Algebra 2 curriculum dump #1

I’ve decided to reflect on the interleaved algebra 2 course that I’ve designed and taught over the last couple of years. My hope is that by taking a more outward-facing approach to the curriculum, it will help me make it better. This is the 1st post in the series.

In my last post, I overviewed the course. I’m going to dig into the units now. Yay! Here is the working problem set for the course.

Unit 1 (Problems 1-24)

  • Function notation & analysis
  • Domain & range graphically
  • Function arithmetic
  • Trigonometric ratios & circles
  • Factoring the greatest common factor

Ah, the opening unit. The point where summer fades away. Students often find graphs to be their mathematical friends, so I lean on them a lot, especially in the first few units. I pair that with some analytical stuff by getting into function notation, function arithmetic, and factoring with GCF. Over 90% of the kids come to me from geometry (the others are taking it concurrently), so I also like jumping right into trig. The ratios are fresh in their minds and they get excited when they see them in the first few problems (e.g. 2, 8, 18). The problems reintroduce sine, cosine, and tangent and we begin thinking about the ratios in the coordinate plane with circles. It’s not explicitly stated in the concepts, but I also go hard with using interval notation when representing domain and range; it sets up so much of our future work on intervals. It’s a small thing, but I like how I frame factoring as “rewriting as multiplication” (problem 7).

Two things that I’m left thinking about: I don’t focus on domain and range in any other representation other than graphically. For this unit, I think this is ok, but as of now, I don’t devote any problems to it later on, either. This is an issue. Also, sadly, the function arithmetic that we do sort of dies at the end of this unit, as I haven’t yet included problems that loop back to it in future units.

In the end, this unit is full of pretty straightforward concepts and the students perform well. This is by design. With such a big contrast in how they learn math as compared to prior years, I find it helpful to build success in for them at the start. It’s a needed confidence booster for the long year ahead.

Unit 2 (problems 25-50)

  • Intervals of increasing & decreasing
  • Extreme values
  • Visual sequences & sequence notation
  • Factoring the difference of perfect squares
  • Trigonometric ratios & circles
  • **Systems of linear equations (2)

By studying where a function is increasing and decreasing and its extreme values, we continue the work on graph analysis that we started in unit 1. Again, interval notation comes in handy here. This problem set also introduces sequences. The intricacies of sequence notation can be tricky for students, but the few problems that I begin with (37, 38) have combated that nicely through the years. We also continue our factoring journey as we study the difference of two perfect squares. Now that I think about it, I wish I had included some difference of squares expressions that had a GCF. And I still don’t know how I feel about obscure expressions like (2x-1)^2 + (3x+8)^2 and asking students to factor them, but hey, that’s why I march to the beat of the Regents exam.

This is an important unit in the development of trigonometry because the kids learn about a unit circle along with defining sine, cosine, and tangent in terms of x and y. We also explore the signs of the three ratios depending on where the terminal side of the angle lies when it’s drawn in standard position. (Annoying: so many kids at this point start using the mnemonic ASTC because of the DeltaMath problems I assign.) Speaking of the terminal side of an angle, lost in all this trig work is the explicit learning of how to sketch an angle in standard position. I assign them problems from DeltaMath, which are really nice, but there’s nothing in the problem set that targets this skill. This results in me creating an opening problem to discuss it with the students at some point during the unit. This is haphazard and sloppy.

**Throughout the course, we come across topics and skills that are included in the problem set, but are not tested on that unit’s exam. I’ve found that there’s no need to test on EVERYTHING that we study during a single unit. Certain concepts, and certain problems, are stepping stones to broader ideas. I simply use them to navigate us to much deeper, more important parts of the curriculum. In this case, wrestling with systems of two linear equations will, over time, eventually lead us to solve systems of three linear equations (along with non-linear systems) in later units.

Unit 3 (problems 50-77)

  • End behavior
  • Function translations
  • Recursive sequences
  • Difference & sum of cubes
  • **Trigonometric ratios & circles
  • **Experimental Design
  • **Systems of linear equations (2)

In this unit, we continue our work on graph analysis, sequences, and factoring. Naturally, I am able to spiral in our previous graphing work (domain/range, intervals of increasing/decreasing, extreme values) in almost all of the end behavior problems (e.g. 57, 65). Factoring the difference or sum of two cubes always seems so formulaic and boring to me, but I know of no better way of approaching it. That said, this year I changed problem #59 to include an example of using the formulas, and it really helped. The students also get their first taste of a reciprocal trig function when they learn about cosecant. I define it plainly, which I’m ok with for now. I fold the ratio into several problems in the set to help build familiarity with it, especially in relation to the other ratios. When doing this, we continue to discuss terminal sides of angles and sketching triangles in the coordinate plane. Submerged in this set is rationalization. I touch on it with one problem (61), follow it up with DeltaMath, and bring it in again with problem 74. This basic level stuff is good enough to be able to help us with future problems. We also mix in our first look at transformations of functions when we do problem 53. I use a Desmos activity for this that I like, but I know could be better — especially when it comes to horizontal translations.

The sequence notation that they learned during the last unit is put to good use when I contract a throat virus (which renders me unable to speak) on the day they must investigate recursive sequences in problems 67-69. It’s a fun, memorable lesson. I drop two more linear systems problems in the set for good measure (60, 73). On a related note, I really like problem 72 because it combines function transformations with systems of equations. The equation f(x)=g(x) gets some attention, which is a nice.

I’ve sprinkled in a couple more problems on bias during this unit, which seemed like second nature to the kids. I feel the problems are useful, but they seem to linger with no stronger connection anywhere in sight. This will soon change when we start getting deeper in statistics in upcoming units, but maybe I can adjust for this next year?

 

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Algebra 2 curriculum dump #0

I’ve decided to reflect on the interleaved algebra 2 course that I’ve designed and taught over the last couple of years. My hope is that by taking a more outward-facing approach to the curriculum, it will help me make it better. This is the 0th post in the series.

Two years ago I decided to turn my algebra 2 curriculum upside down. I did away with the thematic units that I used for my entire career and instead structured the course around interleaving topics. This was quite a shift for me, and I’ve written several posts about it, including a recent one about the mess of it all and why I love it.

Up to this point, I’ve never felt the need to formalize the curriculum. In year one, I didn’t know what the hell I was doing and developed all of it on the fly. It was an experiment. Last year, I heavily adapted much of what I did during the first year and reached a place where I could actually talk about what was happening in my classroom. This year I’m refining it. I’m even trying to integrate a book study. Since my head is now above water, and things haven’t been a complete disaster thus far, I’m feeling the need to take a more critical and reflective look at the sequencing and structure that I’ve developed for the course. To thoroughly think through my clutter of concepts and numbers and spreadsheets, to piece together a narrative from this tangled mess of a curriculum, is what I need to do to help improve it. For me, this means that I have to write about it. (It also wouldn’t hurt to have a coherent representation of the curriculum that someone else could understand.)

For fear of going mad, I’m breaking up how I reflect on the course by marking period. At the end of each marking period, I’ll write a narrative detailing the units that we studied during that time, my thoughts about them, and how they fit together.

Since this is the initial post, I’ll attempt to write an overview that I know is rushed and incomplete.

Course Overview

The course is based on the algebra 2 Common Core standards and taught through many non-thematic units. This means that, instead of being topic-centered, with concepts being taught in blocks of related content, concepts are problem-centered. This places emphasis on problem-solving, yes, but it also means that concepts throughout the course are interleaved. Instead of being introduced consecutively, related concepts are spaced out and learned over long periods of time.

To accomplish this lofty goal, each day we explore problems in class. Some days, there’s a singular focus, where all of the problems are related to one another, but most days we discuss multiple concepts that may or may not be directly connected. Disclosure: This doesn’t mean the problems are especially good, interesting, or challenging. On the contrary, because I teach a Regents course, the problems are quite boring.

What the course does well, I think, is challenge the notion that math needs to be taught linearly (like it is with traditional, thematic units). The cumulative nature of a course’s sequencing is overrated and less important than I think most math teachers realize. For example, early in the year, students in this course learn about factoring, graph analysis, and trigonometry concurrently. While these concepts are not completely distinct from one another, they probably wouldn’t be taught at the same time in a standard algebra 2 class.

As the year goes on, topics are revisited many times. We take bite-size chunks of content, digest them, wait a little while, and then go for seconds, and thirds, and fourths as the year progresses. Trigonometry is the best example of this. As I mentioned, we kick off the first unit studying trigonometry and we’re still learning (not reviewing) new trigonometric concepts in April. By design, it takes a while, sometimes months, for big ideas to mature in the course. Naturally, this development occurs a lot in the spring as we approach the backstretch of the school year. We’ve been studying certain topics for a long time, and it slowly starts to come together through the dozens of problems we have solved. As topics are woven toegther and learned little by little, finding connections between problems a foundational aspect of the course.

 

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My two cents on letters of recommendation

For those of us that teach high school, we know this time of year well. ‘Tis the season for letters of recommendation! I’m feeling rather festive this year, and I have 7 of them patiently waiting for my attention as I type this sentence, so what am I going to do? Forget all about writing them, forget all about being productive. Instead, I’ll just ramble.

Where do I start? Well, when I think about writing a student a recommendation letter, it all begins with the request from the student. How a student asks is important to me. Other than the student being graceful and intentional, which is a given, I’m a stickler for facetime and eye contact. I don’t like it when I get requests by email. When I do get one like this, even if I don’t have them in class anymore, it’s almost always from a student that I see on the regular. In my head I’m thinking, just come ask me! (I’ve long believed that email is abused, especially when it comes to interpersonal relations, but I digress.) When I get a recommendation request by email, I always ask the student to see me in person, to make time to look me in the eye, even if I know that I plan on writing it.

Next comes the decision to actually committing to writing it, saying yes. In my experiences, it seems like some teachers feel obligated to write a letter of recommendation when a student requests it. It’s like, they have to ask and we have to say yes, but that’s just a formality. I get that we want to help students as they begin writing their next chapter, but when did helping out a student turn into me agreeing to camp out in front of our computer for hours and hours to compose dozens of recommendation letters?

When it comes down to it, I don’t mind saying no. If I don’t feel connected to the student, if I don’t feel like I’m the right person to tell their story, then I will simply tell them so. This might result in awkwardness, but I’d rather be upfront and encourage the student to find someone else that can better capture them through the letter. If I know that if I’m not thrilled to write it, to be able to go on and on about the student and share personal stories, then I’m not the right person for the job. Besides, if I did write it, I’d most likely be dragging myself through it and the letter itself would probably end up being mediocre at best. In addition, there’s another reason why I will tell a kid no: I must have taught them for at least a year. If this is not the case, then I won’t even consider writing it. It’s not personal. There are lots of highs and lows in a school year and observing a young person navigate those peaks and valleys is critical to me being able to endorse them without reservation.

While telling a kid no is not my favorite thing to do, I also go the other way and say yes without even being asked. In recent years I’ve requested, and lowkey demanded, that I write a kid’s letter of recommendation without them even thinking about it. I’m so proud that I take ownership of them and want badly to be their advocate. It’s an honor that I want to take on.

When it comes down to actually writing the letter itself, I used to have a canned letter that I would modify a little for each student. It was bad. That was a while ago, and I’m so not proud of myself for shortchanging those kids. They deserved way more than how I chose to represent them. I respect the process so much more now, in large part because of Sam Shah. Several years years ago I read his post on recommendation letters and totally stole the questionnaire that he asks his students to complete. I’ve modified it some, but this questionnaire is by and large one of the best resources I have when it comes to writing the letters. Once you read through it, you’ll know why. It lets them tell the story. Couple this with personal anecdotes and other written reflections from students, and the tone is set.

Through all this, I’ve even begun tinkering with the structure of each letter depending on the student and how I feel about them. For example, my letters these days often don’t start and end the same way. And playing around with these aspects of the letters makes them way more fun to write. Must they look and feel the same? How can I get the letter look and feel like the student that I’m writing it for? Rethinking these sorts of things makes writing it more of a creative act for me. Even if they never get to read it, it also helps me pay respect to the student and all that they are and hope to become. This jives well with my post last year about being selfish when it comes to writing.

Interestingly, my school recently brought in someone from a top university to give us tips on how to write an effective letter of recommendation. I couldn’t attend, but I really wish I could have.

Ok, ok, enough procrastinating.

 

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A five-minute conversation sparked so much sidewalk math

I’ve been really into sidewalk math for a while. It started with me laying it down with students at school to doing it at local parks in the Bronx, where I live. All of this ballooned into my sharing it last August at TMCNYC as a My Favorites.

After I shared it at TMCNYC, Michael Pershan came up and asked me whether or not I had thought about sharing sidewalk math with a larger audience. Other than Twitter, I hadn’t. He mentioned that maybe I should, that it might be well received by the larger teaching community — specifically those at math ed conferences, like NCTM.

Michael and I spoke for a mere 5 minutes, but our conversation inspired me to write a post where I rethought who benefits math ed conferences. Need it only be the attendees? Can we math teachers leave something to a local community to engage with both during and after we leave their space? Can we leave the host city better than we found it mathematically? Though there are many ways to do this, I was thinking of hosting a session at conferences, gathering teachers, taking to the streets, and doing sidewalk math. It’s not going to change the world, but it could help move the needle a tad bit towards spreading math and including more people in conversations about math.

Last winter and spring, I began pitching my idea. I wanted to do sidewalk math at conferences. I didn’t know any better and didn’t really want to spend buckets of dollars traveling halfway across the country, so I proposed my idea to five local conferences. All but one was in NYC.

I guess Michael was right, there was a thirst for public displays of math. All of the conferences let me host a sidewalk math session. Yay!

The odds are pretty good that I’ll never be asked to present anything this much ever again, so now that it’s over, I want to bottle up some of my experiences over these last couple of months.

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June 6, 2019: EdxEdNYCThis was the first one and by far the most nerve-racking. Having never have presented at a conference before, I was deathly scared about having an adequate amount of material for the time slot. Plus, because I built in time to actually going outside to do sidewalk math in the local neighborhood, if it rained, I didn’t know what the hell I was going to do. I was so paranoid that I literally checked the weather every few hours in the days leading up to the conference. In the end, it didn’t rain and the timing worked out really well. I had a couple of former colleagues in the session, so it was also great to able to share it with them. It’s a small detail, but I made an effort to stand outside the room and greet folks as they walked in. I was not only hoping to welcome everyone to the space, but also ease my nerves by connecting, however briefly, to each attendee individually. It really worked. I made a point to do this at all of the upcoming workshops, too.

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July 10, 2019: Math for America Summer Think | This one felt the homiest to me. Also, these teachers were probably the most eager and able to do sidewalk math. There were about 20 people in the session, which was about half of the EdxEdNYC group. And as someone who has been part of MfA for several years, I knew more than a handful of the folks in the audience. There was even a couple of science teachers there, who so eloquently created some sidewalk science. All in all, we fanned out and chalked up the Flatiron district pretty good. There were definitely some sidewalk math problems put down that I pocketed for future use. Interestingly, I learned that the park rangers at Bryant Park are not fans of sidewalk chalk. Boo! Anyways, I was 2 for 2 on avoiding the rain.

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August 5, 2019: Museum of Math MOVES Conference | Of the five sessions, this one was the most contingent on the weather. The session had no inside component. Thankfully, it didn’t rain! (Up to this point, I couldn’t believe the luck I was having. The rain gods were sparing me.) We met outside in front of the Museum of Math and went to town. The Museum of Math is across from Bryant Park, so I didn’t dare bring my chalk inside of the park for fear of a garden hose. I spent the good part of an hour mathing up the sidewalk around the park. I put down close to 20 pieces. Ironically, I talked about sidewalk math to more strangers than I did attendees. This was pretty cool in its own right. One of them even tweeted about one of the problems.

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August 14, 2019: TMCNYC | This was where it all started! Bringing my chalk to BMCC in lower Manhattan was like coming home. The organizers managed to allot the session a full hour, which meant that we could take to the streets. There was a threat of rain during the afternoon, so Michael Pershan and I went out early and scouted for covered areas. Thankfully, it didn’t rain until the end of the day. I was now 4 for 4 and I was convinced that fate was on my side. I brought so much chalk that the sidewalk math continued into the second and third days of the conference. I mean these people brought it! I swear one of the teachers even had a makeshift lesson going on in front of her sidewalk math. On my last day, I went all in and created some bulletin board math at a local restaurant.

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September 26, 2019: NCTM Regional Boston | This was the grand-daddy of them all. It was the last one of the sequence and by far the largest. I was uncertain about how many people would elect to come to the session, but there ended up being over 100 people jammed into a windowless room that was desperate for fresh air. On top of that, it was the last session of the day. Called a “Burst,” it was only 30 minutes, which was nice because I didn’t really have to worry about rain since we wouldn’t have time to go outside. This turned out to be great because right after the session, Boston was flooded with the heaviest rain I have ever seen in my life. I’m most proud of the fact that no one left the session while I was presenting. When it comes down to it, that’s all you can really ask for. I was also happy that I decided to give away several boxes of sidewalk chalk during the session that were used to create some sidewalk math in the days following my presentation. I wish I had thought of this for the earlier sessions I had during the summer. Who doesn’t like free stuff?

 

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Haiku #5

As an alternative means of capturing my thoughts and reflections, I’ve been writing Haiku about my teaching practice. This is the fifth post in the series.

As my teaching has slowed through the years, I’ve been paying more attention to the furious pace with which new teachers experience their students, their pedagogy, and their practice. My awareness of these early-career teachers has matured a lot lately. And maybe I’m just getting old and doing what old people do, but I am feeling more responsible for these teachers these days — even those of whom I don’t work with directly. I love listening to them.

I had a recent conversation with a first-year teacher that struck me for a lot of reasons. It inspired this Haiku.

Teaching, you’re new here

A place where a week feels like

A lost lonely year

 

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