Sunset and Slope

(Cross-posted on my place.)

I wanted the kids to understand slope as a rate of change, and I also wanted them to use slope to understand something interesting. I also didn’t want class to be boring. It went OK, but I still feel as if the payoff is a bit weak, and I’m hoping you all can help out a bit.

Last week we developed a metric for slope using a version of Fawn Nguyen’s (version of Malcolm Swan’s) slope activity. (Note: talking about sports statistics with these 11 boys helped the idea of defining a metric go down smooth.) I also showed them mountains and asked them to rank those, and we considered the various advantages of measuring steepness as “width divided by height.”

Today we were going to study sunset at different times and places, and I just wanted something cool that would get kids ready to think about astronomy and stuff. I ended up with this barely related video:

It’s great and beautiful and I showed it to the rest of my classes today too.*

Most wanted to know what the green glow is, and I don’t really have any idea how that stuff works. One kid had a pretty good explanation along the lines of “something something force shield.” He also knew about solar storms. I’m getting off track here, but kids are tons of fun 95% of the time.



From there I asked them what they knew about sunset. They knew that it happens, that it gets earlier and later depending on the time of year.*

* Pro tip: When living in NYC, don’t assume that the kids know anything about nature.



I asked them how much it changes per week. Their answers ranged from 1 minute to 7. I asked whether that rate was the same all year long. It took a few tries, but I finally got the question across to everyone, and there was a bit of disagreement.

Using the USNO site I made them a bunch of graphs, and asked them to find the slopes between the points. Here’s what they got:


We needed to remind these kids how to calculate slope, and they moved pretty slowly, so most of them only got through 3 of the graphs today. Some kids had trouble finding the height at first, reasoning that the highest height on the graph was the height we needed for slope. (“Look, I’m holding this paper 7 feet in the air. Does that mean the paper’s got a length of 7 feet?”)

I was fairly happy with the way class went, though I was worried by the fastest kid who made it through a bunch of the slopes and told me that he didn’t see any patterns or interesting stuff emerging. And while the kids were doing better on slope and making progress on interpreting the numbers as a rate, there were warning signs as we tried to wrap things up. (Warning sides include, boredom, confusion about the questions I was asking, difficulty interpreting the units involved in the rates.)

So, the follow up is tomorrow in class. At the heart of this lesson is a really cool idea: that where you’re living on Earth radically impacts the patterns of your life. How can I make this pop, while giving my kids good practice with their skills?

Caveats:

  • Yeah, I know that it’s a bit false to ask for the “slope” when we’ve got non-linear patterns. But we talked about it, and we agreed to just search for representative points.
  • It seems to me that there aren’t a lot of good problems hanging out there for kids who need a bit more practice with the connection between slope and rates. We’ve talked about speed, and they’ve looked at graphs. We’ll do more of that. But I was searching for something a bit more, I dunno, worldly and interesting.
  • This lesson is a spin-off of an Exeter problem.
Oh, and one more thing:
My kids are having a really hard time solving systems of equation through substitution. I tried approaching this really slowly, but we’re stumbling on the landing. Any ideas?
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