Monday, March 4, 2013

Slope and Sunset

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?

  • 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? 


  1. Cool, Michael!

    You should definitely give them some graphs where the location isn't given and ask them to figure out what the location is. Some in the southern hemisphere!

    And/or give them graphs from NYC, Jerusalem, and Alaska, but don't tell them which, or from which month. Have them figure out the where and the when.

    Show them whole-year graphs of those three locations and ask them to sketch whole-year graphs for other locations including: intermediate latitudes, on arctic circle, the North Pole, on the equator, southern hemisphere locations.

    FWIW. Let us know how it goes!

    1. Wow. These are really solid questions and ideas. I particularly like the "Draw the graph for Miami" idea. I also like the whole-year graph idea.

  2. I like the fact that the graphs are not linear - gives you something to talk about. Could you place the ones you have done on a map and pick another location and create what you think the graph would look like for that location? Minutes per Day is kind of a funky rate to latch onto, though.

    I found that using pictures in place of variables, and eventually moving to variables makes both substitution and linear combinations much more intuitive.

    Here is an example or two for substitution:

    And a bunch of stuff for systems in general:

    1. Thanks for flagging the fact that minutes per day is difficult to grasp. I can nail that in the beginning of class today by giving them a NYC table (instead of a graph) and asking what they notice.

      And thanks for the substitution lessons. In general, some sort of concrete visual aid might be helpful, so I might steal your lessons. Thanks!

  3. You're doing too much. Did you see my modeling post? I realize you probably think it's simple, but it's designed for intermediate algebra (A2). You could make it simpler for Algebra I.

    1. Your lessons are solid, and we've done stuff like that -- Skype plans, plumbers and babysitters who charge for reservations and per hour, lemonade stands that charge for the cup, music subscriptions, visual patterns where they model the pattern, speed of two people biking, cup stacking ...

      And, would you believe it, they still don't really get the connection between the slope of a graph and a rate? They don't get that the steepness of a graph tells you something clear and quantitative about the way something is happening. Plus, they don't read graphs particularly well.

      I definitely don't think that your lesson is too simple. But I also think that we're at a point in this class where we've burned through a lot of the simple things, and these kids get bored or distracted by just anything. That's what's pushing me towards a more involved lesson on slope.

  4. I'm curious about what your pupils find difficult about the substitution method. Could you elaborate, please?

    1. They don't understand what's meant by substitution.
      They don't know how to solve for a variable with any reliability.
      They don't understand why this technique would help them solve a system.

  5. Loved this -- especially the idea that getting practice with skills can be the same process as making sense of how geography affects our life.

    If you're not already reading Brian Frank, you might appreciate his ideas about teaching seasons (and what students may think about them... these posts are at his new site). He's also got an excellent post about why some ideas are more "generative" than others and why he avoids the common focus on "misconceptions" (this post is at his old site).

    Same goes for Michael Doyle -- who writes some of the smartest stuff about nature and teaching that I have the pleasure to read.

    My students have similar problems with slope. The time when they are most successful at teasing a meaning out of the story of a graph is when we graph something that has a sudden change in slope. In my world that's a diode's current vs. voltage curve. I don't talk about slope, but they start to struggle to describe it, because it's the most salient feature of the graph and because "something different happens." The LED glows way, way brighter all of a sudden. The current "shoots up." They struggle to describe the different parts of a graph -- they're both "straight", but they're straight in a different way. They find it hard to talk about slope when it's constant; it's too backgrounded. It's like fish having no word for water.

    Finally, Riley dreamt up an exercise that targets space/time graphs specifically but slope more generally. Good luck!

    1. Amazing links. I know these folks, but I didn't know these posts. Thanks again.

  6. I personally hate substitution and use elimination using addition as much as possible. Mt students and I have compared substitution and elimination and they prefer elimination in most cases. We only do substitution when it is super easy and would be too much with to change the equations around. If both equations are in slope intercept form we substitute for example. Substitution just isn't efficient.

  7. My kids were struggling with substitution, too. Then I started asking them what y=whatever really meant. They weren't really thinking about it as an equivalence statement that gives them interchangeable parts. I think they understood it better after that conversation.

  8. I have not read all prior comments in great detail, so forgive me if this is repetitious.

    (1) Love the sunset/sunrise/day length tasks. Lots of interesting relationships to explore and (as at least one commenter pointed out) lots of good challenging questions to be asked by playing with the geography. One other thing to play with is same latitude, opposite ends of the time zone. Say, for example, Detroit and Boston.

    Also good to ask is "Why is there not exactly 12 hours of daylight at the equinox?"

    And I am curious how/whether you look at these tasks differently after being immersed in thinking about functions for two weeks.

    Oh, also, you suggest that it's not really proper to consider slopes of curved objects. But that's all we've got! And calculus demands that we do so. As long as everyone throws around lots of "approximately" and "about", everything's A-OK in my book.

    (2) In my experience, solving by substitution is sensible in "racing" situations, and not so sensible in mystery-value situations. And, as Ms Billings suggests, understanding the equal sign to mean equivalence or is the same as is a key prerequisite.