July 13, 2014

Parametric Functions

It's summer! I can't wait to share all the awesome things I've been up to at PCMI, but first I'm going back to my final unit in PreCalculus which I never got a chance to share. Last year we did an exploration of polar functions but ran out of time to look into parametrics. This year when I asked the calculus teacher what her preference was she said to focus on parametric instead. I chatted with several people on twitter, someone (@dandersod I believe) showed me how to graph parametrics on Desmos and my colleague* shared her awesome materials with me. Put it all together and I got this:

Kids finish a test and the instructions on the board say:
1. Pick up the assignment papers
2. Sign out an iPad
3. Work silently (other people are testing!)

On the desks where I've spread out the assignment papers I write Introduction above the first page (in dry erase marker because I have dry erase desks) and Choose One above the remaining three pages. Most students finish the introduction (about half an hour) so I assign as homework completing that page and making a first attempt at the other assignment. The following class we discuss the intro, why parametrics exist and then they spend the rest of the period working on the context they chose. I like projects where there are similar options because students can still have conversations (they all have to graph and calculate) but each of them has to do their own work.

Parametric Intro

It was important to begin graphing by hand so students had an understanding of how parametrics work. Some students were concerned that the t value wasn't showing up on the graph and tried to include it in some rather creative ways. I've edited the instructions slightly (should've thought to post the original to ask for feedback on the adjustments...) so hopefully that will be a less prevalent error. Other students picked strange values for the second set of equations, ah radians. Two thoughts: 1) I'm glad I was able to incorporate a trig function since we hadn't used them much since first semester 2) I love graphing utilities - I was able to say, "Okay, you're not really sure what this graph was supposed to look like, that's fine. Graph it on Desmos and see what happens!" My box of helpful hints on how to type things into Desmos wasn't as visible in the first version, many kids skipped those steps. They were also unclear on what to type exactly as written and what to substitute with other information. Turns out (-4+3t, 1+2t) graphs just as well as f(t)=-4+3t, g(t)=1+2t and (f(t), g(t)), but I like the way we found to use Desmos as it shows that each value of t gives an (x,y) coordinate a bit clearer.

Choose One:

These activities are designed to be done on Desmos as well. Some students didn't appreciate when I wouldn't help them with technical issues until they got out the intro sheet and put it side by side with their iPad. But, they were able to correct their own issues that way so it was worth the eye rolling. They also struggled with graphing the obstacle/hoop. In response to those questions I asked them, "Where is the obstacle/hoop?" And continued asking variations on that question until they told me "At x=__" At which point I responded, "So type x=__ on the next line." Then I pointed to the next line of instructions (how to restrict the range) and walked away.

Great things:
These context based questions require students to continuously switch among equations, graph and description. They have to know what t represents and what it means to land as well as solve quadratics and estimate values on a graph.
A student asked me if the equation took gravity into account. What an awesome question! I was proud that I remembered my physics to point out the -16t^2 (this is feet based physics, I remember 9.8 for meters even more clearly).
In my opinion this assignment shows why parametrics are useful - you can know horizontal distance, vertical distance and time using one set of equations. I failed to successfully convince my students that this was amazing. They obediently wrote that down and that the parameter allows them to restrict the function. But neither of these facts were impressive to them. Thoughts on how to convince students that parametrics are useful and different?

*Colleague O'Malley - I've yet to convince her to jump into our awesome online math community but she does recognize its power and occasionally asks me to ask twitter questions on her behalf. She found the equations and contexts in McDougal Littell Algebra 2 and then wrote up projects for TI. I only had to modify a bit so we could use Desmos. She very generously allowed me to share with all of you!

June 4, 2014

Trigonometry Updates

When I went to meet with the consultants I brought them materials from my trigonometry unit. they provided me with a variety of suggestions and we had a good conversation. Now that I've taught the unit I have some thoughts on what worked and what didn't.

Make a template for kids to fill out for every trig problem they do.
It was really useful, but I neglected to consider how big kid handwriting in dry erase marker is.
Remake the template so it only has the triangle to label, ratios and cues to solve and check.

Version I used this year (print on paper, slide into a plastic sheet protector, instant dry erase template!)

Possible improvement

Have kids write out all the ratios, then pick the one that's easiest to solve.
It takes forever for them to fill in six ratios. All my students struggled with understanding opposite vs. adjacent at the beginning (even when I had them place a finger over the angle in question) and switching between the two angles was rough.
Undecided. I wanted to believe that with more practice the ratios would come faster but with the number of interruptions we've had lately it hasn't happened yet for many kids. I do want to do all six a few times and have kids notice that sin/cos all have the same denominator (hypotenuse doesn't depend on angle), that sin(A)=cos(B) and that tan(A) is the reciprocal of tan(B). After that discussion I'm not sure it's worth the time. There's value in looking at the information given, looking at the possible equations and choosing which one to set up and solve. I may give kids the option to do that in the future.

Have kids solve for all three side lengths using trig, then hand the paper to a partner to check using the Pythagorean Theorem. Kids won't want to check their own work but they'll happily check each others.
Kids don't finish at the same time. Waiting and interrupting are both bad options.
Some kids wanted to check their own work, which was great. Other kids I directed toward any student who wasn't working (not necessarily their partner), this gave kids who were frustrated with the spacial or algebraic demands a chance to do some computations. A few kids I checked for them. I like checking with the Pythagorean Theorem because we get to talk about equal vs. close. Next time I'll start with kids checking their own work and if anyone is resistant I'll offer the other options.

Having a three day weekend and state testing during the middle of our unit on trig was a problem. There were a lot of interruptions and with a block schedule they only have me every other day to start. It was hard to build any sort of automaticity. Such is life.

I also updated the trig intro investigation. That went well. (Original post has more detail.)

May 23, 2014

Students Who Are Behind

Students who are behind are a big problem at my school. Next year we will have more levels of algebra 1 than I can count. Seriously - it's after 9 pm on a Friday so keeping track requires more brain power than I have; counting levels of a course shouldn't require brain power. We have a tracking problem, but we also have a money problem. Kids who would be placed in specialized schools in most districts are kept in house because it's cheaper. When people talk about mixed level classes I think how wonderful that sounds. But the reality is that we are teaching kids with significant learning differences and kids who are years behind. I don't know what the right thing to do is. Some days I complain that it's torture to make certain kids take the state test. Other days I complain that no one bothered to teach math to our autistic kids (who are fully capable of learning math). I know these are conflicting arguments. I don't know who should be deciding which kids get which placement and what they should learn in each placement. The school committee decided that every 9th grader should be taking algebra 1. No matter what. So I taught those kids who hadn't done math "algebra 1" but that had to look different than the mainstream algebra 1 courses. To start addressing this issue my department head shared an excerpt from Learning in the Fast LaneChapter 1 is free to read online. This one is going on my ever growing list of things to look at 'later.' Whenever that may be.

Today marks 30 days in a row of posting. Expect a drop in frequency around here shortly. Still a few more things to share this weekend before I get lost in end of school stuff. June tends to simultaneously fly and drag...

May 22, 2014

Human Knot Reasoning

Yesterday was our last advisory of the year. We had advisory once or twice a month for an hour. We started this schedule in June of last year and it's had some pluses and minuses. At the beginning of the period I had students reflect on their year - in general, in school and in advisory. I'll probably read and post about them sometime this weekend. Then we played games including pictionary, charades and at the end of the block, the human knot.

There was some interesting reasoning occurring as they worked to untangle themselves in the human knot.

There's an odd number of hands!
They were making a joke when someone couldn't find a hand to grab onto. Love math jokes!

At the end the order will go: me, then A, then B, then C. 
This was at the very beginning where everyone was in a complete tangle. She was tracing the path and envisioning the end result. Which also helped her consider the consequence of moving - since every time she moved she had to drag A, B and C along behind her.

Is it always solvable?
I shared a few examples of results I've seen - one circle, two separate loops, interlocking loops - I wonder what their definition of solvable is. Sadly the bell rang before they were able to untangle and see what kind of result they would have. It was a really tough knot because they were working together and listening to kids outside the knot who had a clearer perspective and it was still slow progress.

I got a few photos and another student took a video that will be fun to share at our first advisory next year (they're sophomores now so we get to spend two more years together). Maybe next year we'll solve a knot with everyone participating!

May 21, 2014

Simplifying Looking Back

Part of today's department meeting was getting together with our content groups to reflect on the year. Last year we made curriculum maps for Algebra and Geometry and this year we tried to stick to them, to varying degrees of success. We wanted to decide what units to move, which to shorten and which to lengthen. We'd been having this conversation throughout the year but today was the day to hash out the details. Hashing out details that included first quarter is difficult to do. Luckily I've developed a system where I save my smartboard files for each class in a public google drive folder with the date. The goal is for kids to have a place to check if they're absent or forget to write down the homework or need help with their homework. But it worked great today when we needed to figure out how long we'd spent on each unit.

All of my geometry notes for the year, in order, in one place.

I sent an email to the high school tech person after I found out about Classrooms because I want in, and more than that, I want us to switch over to google apps for education because it would be soo much easier than what we currently have. We use FirstClass for email and last month got an email telling us that we wouldn't be able to email anyone outside of our domain for a while. There was a spam issue they couldn't solve. Teachers are starting to use Drive and expect kids to submit assignments online, but kids don't have email addresses through the school and don't have google accounts. I have to wonder if the district used google apps, would the students look back at their google drive as they studied for finals? That could be a cool assignment that I can technically do because my drive is public... Some of my classes are small enough that each kid would get a month to summarize. I'm going to ponder this idea further...