Today was my last day with the 5-7 graders. We spent most of the time learning about beaches: how sand gets there and how sand moves once it's there (California grade 6 standard 2c). It's a lot more interesting than you might think, and it's explained well in this video. Normally I show just short clips of videos, 30 seconds or a minute here and there to support whatever I want to talk about; a lot of "educational" videos have a lot of fluff surrounding the critical part(s). But I found this video to be packed full of good visualizations of what's going on with beaches, far better than I could set up myself, and very little fluff. So we watched all 20 minutes (ok, I skipped the fluffy first 80 seconds), and I highly recommend it for parents too. Aside: It's from the 1960's, and told in the "voice of God" style strongly reminiscent of the films I was shown when I was in elementary school. Science videos today are quite different, typically with a friendly host just like us who wants to take part in experiments just like we do. That's probably an improvement on average, but I experienced some nostalgia for the "voice of God" style as I watched it.
After the video, I set the kids to work on the "Rollin' Down the Sand Highway" activity from this packet (the last activity in the packet). I didn't provide maps, but just looked them up online as needed. Some kids had never seen a topo map before, so I explained that in context. But for most of the time most of the kids were stuck on the math, which surprised me because the math is pretty simple. I guess it's a question of applying math outside of math class! It's always easier to apply a concept when you've just learned it and you know that the problem you've been given can be solved using that concept.
More specifically, the students did not have a clear idea of how to go about converting cubic yards of sand per year to dump trucks per minute. I led them through the easy step of converting cubic yards per year to dump trucks per year, and I thought this would give them the boost they needed to complete it on their own, but I was wrong. As I circulated around the room helping students, it came out that we would need to know how many minutes per year, and the students were able to come up with that number (although they may have Googled it on a mobile device behind my back): 525,600. But there was a huge amount of confusion regarding whether they multiply or divide by this number, and whether the result would be dump trucks per minute or minutes per dump truck. I walked them through how I think about it, and they stared at me totally lost; I stared back wondering how they could not have seen this before. So I backed up and (much wailing and gnashing of teeth omitted here) found a way to get it across.
Here's what worked: let's say that you are asked to compute 3 times 4, divided by 7 times 3:
3x4
---- = ?
7x3
The kids universally said the following: multiply across the top and also across the bottom:
3x4 12
---- = ---
7x3 21
This surprised me because it's not what I would do, but once I figured out that one kid was thinking like this, I repeated it for all the kids. Although the answer surprised me, it's not wrong, so let's continue along these lines and see what happens. The natural next step is to simplify the fraction 12/21: is this its simplest form? The typical answer from a student was: ...um...well, I don't see any common factors. And of course it's hard to see the common factors when you're staring at 12/21. But if instead you look at
3x4
---- =?
7x3
the common factor of 3 is jumping up and down screaming "I'm a common factor!" So cancel the 3's and you immediately get 4/7. This is not only much less work than writing 12/21 and then trying to simplify; it avoids the potential for a lot of mistakes. Although this kind of thing is second nature to me, it was not natural for the kids, who were intent on following the specific rules they had learned about multiplying fractions.
I had to go through all this just to get to the main idea: we can do the same kind of thing with items like dump trucks and minutes instead of specific numbers. We are given dump trucks per year and we want to get dump trucks per minute, so we can represent the problem like this:
dump trucks ? dump trucks
--------------- x -- = ---------------
year ? minute
We have to get rid of years and introduce minutes, so if we put years per minute in the question marks, we get:
dump trucks year dump trucks
--------------- x ----- = ---------------
year minute minute
The years on the left cancel each other, leaving dump trucks in the numerator and minutes in the denominator. If we had instead tried:
dump trucks minute dump trucks
--------------- x --------- = ---------------
year year minute
this equation is manifestly false; the right hand side should contain dump truck minutes on the top and years squared on the bottom. This kind of thinking seemed to be new to the 6th graders, and I'm glad I did it because it's really important. It provides a system for making sure you do the right thing. Don't know whether to multiply by 525,600 or divide by 525,600? One system popular among the students was to just try one approach, and then if the teacher says it's wrong, just do the other! But here's a system which makes clear that we have to multiply by years/minute, or 1/525600. And not sure if the resulting number represents dump trucks per minute or minutes per dump truck? Again, the system makes clear that the result is dump trucks per minute.
Another thing the kids need to internalize much better is sanity checking. If you multiply 722,222 cubic yards by the $5 per cubic yard it costs to remove, you should get a number bigger than 722,222, not less than 722,222. The kids didn't apply this kind of sanity checking to any of their results, and therefore didn't catch any of their mistakes before showing their answers to me. This was the first math-based activity I had really done with the upper-graders, and I was probably naive to expect that they could apply math outside the context of a math class. I should have given a little primer on how to estimate before calculating, how to check that your answer is right after calculating, etc. This is not really math; it's metacognition in a math context, and I'm now kicking myself for not emphasizing metacognition throughout this trimester with the upper graders.
In any case, we spent a lot of time on this activity: 1 hour, including the movie, before break; then maybe another 20 minutes after break. It was worth it to work through these issues, but then I did have to cut down on my planned post-break activity. I'll dedicate the next blog post to the humidity-related activities we did in the last 30 minutes of the morning.
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