# Category Archives: Precalculus

# in which the unit circle gets its due and movement is noted

A colleague of mine who is teaching Precalculus this year asked about a Unit Circle Project that is mentioned in my outline from when I taught the course. I realized that it’s something I’d never written down as I learned it at a T3 conference in Seattle (from Rhonda Davis) but I didn’t get the handout due to the over-full session and the presenter does not exist online from what I can tell. Over the years I’ve done it my way and like to put it right near the start of Precalculus.

Students in the district see a pinch of trig in Geometry and maybe a small bit in Algebra 2 if they took honors, but Precalc is the real beginning for them. I’ve scribd a teacher’s guide for the Unit Circle Project and a student worksheet that goes with it for anyone interested.

In other news, I am now living on the East Coast in Maine and working as a mathematics consultant for a few groups (teaching jobs are not to be had here due the remote nature of my new location). Hopefully this will mean I’ll get to clear out and post several half-done blog entries over the next while. Unpacking comes first, though. And if anyone is going to NCTM Hartford, let me know as I also plan on attending.

# in which my mind is blown by polynomial long division

My love of polynomial long division is already documented, but now I’ve found more reason to love. To give some context, I am working on plans for my Intro to the Calculus Unit for my Precalc kiddos. I do it right at the end of the 1st semester followed by an Introduction to Statistics Unit because I want my Sophomores and Juniors to make an informed decision about whether to take AP Calculus or AP Statistics next year.

The Calc unit is fairly straightforward: here are some of the big ideas, hey lets learn some notations, oh noes division by zero?!, take some deep breaths it’s just a limit stop freaking out type of stuff. After doing some work with the delightful Chris Sangwin, I have chosen to play around with the following piece of information (and hopefully some geogebra modeling) to take polynomial long division to a new level in precalculus:

*Given a polynomial P with degree n ≥ 2, the remainer when P is divided by (x – a)² is the equation of the tangent line to P at x = a.*

Boo Yah. Go try a few examples–it’s rather fun. I am going to set up the kids w/ the long division and then have them graph the original equation in geogebra along with the remaider equation and then have them write down what they notice. I have yet to come up with anything practical as it’s not all that useful for finding extrema, but I need to think about it some more.

Oh, and please feel free to shoot me down if this is wrong. My working knowledge of Taylor Series is rough at best and that’s where this idea comes about.

# why polynomial long division is awesome

A week ago I had plans for Precalculus. I liked them alright, but I felt they could be better so I did what I always do when I need my brain to percolate over ideas: I start catching up on my reader. I ran across Jason Buell’s post about the IMPROVE model (you should go over there to read the details of that model if you’ve never heard of it). I then scrapped my plans and started re-writing them. That was Sunday night. *shakes fist at Jason and his inspirational posts*

Before we get into the week, let me be the first to say that I have no grandeous dreams of students going forth into the world and using their ability to long divide polynomials or sketch them from factored form by hand. Polynomial Long Division (henceforth referred to as PLD) is a way for me to do the following:

1. Check on and improve student ability to add/subtract positive and negative numbers as well as distribute them

2. Improve mathematical endurance (why yes, those last two problems may take up a whole page each)

3. Improve student understanding of how the equation, the factors and the graph work together with higher order polynomials

4. Create an opportunity for my studens to feel like mathematical Rock Stars

Let me explain that last one. It goes back to my drive to help students see themselves as being ‘good at math’. PLD is impressive looking. I mean, really impressive. Especially when you start with a 5th degree, have students figure out the zeros, create the factors and then do multiple rounds of long division in order to come up with the factored form which they then must graph by hand labeling all intercepts. Then they can check their work with the calculator. Yay for instant feedback.

At the start of last week the kids had only worked a day or two with PLD. By then end of the week, with some group assistance, they were all working through these massive problems. They’ll whine that the problem’s huge and takes up too much space, but they’ll do the whole thing and then just kind of sit back and stare at it with this look of ‘did i actually just do all that!?’ on their faces. There were lots of “got it!” mumblings while they worked. It’s not the crowing cheers you get with, say, WCYDWT problems. I liken it to the satisfaction of a job well done. Because what is the true hold up for a lot of students in math? The concepts, or the grammar? Is it that the kid can’t understand what’s going on, or that they are so hung up on the symbols and the difference between coefficients and exponents and plus’ and minus’ that their brain is spinning? I think it’s the latter much of the time that shuts off the student mind.

So here’s the worksheet given out to each table group. I put them in a plastic slip so the four students in each group would have to share. Maybe even, I dunno, *read* a problem to the group. Work *together*.

# WCYDWT Response – Boat in the River

a.k.a. Where is the boat? Find the river.

**Warning:** This is a 4-day lesson plan/description of how it went for my three precalc classes. In other words, it’s a bit long. Just letting you know.

**Intro: **All teachers have had horrible days. No matter how well planned you thought you were, some days are just epic fails. Technology can fail. Students having bad days can lash out. You can notice gaping holes in your Plan of Awesome™ as you are teaching it. All of the above have happened to me. This is not about one of those days.

I was fortunate enough to take part of Dan Meyer’s second DimDim session surrounding the book/movie Holes. Being able to experience live his style, seeing a video, and thinking about the questions that could be ask I found myself most interested in the variety of questions the participants came up with. So many of them could be followed to interesting mathematics and I was a bit sad to not have more time to do so. When the Holes session was over I realized there was a whole other DimDim session I missed out on and there was a boat and a river involved. Clearly I wasn’t checking my reader enough this summer. I blame[thank] PCMI. And Hiking. Lots of Hiking.

After actually watching the video and reading the post I realized there was no boat or river. Thanks to Jason Buell I was able to watch a screen capture of the whole thing and it set my brain ticking about having the audience propose the questions from the video. Dan had a virtual session full of teachers do it and most came up with “how long will it take Dan to get up the down escalator?” The question seemed obvious when you stopped the video as Dan’s a few seconds up the down escalator. Obvious to me as a trained educator. Apparently it wasn’t as obvious to a live session of high schoolers.

**Day 01**

The first day of school. I have three Precalc classes back to back at the end of the day. After the regular hullos from myself and my Teacher Candidate (TC, think student teacher but better trained and using a co-teaching model from the start of the year), we introduced them to Dan and then had them watch the video twice—stopping when Dan’s about 4sec up the down escalator. The first time they were instructed to watch and think about what type of questions they could ask. The second time we asked they jot down observations about the video to share with their table group. My classroom has eight 4-seat groups that we put a quarter sheet of paper on and asked that in 5 minutes each group write down the question they would ask and the information they would need to answer the question. This is my version of having people type their question and all hit enter at the same time.

During the discussion time we walked the room and checked in with groups. There were a few who admitted being baffled at what they were supposed to do so we clarified. There were groups who said they couldn’t think of a question so we pressed them on what they observed and when they couldn’t explain the why we suggested that maybe they did have a question. One group wrote down the name of the song and the artist under their question and was pleased with Dan’s taste in music. There was definitely some non-math chatting going on throughout this but the directions got them talking with their groups and sharing ideas, which would help to set class norms for the year. I’ve found that if I start them off discussing (even if it’s not 100% maths discussions) it is much easier to foster discussions for the rest of the year. I want my room to be a place they expect to have discourse and debates in and a room where they feel safe doing so. That is my Big Goal for the year.

After we collected the slips, I shuffled them while explaining that we would take a look at the questions as a class and see what similarity and differences the groups had. The following is a sample of what some of the quarter slips said from the three classes. Almost all of the questions were repeated in some form and the classes were fairly homogeneous with their questions.

-“If he stood on the escalator, would the escalator still be faster than the stairs?”

-“Was there a significant difference between how long it took him to go up the steps versus going down them?”

-“Why does he put headphones in?”

-“How fast was he going?”

-“How many beats per second are in the music?”

-“Was the time it took to reach the top different from the time it took to reach the bottom of the escalator? On the stairs?”

-“What speed is Dan going up the escalator?”

-“What was the speed of the escalator?”

-“Was his speed the same on all trials?”

-“How far did he walk?”

-“Where is the boat? Find the river.”

The speed question was most common followed by distance questions. Exactly one group out of 24 asked “How much time does it take to go up the down escalator?” One. It was not the result I was expecting.

The last question listed above was seconded by a lot of students and gave me a heads-up that they pay attention to file titles. I evaded by nonchalantly saying that there is a collection of math problems referred to as “boat in the river” problems and that the one in the video is just one version while silently praying none of them Google it and ruin my punch line. The rest of class was spent discussing the music and I was surprised at how few students caught on that Dan was walking to the beat after two showings. Two of the classes asked to see the video again and I obliged (love being able to do that). The classes agreed that he started at the same time from the top and bottom, but even though consensus had been reached that he was walking to a beat, a good portion of the classes didn’t think that meant Dan would end his stair walking at the same time. Some students proposed that you go faster going down the stairs than up so the Dan walking down the stairs would reach the bottom a bit before the Dan going up. Counter arguments were made about the beat of the music and then class ended so we shelved the discussion until the next day.

After school my TC and I sat down with the questions each class came up with and decided that since the big question we expected barely happened, we would have the classes answer their own questions in a logical order. We would start with figuring out the distance, move onto speed of Dan, then find the speed of the escalator, and then finish with how long it takes Dan to go up the down escalator. Before any of those questions could be answered we came up with a field trip for the next day to try and clarify the walking to the beat questions.

**Day 02**

I enjoy online videos a lot. I’m not a YouTube person as I find it cluttered with too much stuff I’m not interested it. I prescribe to Video Sift as it leans geeky and clips from QI are regularly tossed up and I ❤ Stephen Fry. While thinking about the escalator problem and walking to a rhythm, I was reminded of a clip from a few weeks ago billed as “Synchronized Walking Competition” (SFW, but the comments/ads may not be).

The day started with telling the students we needed to get some better visuals of walking to a beat going and that we would watch a video and then take a little field trip to a staircase. Students were highly unimpressed with the start of the video especially as the only intro I gave was asking them to check out how the gentlemen were all walking to the same rhythm and had trained to walk the same sized step. At about 1:50 into the clip they perked up and a few even applauded. I paused when they do the ‘synchronized sitting’ bit to keep class moving.

Next we laid out the plan for the field trip. We would run 3 trials with two students per trial where one would walk up the staircase and the other would walk down stepping to the beat of Daft Punk’s Around the World. My school is two stories and we are lucky enough to have a staircase wide enough that students could line the sides for a good view and still have enough space for the two walkers. Music volume was a bit of an issue, but we made it work. Next year I will have students snap to the beat to help everyone hear but not disturb other classes. I demoed what it would look like and then had ‘volunteers’ line up at the top and bottom (in one class we had more than enough offer to help, in another no one offered so ‘volunteers’ were selected). With an 8-count call out to make sure they would both start at the same time, the classes got a nice visual of what it means to walk to a beat and, outside of a few missed steps, every trial ended with the two students finishing at virtually the same time. When that didn’t happen, students were quick to point out “He started late!” or “She took an extra step!” As a bonus, I was able to corral a passing teacher in one class and a passing administrator in another class as ‘volunteers’ which the students found hilarious.

The field trip took a good 15 minutes out of class, but helped get all students on board with what the beat in the video means in terms of Dan’s walking pace. It’s also amusing to listen to students’ whispered excitement about taking a field trip in a math class and how they’ve never done such a thing before. I’ve found that by doing unusual things at the start of the year I get the students used to expecting me to be a bit odd and thus have more capital to try odd things without them resisting.

Back in the classroom we put all the question slips from the previous day that dealt with speed under the document camera (mmm, student ownership) and announced that we would be answering their questions and that Dan’s speed on the stair and on the escalator would be the first. Next we reviewed the video and asked if they wanted to focus on Dan going up or going down? Thankfully, multiple students pointed out with backup reasoning that it shouldn’t mapper which left me feeling positive about the field trip. They choose to focus on the ‘up’ portion of the video because you can actually see Dan’s feet when he hits the top.

Using the same quarter sheet system as on Day 1, we gave them a few minutes to discuss with their group what information they needed to calculate Dan’s speed and then collected. The responses fell into 4 categories:

-number of steps

-length of a single stair

-start and end times

-beats per minute

That last one is something we’d decided to save as a post-activity problem so we focused on the first three. Enough groups saw the extra-long landing in the middle of the stair that it was brought up before we showed the pictures of the steps. There were some great discussions here about accuracy and if it was okay to round off to the nearest inch or half inch. I enjoyed seeing who wanted it ‘perfect’ and who was okay with the rounding. One of the classes had multiple students asking for the more accurate read on the tape measurement so they could work with those numbers. Counting the number of steps was harder than it should have been due to a dying projector bulb but they were able to do it. At this point class was at an end so we assigned calculating the distance Dan walks up the stairs (not the escalator) as homework.

When students had left we planned out Day 3 with hopes of getting to the final question by the end of class. Ah, hopes.

**Day 03**

What we thought would be a 5-10 minute check of student work turned into an awesome conversation about distance, accuracy, and sharing of student ideas. We asked the groups to share their calculations and be ready in 5 minutes to provide their answer to the question “what distance did Dan walk up the stairs?” We called each table and wrote them all down on the board. We marked matching answers with matching symbols and asked the groups with the largest and smallest numbers to share their thinking.

Answered ranged from 200 inches (whoops, missed a button on the calculator) to slightly over 700 inches. Most landed around 530 inches. Two groups in two different classes added all the heights and all the lengths. Many used Pythagorean Theorem which caused some students to facepalm that they hadn’t. A few broke the stair into three pieces and added two diagonals and one horizontal together. Even had a student volunteer to come to the board to draw a picture. She was rewarded with whispers of “wow, nice” and head nods from other students followed by class agreement that that was probably the most accurate calculation on the board. Other students pointed out that her calculation is within a foot of their Pythagorean calculation that ignores the landing and just takes total horizontal and total vertical in one triangle. I made note that a bit more work led to a slightly more accurate number and that sometimes it’s a judgment call when to do more work and when to say good enough.

We moved onto the escalator distance calculations and had a bit of a struggle counting the number of steps. Watching my computer screen I’m pretty sure there are 29.5 steps, but one class went with 31, the second went with 32, and the third said 33. Again, the projector resolution was not up to the task. Students quickly pulled out Pythag and consensus was reached in each class for the distance Dan travels on the escalator.

Figuring out the times was something we knew would be a bit of an estimate. To try and get better accuracy, we split the class and had half look at the escalator and half look at the stairs. I asked how we should figure out the time.

Class: Take the average of our numbers!

Me: Which average?

Class: The average.

Me [confused voice]: Which average?

Class [getting annoyed]: THE average!

Me [confused voice]: Which one?

Class [convinced I have hearing damage]: the AVERAGE!

Me: Which. Average?

Student: The Mean!

Class [groans]: The Mean!

Me [sweetly]: Just checking.

Each group averaged their scores and my TC and I snagged their averages and put them on the board. And then class time was up so we sent them home with an assignment of calculating Dan’s speed on the stair and the escalator.

After school we poured over the differences between the data each class was working with and pounded our heads trying to smooth the connection between the escalator speeds and vectors that we wanted to make.

**Day 04**

Timing was thrown off by needing to get books checked out at the start of class. The process cost us over 15 minutes, but gave me time to chat with students in line and learn a bit more about my classes. Returning to class, we got to work summing up the speed calculations. The average between the classes put Dan’s speed up the stairs at around 40 in/sec and Dan’s speed up the escalator around 60 in/sec. When doing a quick start-of-class check in with the groups a student asked me if the math was supposed to be hard as he’d been working with s = d/t since middle school. I agreed that there are probably kids in elementary school that could calculate speed, but I wondered if they could explain the ‘why’. I reminded him of all the ‘why’ questions that had been posed the past three days and he agreed that no, his middle school self probably would not be able to justify as well as his current self. I also told him that part of the project was to get everyone used to talking with others in their group, sharing their ideas and calculations with the class, and helping me learn more about the personality of the individuals in the class and the class as a unit. He seemed appreciative of the plan and then asked if they would ever get to see the end of the video. I promised he would see it by the end of class. I heard a few muttered prayers of thanks from eavesdropping students in nearby groups. I added:

Me: “It’s a guy walking up the down escalator. Why do you want to see the end so badly?”

Students: “We just want to know what happens!”

They want an ending. They want closure. We haven’t even proposed guesses for how long it’s going to take to climb that escalator, but they want Dan to reach the top and to see him do so with their own two eyes. I let them know we have two more questions to answer before they get their wish.

On the front board is a summary of student calculations/estimates for distance, time, and speed for both the stair and the escalator. I throw down the first gauntlet while holding up the quarter slip one of them originally wrote it on: how fast is the escalator moving? Some students have total idk looks on their faces and others are muttering that it’s easy. We let them know they have about 5 minutes to work at their tables to find a solution and that they need to be ready to explain their answer as we will call on a few groups. I’ll note that since there are two of us in the room we’re able to give the groups good attention. We don’t offer a lot of help here—they have the information they need, they just have to think through the ‘why’ in a logical manner. Several of the groups I hit up first have a number, but are fumbling with the why. I push at getting them to say what they know and ask questions about where the numbers we have already calculated come from. I ask about our field trip. I have them remind me why he had headphones on. I sneak away when they start debating the wording of their argument.

When we have the groups share out (closer to 10 minutes later than 5) I am looking for the key point that Dan is walking the same pace on the stair and the escalator. Questioning techniques is definitely a weak spot of mine (as NBCT proved to me last year) so my brain is flying here. In the first Precalc class of the day after a few groups use some great language about pace and how if you remove the ‘Dan’ element from the ‘Dan’ + ‘Escalator’ speed you should have just the ‘escalator’, a student puts forth that she doesn’t think you can just subtract the numbers since the escalator steps are larger than the stair steps. Half of me freaking out about the monkey wrench she just threw into my head as I never thought about that question and I have a sneaking suspicion she’s right and the other half is cheering on her thoughts because it’s a great point to bring up for clarification.

Keeping the straightest face I can, I repeat that the step-height is different and then ask the class if that’s going to affect our current calculation. The class splits here between yes, no, and not quite understanding the problem. We let them talk it through for a few minutes. I interject a reminder to think about how we calculated speed. It’s noted that speed is *total* distance travelled over *total* time. The class concludes that the difference in step height has already been taken care of when we calculated the distance travelled. At the time I was with them, but there was nagging doubt.

Pondering the question now, if the escalator was frozen and Dan walked up it and the stair at the same pace, he would hit the top of the escalator first since he’s been travelling slightly further per beat. If you run the math, Dan is going about 14.8 in/step[beat] on the stairs and 17.4 in/step[beat] on the escalator. The quick and dirty subtraction rests on the assumption that the steps are the same size. The difference isn’t huge, but it’s there and something I’m going to have to account for next week. On the bright side, if my excel work is correct it explains why the students have an overestimate. I’m hitting myself upside the head for not considering this beforehand and being ready to help students think about the problem or propose it in any class that didn’t consider it. I owe that student some cookies.

We moved onto the last question: “How much time does it take to go up the down escalator?” There are only about 20 minutes left in class at this point and ending with showing the video answer would be sweet and is my goal. Students set to work and this is our chance to probe the thinking of some of the quieter students. As I check in with groups, when one student offers up a number I ask a different student at the table for the why. The Quiet Ones always seem shocked that I spoke to them, but every time I did this they were able to answer fully, if not very loudly. We fielded a few out-of-the-box ideas (some really want to use the beat/sec but I asked them to skip that idea for now as I wanted to follow up next week looking at that). In general, the groups took Dan’s stair speed (40ish in/sec), subtracted the escalator speed (20ish in/sec, it had been concluded earlier that the escalator moves the same speed up and down), and then divided the distance by the speed to get a time (around 24 sec).

Given the timing, my TC and I were able to chat with each group but we didn’t have a chance to have them share out and show the video solution so we chose the latter. Before starting the video we put on the board the guess the students came up with and I told the class to say bye to Dan (which they did). We also asked if they thought their time would be perfect (“No!”) and why (“We rounded!”).

We started the video from the beginning and when Dan started up the down escalator every set of eyes in the class was glued to the screen. At first there were mutters for him to go faster as they seemed to think he would go over 24 seconds. When it became clear he would finish before 24 seconds they called for him to slow down. When one class realized they were within two seconds, they cheered. They. Cheered. There were high-fives. Students were congratulating their group for being awesome. The bell rung. Great first week everyone. Have a good weekend.

**Bonus Round**

My school is a PLTW school and a film crew from South Korea was in the school filming for a documentary on different education practices in the US, Japan, and elsewhere. They have this entire lesson on film and I am trying to see if they will let me have a copy because that would be awesome. The camera guy went right up close and it made the kids both nervous and amused. I am curious to know what they thought of the classroom.

**End Notes**

No worksheets were used during this process. Where we ended up is where I hoped, but it took longer than I expected and didn’t take the path I expected (really, only one group is curious how long it’ll take him to get up the down escalator? Really?). A worksheet would have locked us into certain expectations and I didn’t feel it necessary. I plan on collecting the scratch work students did (if they have it), but I didn’t demand any records be kept. Lastly, it was interesting to walk around and see how different students organized their work—the carefully organized with full sentences versus the vague sketched out numbers in random places on a page.

Will I do this again? Yes. I had fun discussing the problems with students and I felt it had a lot of entry points for students of all levels. Everyone has been on (or at least seen) an escalator. Speed = distance/time is straightforward. Pythagorean Theorem is comfortable.

**Next Steps**

Though we know it’s not a perfect fit, we’ll be using the ideas students discussed surrounding adding and subtracting the speeds to intro vectors and we’ll even have them doing a “boat in the river” problem to kick things off. Vectors will lead into triangles and angles which will lead into Trigonometry. Trig is the main goal, but vectors are nice because (a) a lot of these kids are in physics or will take it in the next year or two and, (b) historically students suck at notation and vector notation will help prime them to pay attention to details (I hope).

**Lastly**

If you have any ideas/comments/questions/quandries, please feel free to post them below. I have some pondering to do about this lesson, but I need some distance from it first and commentary will be super-helpful to look back on.

Big thanks to Dan Meyer for the awesome video. Thanks to everyone who was in the DimDim session that I’ve watched more times than is probably healthy. Thanks to Jason for the screencast. And thanks to my TC for being awesome and supporting my sometimes crazy ideas and being way better versed in vectors than I am.