Intro to Instantaneous Rates

I've been getting really good teaching mileage out of my Pringles cannon video. First, I taught my H. Geometry students how to graph that parabola in the calculator and how to use the graphing calc to find and analyze the maximum. (Since I had to teach everything from scratch to my Precalc kids this year, I'm extra cognizant that if I can squeeze extra algebra/calculator skills into earlier math classes in context, the better off they'll be in future years. Plus, it was actually a video from an activity they did in class, so I figured it's as non-pseudocontextual as it gets.)

They loved it! So far, they can use their calculator to: graph a function (or multiple), find intersection of graphs, find min/max, interpret the min/max given a simple situation, and adjust window ranges. Not a bad basis to have before going into Alg2 next year. We've also reviewed fractional math and reviewed solving systems of equations in context of finding non-integer circumcenters, so I feel like I'm doing an OK job preparing them for the algebra that is to come. For these kids, I also hope to get to maximizing volume problems before the end of the year, in between teaching them proofs and doing 3-D stuff, which I'll be doing with all of my regular students as well.

Anyway, on Friday I introduced the idea of instantaneous and average rates to my Precalc kids using the same video*. The Do Now was a simple review of speed and acceleration terms and making tables of values based on a simple description. (As usual, I was surprised that some of them couldn't do a problem like, "What is the average acceleration per second for a car that starts off at rest and reaches 60mph in 3 seconds?") Then, I reviewed with them how to set up the quadratic equation for projectiles before watching the Pringles cannon video*. They noted that the ball was in the air between t=4 and t=9, so I had them graph the function H(t) = -4.9(t-4)(t-9) in their calculator, and they then used the tables in the graphing calculator to fill out this table below. For t values, they used t=4, t=4.5, t=5, etc. ...going up in half-second increments, all the way through t=9. I had to guide them through how to do the averages, while they worked in pairs.



Once they filled out the tables, natural questions arose as they compared their results for the last two columns. I went around to probe why each pair of kids thought that the half-second "speed" (I guess I should have said velocity, to be more precise) is decreasing and becomes negative in the right-most column. They were excited to be able to figure out why all on their own! It's physics in action!! And then, as a class we discussed why the "average speed since launch" actually becomes zero the moment that you land on the ground. One kid raised his hand and explained to the class that it's because your overall change is zero at that point, so therefore your average speed is zero. --Awesome!! In our discussions, I used the term "instantaneous" loosely to describe what's been happening in the last 0.5 seconds (4th column), to distinguish it from what's been happening since the beginning (3rd column).

It was awesome. Later on I drew a graph on the board like this and asked them to describe the "speeds" between different points AB, CD, and AD. The class comfortably told me that AB has positive speed, CD has negative speed, and AD has zero speed. Good conceptual basis for next week, when we move into the actual algebra of finding instantaneous and average rates! I hope that this activity will "stick" with them, so that when they look at graphs in the future to analyze rates, it won't just seem like some abstract concept.


* Most of them had built and shot their own Pringles cannons in physics this year, so I felt like it was OK to just use the video instead of wasting time going outside to do the same thing again.

PS. I suppose I should mention that Obama is coming to El Salvador. School's not in session on Tuesday and Wednesday for street closure reasons. Selfishly, I feel a bit of relief that this comes 2 days before quarter 3 grades are due, even though it means that there is some instruction time lost, obviously.

Math Writing Project Brainstorm

In keeping with the theme of reinforcing language skills in all classes, I have been brainstorming ways to implement a writing project in Geometry this year. The only real research/writing project I have ever assigned was a long time ago, and even though some of them had hated the idea of writing in a math class, I was adamant that they needed to be practicing research and writing skills in all classes -- including mine. I think it's about time to do another writing project with my current group of students, but I just need to figure out how.

A history teacher at our school recently did a scrapbook project, the results of which are beautiful and are on display at the library at the school. Here's the description of the project, as sent out by our awesome head librarian:

"After reading historical fiction novels, tenth grade Latin American History students crafted together scrapbooks that included a historical analysis of the novel, student artwork based on the novel, creative writing, documentary evidence, and literary analysis."

I went and took a peek at those projects. Each page was beautifully pop-uppy and 3-D, with tons of student writing to boot! Can we do something like this for math? Can I make my kids research about fractals, Pythagoras, or trigonometry (both the history and the modern applications) and to write about it and to illustrate their work? What if we made a single-issue Geometry magazine as a class? (I have at least one kid I know who is in Journalism and is supposed to be a good editor.) Would we have enough non-duplicate material to include in order to making this a success?

Thoughts? Have you ever done something like this in your classes?

Flexibility

If there was one thing that I wish I could do better with inside the classroom, it's to bring in the sense of play into more lessons. How do you do that consistently? Math, to me, is such a beautiful subject, because it's mostly a series of puzzles, one wrapped inside another. In order to solve those puzzles, you need to have some pertinent skills and some base knowledge, and to be able to use them flexibly.

But, how do I introduce that sense of play on a daily basis? I can't help but feel that I should be doing more of it in Geometry, since Geometry is such a visual subject.

I've noticed time and again that there are problems that showcase how even many of my honors Geometry kids lack basic intuition when looking at a diagram, and they end up way over-complicating the situation. Take a look at the following problem, given on my most recent H. Geometry exam. The directions were deceptively simple -- to find the area of the concave quadrilateral ABCD.



I made this problem and envisioned that many of the kids would quickly get it solved, after our foray into quadrilateral areas. To hint at the fact that they didn't need to cut ABCD further into smaller parts, I even led into the problem with, "Given that Area of ABCD + Area of ADC = Area of ABC, find the area of Quadrilateral ABCD." ...A dead giveaway?!

Anyway, perhaps predictably, some of the kids struggled on this problem. (To their credit, many others did brilliantly, including a girl who was absent for several days during the unit and had to catch up belatedly on all of the heavy-duty trig content.) Mostly because those kids made some false assumptions, such as assuming that Segment BD would bisect Angle ABC. But, even some others who were able to solve the problem correctly took some detours to get there, such as cutting ABCD up into a right triangle and a scalene triangle. It's clear to me that most of them still lack the ability to zoom in and zoom back out on a diagram -- which is what Geometry is all about!!

I've certainly given them "similar" problems to struggle through in class, but especially as honors students, I also expect them to have a level of ability to apply that knowledge to new situations, on the day of a test. But, how do I teach that flexibility?

Anyway, I really feel like I should be doing a better job helping them develop a better sense of spatial intuition. I just don't know how. :(

Course Advisory

I'm advising kids on courses to take for next year, and here are some thoughts on my mind:

* If a kid is getting a high 70s grade currently in my honors class, would that kid benefit from dropping down to regular math and getting more in-class reinforcement / practice? (We can only do so much practice of the same thing in honors class before all the other kids get bored.) What if that kid enjoys the pacing and rigor of an honors class, but just has trouble mastering everything at that pace?

* If a kid is consistently acing my regular math class, but isn't much of an adventure-seeker in math, would they be suitable for honors?

* If a few kids who are pretty sharp in honors geometry wish to take Algebra 2 over the summer so that they can "move along" on the track and end up taking Calc BC in their senior year, is that a bad idea (or is that just my old-school opinion that no one should be squeezing a year's worth of algebra foundation into 6 weeks)?

* If our policy is that any senior who fails a course won't graduate on time, how can I encourage my struggling juniors to take non-AP Calculus next year to improve their college-readiness?

Thoughts or recommendations? I worry about the best placement for every kid, because long after I am gone from this school I would still want them to be properly challenged and to be able to enjoy math at the same time.

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Addendum: The kids are coming around to agreeing that they shouldn't be squeezing Alg2 into a 6-week summer course. It helped that I said, "This other math teacher who graduated from MIT and teaches at a very prestigious private school in the States thinks it's a bad idea," and it also helped that my former Alg2H kids (now they're in Precalc honors) walked in on the conversation with one kid, and they collectively shook their heads to say that it's a BAAAAAD idea and that everything you see in Alg2, you will need in Precalc. Also, one of my current Geometry Honors kids told his friends, "Remember how so-and-so took Geometry over the summer to skip ahead? I gave him an easy -- facilisimo! -- Geometry problem the other day, and he couldn't do it. Taking math in summer school's a bad idea!"

On Raising the Bar for Tenure

I feel very emotionally affected by the mess that is the US* politics of education. I hate the thought that I could be forced to stay in international education for years because my job in the States would be too unstable once I re-enter the public education sector. I got into the business to help kids, and although there are kids who need help everywhere in the world, the places where I can make the most direct impact are in inner-city schools where I could potentially really change some of their lives. And I would like to think that at some point in the future, regardless of where I've been or what I've learned, I would return to do my part. And I would also like to think that I would be able to find an urban school with like-minded teachers and administrators -- one where I could potentially stay happily for a long, long time.

And God knows that when I do find myself in such a situation, I wouldn't want to be additionally dealing with the uncertainty of whether I would be able to keep my job the next year. In enforcing unpopular policies, the States* is definitely not encouraging teachers like me who have the option to work elsewhere, to return to the States. (And I'm only a young-ish 5th-year teacher; think about those who are much more qualified/experienced than me, but who have a family to raise and therefore have much more to lose. Would you return to the States right now with your family amid such a mess, in hopes of saving the world, one child at a time?)

All of that is a preamble to this: I really resonated with this blog post, which I think offers some great suggestions for balancing how much we protect young teachers versus how much we protect the more experienced teachers. If we raise the bar for receiving tenure, then we would less likely have to get rid of qualified teachers from one school simply because we need to find places for other tenured teachers. It would give everyone hope for working towards that level of job security, by actually doing a good job.

Am I just being naive and hopeful that there could actually be a solution?

*Obviously, the policies vary per state, but it seems like it's the same problem all over -- not enough union or too much union. Either way, as a young teacher but also someone who intends on sticking around in a system for the long haul, I would be screwed either one way or the other.

Learning from My Mistakes

Both last year and this year, I have taught various methods to finding the circumcenter of a triangle. The reasoning is this:

1. It's interesting. Circumcenters are equidistant from original vertices A, B, C, so they give rise to certain problems such as "where is the best location for placing a new hospital/communication tower?"

2. The geometry of circumcenters is beautiful; it can be found by folding each point on top of each other point* (thereby creating 3 perpendicular bisectors kinesthetically), and afterwards its location can be verified by drawing a circumscribed circle that goes through all original points. This also helps to reinforce the circular property of equidistance.

*And this year, since my kids were the ones that came up with the "folding" algorithm, it makes total sense to them why that line is the locus of equidistant points from vertices, and why the circumcenter must lie on the point of concurrency of all those perpendicular bisectors.

3. The algebra of perpendicular bisectors is a nice way to loop back to line equations, midpoints, and perpendicular slopes. For the more advanced geometry kids, the algebra of finding a non-integer coordinate circumcenter also brings back systems of equations algebra, since that is the point (x, y) that needs to satisfy all perpendicular bisector equations.

But, last year my students had a lot of trouble with the algebra part. This year, we're done with the kinesthetic part and they don't seem to have any trouble with the overall concept. We started looking at the algebra, and originally I started it the same way I did last year -- by running them through the list of properties that a perpendicular bisector should have, and using those properties to help us write the equation -- which sounds good in theory. These regular Geometry kids can follow conceptually what I'm saying (since they have a strong conceptual understanding of circumcenters through our various activities and demos), but then when I let them follow up on the algebra example by doing one of their own, all hell broke loose.

Naturally, I thought in my head: I need to back the heck up!!

So, I made the following worksheet for them yesterday (ouch, algebra on a Friday!), and it went really well. The worksheet had different problems of varying difficulty; initially I would give them either the original slope or the midpoint already found (or both), and expect them to find the missing pieces and to find / graph the perpendicular bisector equation. (They graphed to check visually whether their perpendicular bisector equation "looked right" relative to the original segment; I refused to tell them whether or not their equations were correct on the first page.) Then, the worksheet scaffold up to them doing the whole process of perpendicular bisectors by themselves, and finally to finding the circumcenters using the intersections of those graphed perpendicular bisectors.





I found that by breaking it up like this into little pieces, it finally started to make sense to kids and they were able to "see" how the perpendicular bisectors, once they had finally understood how to find them, would lead them graphically to the circumcenters. (To be sure, these regular kids were having a lot of basic algebra issues. This exercise also helps them to zoom in on those, because it makes it relatively easy to figure out which part your mistake must be coming from, if half of the problem was already done for you. Because I'm trying to be less helpful and to force their independence, I also told them they needed to find their own algebra mistakes and not rely on me. Only very seldom would I help a kid diagnose that their midpoint wasn't correct, for example, by asking the kid to find that point on the graph and telling me whether that location appeared to be the correct midpoint.)

--Score! ...You know, it's funny. These are the traps I should have been able to avoid even as a first-time Geometry teacher last year. You can't teach kids a whole algebra process at once, even IF they already have the conceptual foundation. They need to already have a solid understanding of the individual algebra pieces, before they can start to put the whole process together end-to-end. It's something that I always forget the first time I try to teach something. But for some reason, it never occurred to me last year that this was what I was doing wrong. More practice doesn't automatically lead to more understanding; better and more thoughtful practice leads to more understanding!!

A Soft Approach to Holding Kids Accountable for Learning from their Tests

I know that a lot of you do the SBG thing so you don't believe in test corrections. Well, I have to admit that I've never been sold on it one way or the other. I've always thought that my job as a teacher is to create as many opportunities as possible for apathetic kids to learn. That means creating insanely accessible and stimulating and (as much as possible) not boring lessons. It also means giving them homework and checking it off for completeness, in order to encourage kids to stay on top of each topic. That also means consistently giving kids practice quizzes and tests with numeric solutions and time provided in class for questions / feedback, so that the material on the upcoming assessment seems concrete and doable for every kid. It also means allowing them to do corrections on quizzes and tests for partial credit, so that they'd go back over their own mistakes. And it means making videos and uploading them online, even if I'm not sure how many kids are going to be using them.

But, all this, for what?? NOT because I'm trying to reward kids for some sort of "nice" behavior in some sort of points game called school. I do it because the points don't matter to me; I don't care if kids "win" and come out of the other end averaging 70s or 80s or 90s in my class, as long as it means that I've successfully tricked them into doing hard work for me the entire time. I don't care if they got some buffer points for homework, if that means that their overall understanding went up and we can get to some trickier material during the unit. I do those things because I'll do whatever it takes for them to be thinking hard daily in my class. I do it because I want to create as many opportunities as possible for them to learn; it's like I'm leading the horse to the water. Quiz and test corrections, for example -- why are you against assigning points to them? Is it because you think that when kids correct their tests at home, it doesn't show real mastery? For me, I allow them to do corrections because I want to give them an extra reason to be sitting down with their quiz or test, and that incentive had better beat out all of the social network and Blackberry temptations. If they're irresponsible enough to copy off of their friend's quiz or test answers without even bothering to look and try to somewhat understand the answer -- well, that lack of effort will come back to bite them later on in my class, and in life. You can be sure of that. (I think of it as a karma of intellect. I help enforce the karma of intellect by making sure that every week in a unit, the level of difficulty of the material is ramping UP in my class. If a kid is happy with a 60% on a couple of quizzes and doesn't try very hard to look them over, he or she'll definitely get a 40% or lower on the test. Almost guaranteed.)

But, anyway, all of that said, I admit that quiz and test corrections are not bullet-proof. They're just a way to encourage all kids to remediate the material without risking lowering their grade further. (Most of my truly struggling kids are NOT confident enough to show up for a re-test, even if they might be able to do better the second time around.)

This week though, I think I've found the perfect complement to this test corrections thing. The kid has to come and explain the work to me, line by line. I swear, I did that with my 11th-graders (2/3 of whom had failed the last test, remember?), and each one of them rattled on beautifully about math for 15 minutes while walking me through their self-corrected 2-day-long chapter test. I would periodically stop each of them to ask questions, and for the most part I was extremely satisfied with their improved understanding. One kid said after I accepted his corrections, "I learned a lot from this!"

So, I don't care that the kid got those points through corrections. The worth is in them sitting down with me and explaining every problem. Even if they might not remember all of this material a month from now, they'll remember the sense of confidence that came with their few days of hard work in correcting this exam, and that'll do good to their relationship with math.

So, I'd say that my chat with my juniors is working. In fact, I've never been prouder of them since August! :)