Author: samjshah

When I’m busy, tired, stressed, I turn to math

This has been a stressful few days. In my school, we have to write comments on each student (about 1/2 page single spaced) and those are due on Thursday. In addition, with the end of the third quarter, there has been a massive grading effort on my part to finish up the video projects I assigned. Plus, that hiring committee I was on took a ton of my time.

Still, even though I have bloodshot eyes and am crashing at random times during the day and evening, I couldn’t help but get addicted to working on Blinkdagger’s Monday Math Madness problem. I think I found a solution, but the great (and cruel) thing about optimization problems is that you can’t check to see if you have the best solution without proving that no other method will beat the method you came up with. Which I’m not going to do. But not knowing if I have the optimal solution is frustrating, but also exciting, because it’s a different experience to not know with certainty if I’ve finished the problem right or not.

I have to say that this problem, more than any other problem I’ve encountered in recent memory, has gotten my brain to work in such a different and crazy way that I have to recommend it to everyone. I think those in computer science (read: not me) will find it easier than I did.

With that said, the hardest part of this wasn’t actually coming up with my solution, but it was writing it in a way that could be understood by someone else. I wanted a student from, say, my high school math club to be able to follow the logic. But whether I succeeded or not, that’s a question that will have to wait until the contest is over and I post my solution.

With that said: check out the problem, test your mathematical mental mettle.

Thinking!

I love it when my students think for themselves.

When learning the law of sines and law of cosines — and when it’s appropriate to use one or the other — our textbook gives pretty prescriptive directions. For example, when you are given SSA (a side, a side, and an angle opposite one of those given sides), you are supposed to use the law of sines. And depending on these values, you can actually get two possible solutions!

Let’s work this out.

If in triangle ABC you’re given that side a=6, side b=8, and angle A is 40 degrees, then let’s solve the triangle.

Using the law of sines, \frac{6}{\sin (40)}=\frac{8}{\sin B}. Rearranging, we get \sin B=0.8571. But we know that sine is positive in quadrants I and II, so we get for B=58.99 or B=121.01. And hence, we have two possible angle values for B, which leads to two different triangles. [For a more detailed explanation, see here.] Another quick application of the law of sines yields that c=9.22 or c=3.04

And looking at the picture below (cribbed from the site above), you can see that both triangles are possible!

Here’s where student thinking is awesome. The book, as I said, says that everytime you have SSA you should use the law of sines. And I agree, it is easier. But it is definitely possible to use the law of cosines too, as one of my students pointed out to me.

Let’s do it:

6^2=8^2+c^2-2(8)(c)\cos(40)

This simplifies, with some rearranging, to the quadratic: c^2-12.26c+28=0. This can be solved to get the two values for c, which are c=9.22 or c=3.04.

I love that it works, and that the student insisted that we could do it. It might be slightly more work, but not that much more, and the exploration aspect is awesome. [1]

[1] An extension for a project for next year might be: how can we use the fact that we can generate a quadratic help us in determining when we have two possible triangles, one possible triangle, or no possible triangles.

Reorganizing Trigonometry

In my trigonometry classes, I decided to deviate from the textbook ordering of concepts. The other teacher, thankfully, was on board. (And next year, I want to tweak things even more.)

Our textbook presentation of trig starts out with reviewing right triangles, SOH CAH TOA, and in the same section, introduces the reciprocal trig functions (csc, sec, cot). It then goes into application problems, involving angle of elevation and depression. Finally, it throws all the hard but juicy stuff into the next section — a section that took me 4+ days to cover. It included introducing the concept of using trig for angles greater than 90 degrees (a VERY hard concept for kids to grasp), reference angles, quadrant analysis, and a variety of different types of problems that students are expected to do.

But then the book starts veering into radians (which I covered already this year, and next year, which I’ll postpone), the graphing of trig functions, and finally goes into the translation and stretching of these graphs.

I skipped this graphing work and an entire other chapter to get to the law of sines and the law of cosines. Um, yeah, hello? Let’s think about this:

Students start by learning that trig helps with right triangles and they do application problems. Then they learn how we can extend this trig work to angles greater than 90 degrees. Me thinks that it would be natural to show them how trig can help them with all triangles at this point – including obtuse triangles. (Importantly, the law of sines also tests a student’s understanding of reference angles.)

My students seem to find doing problems with the law of sines and cosines very tedious, yes, but they also love the grounding and concreteness of it. They told me that. Which makes me think it’s better to put that sort of thing at the beginning of our exploration of trig. And leave the more abstract discussions to later, when the basics are fixed in their minds.

Rethinking the textbook makes me feel good, because it means I’m paying attention to the flow of the subject, to how I’m presenting the topics, and what students are thinking as they learn trig. And it means I’ve started fulfilling one of the goals I set before school started, one that I decried just couldn’t happen my first year.

[1] For giggles, I want to share what the section after the law of sines and cosines is in the textbook: Graphing complex numbers on a complex plane. Can we all say “jarring transition”?

Students teaching students

In the past few weeks in my seventh grade pre-algebra class, we’ve been working on some hard problems involving inscribing circles in squares and squares in circles and, sometimes for good measure, playing around with equilateral triangles too. Radicals abound. And for the most part, I see them getting it.

But recently (for a number of reasons) I’m being a “teacher centered” teacher. I’m at the whiteboard explaining things or doing problems. I call on them from the whiteboard. I let the kids work and I walk around and check to see how they’re doing. But their eyes are always on me or their work [1]. Partly it’s because I know it works. And it doesn’t take a lot of time. (These kids pick up what we’re doing quite quickly… you can actually see their initial struggle, and their breakthrough… sometimes you more than see it… you hear it… “OHHHHHHHHHHH, Mr. Shah! It’s sooooo simple!”) [2]

But today I had an extra 10 minutes and so I had a student come up and take over the classroom. She was presenting the solution to a problem, and I gave her total control. She could call on students, or be at the board, or do whatever she needed to to explain the solution to the class. And with great poise, she strode up there and started asking good questions (“How do we know what that side length is?” “What’s the area of a circle?”) and adroitly led her classmates through the solution. The other students were into it: their hands waved in the air, eager to answer her, as they are eager to answer me. [3]

The cherry on top of the sundae? Once she finished, about 75% of the kids had their hands up wanting to present the next problem. They want to be teacher. Which made me happy to be their teacher. I’m sad I’m not going to be teaching seventh grade next year.

[1] Okay, that’s not entirely true. They are a collaborative class; they work with each other answering questions, running ideas of each other, and comparing answers to see if they’re on the right track.

[2] Of course, on the other — less happy — side, you’ll get to a topic where a few kids will proclaim their hatred for the subject at hand, at which case I feign (do I feign? or is it real?) pain that someone could say something so awful about something I’ve devoted my life to, and how I want to curl up in a corner and cry.

[3] I noticed that her teaching style was a lot like mine, which means that I’m rubbing off on them.

Advice from someone on the “other” other side

In my last hastily written post, I wrote about the process of interviewing from the side of the interviewers. That’s because I’ve been heavily immersed in it for the past few weeks. But it wasn’t so long ago that I was on the job market myself, writing personal statements, meeting myriad people, and giving my demo lessons. And as a first time teacher, I had no idea what to do or expect. I mean, I had my education classes, but they were ages ago (I went to grad school after getting certified to teach), and they didn’t cover any of the practical stuff on how to get a job.

Fortunately, my sister is in education — she teaches at an independent school in Massachusetts, and she went through the whole rigmarole and gave me loads of advice. She also put me in touch with a number of her teacher friends who I asked tons of questions to. I didn’t know the wide diversity of independent schools until I talked with them. Boarding? Day? Mixed boarding and day? How many classes is it normal to teach? How many different preps? What does a dorm parent do? etc.

I also did a lot of library/book research to find out salary information, how to write a good resume, and where to look to find public schools with openings (I’m certified in Massachusetts), and how to find independent schools with openings.

In other words, it wasn’t an easy process. But I learned a number of things along the way that I would recommend to anyone going on the job market. I think most of these mainly apply to independent schools, but I’m not sure…

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Advice from someone on the other side

This year I’ve been intimately involved with the hiring process. With two of our veteran (and funny, and quirkily snarky, and supremely excellent) teachers retiring and our department head moving on to bigger and better things, there have been a number of math openings to fill. I’ve now seen a lot of demo lessons, had lots of conversations over lunch about teachers, and read a fair share of resumes, recommendations, and personal statements.

In addition, there has been a reorganization of the Upper School administration, and we’re switching from two academic/social/disciplinary deans (one for 9th and 10th grade; one or 11th and 12th grade) and four grade level deans (that deal with attendance, keeping advisers on the same page, and lead the big grade level project) to four “everything” deans and one new “assistant head of the upper school” (assistant principal). These deans travel with the grade (so they work with the same kids each year) and deal with anything and everything regarding the students.

A consequence of this transition is the search for these “everything” deans and the assistant head position.

And I’m on the search committee for these candidates too.

So I’ve been really involved with watching demo lessons, asking interview questions, and evaluating. Because these hires affect me significantly.

Things I’ve learned, observations I’ve had, advice I can give… But this seems very specific to my school, or similar independent schools…

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