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.