3d!!!

I am pretty concerned about this chapter on surface area and volume, some students are really struggling.

I spoke to some of the other math teachers about it and they made some interesting points.

Some of the students have a hard time understanding the line drawings as 3-d objects. To me that comes so natural, it hadn't even crossed my mind that they might have that issue. But now that they mention it I can see their point: it may be very confusing to look at a series of black lines on a page and try to move that into 3-d space, especially when you can't rotate the drawing or see it from different angles. And the real kicker is I don't know how to explain it: how you teach someone to visualize a 3-d object based on a line drawing? How you do you lead someone to "see" something, or form a mental image?

Also, composite objects are always a problem. A shape made up of a cylinder, a block, and a hemisphere will freak them out every time.

The other teachers suggested using lots (and lots) of real 3-d models in class. And to work at it 'backwards': instead of starting with a composite object and pulling it apart into simpler shapes, start with simpler shapes. Get them to buy into the idea that this one has volume V and that one has volume Q. What's the volume of both of them together? V + Q. What if I physically stick them together, now what's the volume? V + Q. What if I move the pieces around and stick them together in a different configuration? Etc.

I tried this today and it had some success. More people seemed to be getting it. I took a cyclindrical can of wipes and taped two hemispheres to either end (rougly same radius) so I had a capsule shape, and we talked about finding the volume and surface area.

Which brought up another tricky point: with volume you can just break into parts and add: Part 1 + part 2 + part 3. But for surface area it's a mess because when you stick parts together, things that were on the surface are no longer on the surface. One student in particular was having such a hard time with this. They could not understand why the lid and bottom of the can of wipes didn't count towards the surface area of the whole capsule (probably didn't help that the hemispheres were see-through, so you could still actually see the bases of the cylinder when they whole capsule was assembled.

The good news in all this is I am feeling more free to slow down. We have a final at the end of May and this is the last chapter of really new material. The next chapter is more prep for the SAT (review) and prep for the final. So I feel like I can lose a day and just slow down to review things, play with models, try to dig deeper into existing subjects. If we were in the middle of the year I might have been more uptight about spending so much time today on just the one capsule problem, but I think it was time well spent.

Because I wanted to spend time on the model business, I was trying to rush in just presenting the new topics: surface area and volume of sphere. I was really tempted to just give them the formulae and call it day, no explaination of why it's true. This class in particular seems to not really care "why". When I covered surface area I gave a handwavy explaination that you can imagine peeling the cover off a baseball, it makes two peanuts, each lobe of each peanut is roughly a circle of radius r, total is 4 * pi * r^2. When I got to volume of a cube (4/3 * pi * r ^3), I didn't even bother with a 'why' because the 'why' is pretty complicated. I thought they wouldn't care.

Then this one student, who has really struggled and generally been pretty volatile at times, raises their hand with a big smile and says "Why is that"? They really seemed to want to know. So I took a few minutes to explain it all. Not sure it registered but it was touching to me that that student in particular proved me wrong: they are curious.

Inertia

The new chapter has to do with finding the surface area of geometric solids. Started with prisms, and that involves finding the area of the polygon at the base.

We had several lessons on finding the area of polygons back in Chapter 5 (this is chapter 10).

I was surprised at how they handled this on the first set of homework. Maybe not so much surprised as disheartened. On so many papers I see the evidence of the same pattern:

* I don't remember how to find the area of a polygon.
* It's too much work to look back in the book or at earlier notes.
* I will just copy the answer out of the back of the book (evident because the entire answer is just a number, with no indication they did any actual calculations).

The first point is a bummer but there you go: for those who approach this as "memorize random facts without really understanding why", I can understand that those facts will atrophy pretty quick.

But the second and third ones are even more discouraging. Rather than do even a little work, I will resort to trickery. This homework problem is worth a tiny tiny portion of my grade, but I'm going to take a shortcut here that guarantees I won't really understand anything.

Grumble grumble.

I am very tempted to give a quiz, just to the ones who clearly did the copying, where I ask them the exact same questions.

Rounding

I just graded 2/3 of the trig tests. So far so good: scores are generally higher than they have been. And in some cases, students who have really been struggling got A's, which is very gratifying.

But...

There was a lot of trouble with rounding. Again, I keep getting surprised by what they stumble over. They are doing trig like nobody's business, setting up the ratios correctly and all. But all kinds of trouble comes up with rounding the answer.

The test says at the top "Round all answers to the nearest tenth". Some individual questions say "Round to the nearest foot" or whatever. In all classes I specifically called these out, multiple times. "Note that all answers should be rounded to the nearest 10th, unless the directions specifically say otherwise." Still, lots of people missed it altogether, just rounded willy nilly.

Then, there's confusion about when to round. Even though we talked about when to round (at the very end) in class, they are still taking the trig ration (e.g. 0.234576...), then rounding *that* to the nearest 10th (0.2), then using that for the rest of the calculations.

Finally, there's confusion about how to round. Some of them think rounding to the nearest tenth is the same as rounding to the nearest ten. Some think that means using the tenth decimal to round the one decimal up or down. Etc.

The net result being some people lost up to a fifth of the total points on rounding alone. Which I felt really conflicted about. On one hand, the main 'point' of the test is trig, and they are doing the trig basically right. On the other hand, we definitely spent time in class talking about all this (directions, when to round, how to round). And at the end of the day, the answer they are giving is wrong. In the merciless world of standardized tests, there will no grace given for rounding incorrectly. So I want to get their attention now, when it matters less.

Still I feel like an ogre.

SohCahToa

We are covering trig functions. Yet another reminder I have no idea what I am doing. I thought, looking over the chapter, that this would be a complete trainwreck. It requires a certain capacity to visualize and rotate objects in space, to realize that X is positioned relative to Y, to remember certain arcana that this ratio is called sine and that one is called cosine, and to do inverse trig functions when necessary to find angle measures.

I was wrong.

They are taking to this like ducks to water. Several of the students who have struggled the most in the past have called this "easy". They love the SohCahToa memory device (sine = opposite/hypoteneuse, cosine = adjacent/hypoteneuse, tangent = opposite/adjacent). I find it written on everything: quizzes, homework, margin of the book. It's like the new graffiti.

What really baffles me is that these are the same kids whose brains exploded over 45/45/90 triangles and 30/60/90 triangles. I could not, for the life of me, get them to remember the ratios for these particular instances, but they can remember how to set it up in the general case. *My* brain just exploded.

Spring Break

First day back after spring break.

I have heard this many times and many ways in my life, but over this break it made more sense to me than it ever has: money isn't everything.

I was enjoying the time off *so much*, time to just be still, rest, read a book, do whatever. And I realized I will have this *all summer long* (at least until I start looking for a new job). And that's pretty amazing.

The culture I come from (Silicon Valley startup land) has this built-in assumption that you live to work. A 60 hour work week is typical, 40 is embarrassingly low, anything less is just shameful. Makes sense if you find it all fun, if you'd be doing this whether or not you get paid. But if you're doing it just for the money, it's a pretty poor tradeoff. Yes you can get money that way, but at the expense of something even more precious: your life. The few waking hours you have while you're young.

Again, probably old news to most people, but it really hit home this past week. I just noticed how pleasant it is to have time to myself, and I realized it might be a perfectly legitimate choice to decide to preserve some of that time, in spite of the "live to work" culture I'm in.

In completely unrelated news, over the weekend someone made sandwiches out of leftover pancakes, and they were called either sandcakes (unappetizing) or panwiches. I can't believe one or the other of these terms has not become common in English already.