special nights for all sorts of reasons.

Some Fun Approaches

We have adopted a new schedule at our school and we are on a seven day rotation this year. At the beginning of each rotation, I give my AP Calculus BC students a problem set that is due at the beginning of the next rotation. These are just grab bags of problems that I find interesting. Some are calculus problems, but most are just fun stuff I have gathered over the years. On our most recent problem set (the last one of the year) I gave a problem that I think I found in an Exeter problem set. The heart of the problem was the image below. 

We are told that we are to start at hexagon #1. We are allowed to progress at each step to an adjacent hexagon as long as that hexagon has a number higher than the number we are currently on. So, for example, from 5 you can proceed to 6 or 7 but cannot go back to 3 or 4. The question is to determine how many pathways are possible from hexagon #1 to hexagon #13.

I did not know the answer to this question, but I was confident that I (and my AP Calculus BC students) could find the answer.  I approached this problem the way I do many problems, I wished it was smaller and I hoped to see a pattern emerge. I have advocated this problem solving strategy with my students but few pick up on this. I think that this has to do with their sense of freedom as mathematicians. I think that changing the problem feels like a privilege that they don’t think that they have. Need to work on this…

So, I built up a table and saw that if there was just one hexagon then there is just one path. A boring one of standing there. If there are two hexagons, there is also only one path. Hmmm, not promising yet. Three hexagons? Two paths, from 1 to 3 or from 1 to 2 to 3. Four hexagons? 1 to 2 to 3 to 4, 1 to 2 to 4, and 1 to 3 to 4. Now, I am confident that Fibonacci is hiding here. A quick check confirms this and I was pleased with myself for finding a fun problem that did not have an obvious solution.

I used the word obvious for an in-joke. One of my particularly clever AP Calc students will routinely refer to things being obvious in class discussions. His name is Owen and the way he marked his diagram was interesting to me on his problem set so I asked him to explain this in class. He started essentially the way I did but instead of a chart he simply wrote a 1 in the 1 box for # of paths and a 1 in the 2 box for the same reason. Now, his explanation gets interesting. Next, he mentions that it is obvious that if you get to hexagon 3 you have to have gone through either #1 or #2 so that the total number of ways to get to #3 is the sum of these two other numbers. Similarly, to get to #4 you have gone through #2 (one path) or #3 (two paths) and now Fibonacci is obvious. I was so delighted by his approach to this problem.

So I decided to present this problem to my other classes to see how they might approach it. In each class I explained my result after allowing them about 8 – 10 minutes to share thoughts about the problem with their small group partners. While none of my other students arrived at a conclusion in this relatively short amount of time, they did have some interesting approaches. One of my Discrete Math students tried to leverage what he’s learned about combinations by starting with the notion that a pathway along the odd numbers is six steps. Then he said that we could add one even number and this could be done one of six ways. We could add two even numbers to our path. This could be done in 15 ways (using combinations or Pascal’s triangle) and he wanted to simply add all of these up. A super cool idea but we started to see problems here. For example, if we add 6 and 8 as stops along the way in a row, then we have to skip hex #7 so we started trying to enumerate all of the path restrictions. Similarly, we realized that we’d need to individualize the number of odd hex visits in a similar way. Daunting, but a great example of trying to use knowledge he has gained this year. A group in Geometry recognized that the shortest path had six steps and the longest had twelve. They wanted to enumerate the number of pathways broken into these categories. A great idea and a way to get a handle on smaller cases to imagine. They quickly became frustrated by the daunting task of keeping track of these tracks, but I loved the idea.

It was a fun couple of days batting around these ideas. I have been really thinking about the distinction between ‘problems’ and ‘exercises’ and  problems like this one reinforce the ideas I am wrestling with. I am determined in each of my classes next year to have homework and classwork assignments labeled as ‘problem sets’ or as ‘exercise sets’ and I am hoping to help develop some clear strategies with my students to use when they encounter a genuine problem in math.

Sad Little Girl

Two posts today, both kind of brief. They are about two aspects of my life. The first is my dad post, my Geometry teacher post is coming later today.

 

I have a 14 year old boy and an 8 year old girl. They are very different people. My son does not stress out about school in any visible way. While I hear parents in our community talking about their 8th graders spending hours on homework, I never see that. He is up an down in his school performance but he does not seem to judge himself by these exterior reports on his progress. Sometimes I wish he was just a little more concerned, but he’s doing just fine.

My girl cares deeply about these exterior reports on her performance. She wants her teacher to think highly of her, she wants to please us. I am charmed a bit by this but I also wish she was less stressed about these types of things. This morning she was unusually quiet and reserved before school. I thought it might be her allergies – she has pretty wicked seasonal allergies and spring has just exploded on us here – but that was only a small part of it. She was worried because in PE this morning she was due to take part of the annual fitness test. This made her super sad. She loves to run and play with her friends but she has already begun to identify some friends as ‘sporty’ friends and she does not see herself this way. She had tears in her eyes because of anxiety about a fitness test this morning. This brought tears to my eyes as I thought about other students crying in the morning because of an upcoming Geometry test (or an upcoming Biology test or whatever), I was shook up thinking about the impact on self-image, on feelings of self-worth, on just the general task of living through the day that I saw a glimpse of. I am so sad to think that I am seen as the cause of such stress in my students’ lives. I wish I had some insight into how to battle this for my daughter and for my students. I know with my daughter I can talk to her about how her time in running around a field has nothing to do with how much I love her or how I value her. I try, in an appropriate way, to let my students know that their grade on a paper is not an evaluation of them as people, just a snapshot of them as learners. I need to be more explicit with them more often as I was reminded this morning. I also need to be more explicit more often with my two lil Dardys at home.

Man, what a bummer of a way to start my day today. My next post will be about a much rosier ending to the day (at least the school portion of it.)

Vectors!

A brief post this morning. We are winding down in our AP Calculus BC class and the last topic of the year is a short unit on vectors and parametric equations. Many of my students buy a (slightly) different version of our text book so some do not have the vector chapter. I use a curriculum module from the AP site as the spine for our work through these ideas. I have to admit that I do not have a great amount of enthusiasm for this topic, at least at the level that we work with it. But on Wednesday we had a fun breakthrough in class. We were working on a fairly typical example of a parametrically defined function on the Cartesian plane and found its derivative. The kiddos asked for a picture so we graphed both the position and velocity vectors on Desmos. One of my students expressed disappointment that we did not see the order of the graphs so it was hard to move our eyes from one graph to the other to see how they related. I am more comfortable making GeoGebra jump through hoops so I moved on to GeoGebra and graphed both with sliders and leaving a trace on. The kids seemed to perk up a bit liking this visual better. Then one student asked me to change the velocity vector. Instead of having it rooted at the origin, he asked me to redefine it so that it was attached to the point, so that it would be a tangent vector. I made this adjustment (you can find my GeoGebra of it here) and the kids seemed so much more engaged immediately. The power of seeing the trace points move apart from each other combined with the direction and length of the velocity vector changing along really caught their attention. I want to tweak it a bit still, there was a request for adding the acceleration vector as well. At a time of year when energy is running low, it was a fun blast of energy and engagement here.

 

That’s all for now, just wanted to share something fun.

Looking for a New Teammate

My school is looking to hire a new upper school math teacher beginning in the 2018 – 2019 academic year. I am in my eighth year here as chair of the upper school math department at Wyoming Seminary. We are located in northeastern PA right across the river from Wilkes-Barre. We are a preK through post-grad school on two separate campuses. The young ones – including my two children – are on our lower school campus located about three miles away from our upper school. My son, by the way, will be at our upper school next year. I am equally anxious and excited about this development. Our upper school is a 9 – post grad school that is a mix of day and boarding students. We have three dorms on campus and a number of faculty also live in campus housing. I lived in a boys’ dorm with my family for six years before moving into a campus home. We have a wonderfully diverse student body both in terms of where they come from – we have over twenty countries represented in our high school student body – and what their interests/talents are. We have nationally recognized athletes, we have top notch artists, we have stunning scholars. We have kids who LOVE math and are in Calculus as freshmen. We have kids who dread it and are in Algebra II as seniors. I firmly believe that all of these kids have meaningfully experiences and they grow as students while they are here. We are losing one of our valued members of the upper school faculty and I am sad to see him go. However, this is the time for me to look ahead and dream about what a new colleague can bring to our school. If you are reading this and are contemplating a change of scenery for next year, please reach out to me in the comments here or through twitter where I am @mrdardy or simply write to my work email. If you are reading this and you know someone who might be a great fit, pass this info along.

Our school’s website is https://www.wyomingseminary.org/

 

Greetings, 2018

A non-mathy post for this morning. I feel like I need to clear my head out a bit here.

 

  • Thanks to Meg Craig and the #Fitbos gang for helping to keep me motivated this past year. I set two goals for myself with my trusty fitbit flex. I wanted to accumulate an average of 30 minutes per day at an ‘active’ level. I compiled a total of 207.45 active hours. Last time I checked my multiplication, this exceeds my goal! I also set a goal of walking 2017 miles in 2017. I ended up at 2050.16. I am pretty pleased, but time is still working against me, despite this level of activity I am more achy and a bit paunchier than I was this time last year. Have to ramp it up to fight against Father Time.
  • Thanks to connections that my wife has at her college I was able to score a gig as a DJ at the local college radio station. Almost every Thursday since June, I have had the great pleasure of spending two hours (from 4 – 6 PM ET on wrkc.kings.edu) playing pretty much whatever music amuses me on terrestrial radio. I have been compiling playlists over at Spotify where you can search me up as mrdardy. It has been one of the real joys of my life this past year.
  • Mostly a consequence of my DJ gig, I have listened to more new music released in 2017 than any year since the birth of my son in 2003. It feels great to be reminded of the pleasure of discovering new music again. I still feel a bit overwhelmed when I read Best of lists at the end of the year, but there is a better chance of me knowing a number of items on these lists than I have had in years.
  • At work we have had a couple of important changes. We moved to a new, rotating schedule. We have 7 periods, 5 of which meet each day. In a seven day cycle each class meets five times. Four of the meetings are 50 minute classes (every once in a while an assembly moves that back to 45 minutes) and meets once for a 90 minute block. This has been a great change in our daily lives.
  • In our department we adopted a test correction policy where all students are allowed to earn back points by reflecting on their work. We ask them to submit corrections in the form of pointing out where/what went wrong in the problem’s work and then correcting said problem. I am super excited about this project and I see students being really thoughtful and attentive in submitting these corrections.
  • My life at school has been a bit more hectic than I’d like, despite the change in schedule. I have five classes this year (more often than not, this has been my standard work load here) which is especially manageable in this new rotation. What has been tiring is that I have four different class preps. Keeping all these trains running in my mind, especially since my two Geometry classes are rarely ever aligned anymore, has been a tiring challenge. I think being 53 and having a 14 year old boy and an 8 year old girl in the house has an impact as well!
  • I was able to attend TwitterMathCamp for the fourth summer in a row. As an added bonus, this past year did not conflict with my daughter’s birthday. Another bonus was that Atlanta is the home of an old high school buddy who was also my first college roommate. I had not seen him in years and had a lovely night with him and his family on a warm southern night, hours spent on his porch catching up was a delight.
  • My time at TMC was followed by a trip to FLA that included a couple of nights catching up with friends in my old hometown of Gainesville. Had not been there in a few years.
  • My school is a PK – PG school on two campuses. We live on the upper school campus, the lower school is about three miles away. My son is in 8th grade and he and his pals will be in my hallways in 8 months. Exciting and scary at the same time!
  • My wife is nearing the end of her Master’s Degree program. It’s been fun listening to her talk about her school experiences. It has been ten years now since my course work last ended. I’m a bit jealous, I think.
  • Off to face the new day, the new year, I guess, now…

 

Optimistic

Another quick post. We are in exam week here at my school. I have ALL sorts of thoughts about term exams and why we do them, but those are for another place rather than this public forum. I have written about our department’s decision to move toward a test correction policy. I am so so so optimistic about our exams this week. I really believe that we will see largely improved results because the students have been more actively involved with examining their tests and reflecting on what went right and wrong on their tests. They have been talking to each other and comparing ideas. They have been talking to their teachers about how to fix their problems solving approaches. Our department exams are mostly on the next to last day of exam week. This will work against our students as energy levels start running low. Despite this, I am hopeful that we will see a different level of engagement on these cumulative exams. I will report back either way in a week or two, but I feel good based on what I have seen in the past week of reviews in class and from watching and listening to kids in the hallway as they work together.

Improving a Lesson Plan

My Geometry classes have just finished a few days considering different translations on the Cartesian plane. We are working toward being comfortable with rotations (almost always around the origin), reflections (horizontal and vertical lines and the lines y = x and y = – x), and vector translations. Last week I had a particularly unsuccessful lesson where I tried to help my students discover a pattern for 90 degree rotations around the origin. I want to try and outline my thinking here and I would love love love any insight into why it did not go well and how I can improve this in the future, or even how to go back to cycle and revisit this with my current team of scholars.

So, my idea was this – try to pull together what we know about perpendicular slopes, our developing ideas of vectors as a physical object similar in nature to a line segment, and developing an intuition about the fact that a 90 degree rotation should result in a move of one quadrant in a certain direction. I asked for three coordinates from my students and drew a triangle. I asked them to predict where this triangle would end up after we moved it 90 degrees clockwise. Two of the coordinates given were in quadrant I and one was in quadrant IV. It seemed that my students were happy/comfortable with the idea that the two quadrant I points would live now in quadrant IV and that the quadrant IV point would move to quadrant III. This may have been a tepid agreement in retrospect.  Next, I focused our attention on one of the quadrant I coordinates and I drew a segment from the origin to that point. We talked about the slope of the segment, we compared this segment to a vector, we talked about the length of the segment. I then asked the students to imagine a wheel and I told them that when I think about rotations I think about a bicycle wheel. In my mind I saw this segment as a spoke and I thought about distance from the point to the center of the wheel. Here is one place where I know I failed my students. I did not explicitly stop at this point and discuss distance the way I could have/should have. I also have a handful of students who are still struggling terribly with the idea of calculating distance. We have been talking about it since day two, I have coached them to think about Pythagoras, we have practiced it repeatedly. The combination of squaring, of square roots, of subtraction in one piece of ‘the distance formula’ and addition in another piece of it, comfort with mental arithmetic, all of these factors are working against my students being unanimously comfortable with calculating distances. So, the next step in my plan was to ask them what they recalled about perpendicular slopes. They all should know this and most recalled it pretty quickly. We had a segment in front of us with a slope of 4/3 and my students quickly agreed, maybe passively maybe enthusiastically, that a segment perpendicular to this would have a slope of -3/4. So, the question at hand was now whether the fraction was in the form or -3 over +4 or in the form of +3 over -4. I was convinced in advance of this lesson that this string of conversations would be a positive path to take. I felt that the combination of recalling past slope ideas, looking at the physical Cartesian plane, tying in ideas of line equations, etc. would gel together to make a lasting learning experience. I was wrong. When I prodded them toward the conclusion that -3 over +4 was the conclusion we wanted I saw some uncomfortable faces. When I mentioned the idea of a spoke as a visual to hold on to, I saw blank faces at this point. I got a bit frustrated and asked for my students to describe to me what they were thinking of when I mentioned the word spoke. Nothing. I pulled up a google image of a bicycle wheel and asked them to tell me what the spoke was in the image. By this point their reluctance to engage in this conversation was building, my frustration was increasing, and any positive momentum in building this process was falling apart. My fault for showing my frustration. My fault for stacking up too many ideas at once, I think. When I spoke to my Geometry colleague she felt that adding on the layer of talking about perpendicular slopes was the tipping point of discomfort for my students. I trust her instincts on a number of levels in part due to her experience in teaching our Algebra I course. She knows these Algebra kiddos and knows not only what they know but how comfortable they are knowing it. So, at this point it was clear to me that this was slipping away. We limped to the end of the conversation. Most students were willing to agree that the point  (4,3) would end up in quadrant IV. They were split on whether it would land on (4,-3) or on (3,-4) and it honestly felt like many were mentally tossing coins to make this call. I showed them the conclusion on GeoGebra and we sort of ran out of time by this point.  We have since gone back and tried to reinforce the conclusion we reached and I think that most of my students can reliably answer this question, but I am completely uncomfortable with how we got there. I would love any insight/advice about how to best structure this info. You can certainly drop a comment her or over on twitter where I am @mrdardy