In my Precalc class we were studying average rates of change.  As you do.  Anyway, the formula looks something like .

This lesson was an eye-opener for me.  I had mostly 10th and 11th graders in the class, and I casually drew a distance v. time graph on the board, and drew two dots on it.  I was going to talk about how the slope of the line between the two points was equal to the average speed experienced by an object moving from one point to the other.

I admit that I was naive.  I really thought I could say something like “the average speed experienced by an object moving from one point to another.  Oh btw the ‘points’ are in space-time, not, like, space, ok?  So anyway…”

It turned out, as you may have guessed, that my students were not very comfortable with the idea of a space-time graph.  ”Riley, why did you write an x on the y-axis?” and “What happened to the x-axis?” are representative of the actual quotations I wish I had written down at the time.

This was super-sad for me, because the only thing interesting about the average speed of anything is that it can be represented as the slope of a line through space-time.  Yes, a cop might be able to prove you were speeding if there were two speed traps a known distance away from each other and the cops write down license plate numbers and times, right, but I try to avoid that problem:  they can’t catch every type of speeding that way, and it seems like maybe, instead of writing down thousands of license plates and times, they should just use the radar gun next to their stop watch.

So, I became determined that my students would understand space-time.  Maybe we wouldn’t get to factoring that tiny percentage of quadratic polynomials that are factorable, but then maybe I’d just let them use a calculator on their exams.  Here’s what I did:

I marked up the sidewalk in 1-meter increments, from zero to 60 meters.  I put my students in groups of five – a big enough group that they can support each other in yelling, so they can all yell at once with no one having to yell by themselves.  I gave each group of students a stop watch, and I had one myself.  Already, you’re thinking, “oh, this is going to be great!”

I said to the class, “BEHOLD SPACE, class!” and indicated the length of the sidewalk.  I said to the class, “BEHOLD TIME, class!” and indicated the timers.  I moved to the 5-meter mark and said, “BEHOLD, I am at ’5 meters’ in space, class!”  I looked at my stop watch exaggeratedly and said, “BEHOLD, I am at ’50 seconds’ in time, class!”

And then they understood space-time, and we went back inside.

Haha, no, not yet.  Though yelling imperatives about understanding does usually work well for me, in this case the activity called for something a little more interactive.  I actually said, “Group 1, please move as a group to the 15-meter mark, and be there at 120 seconds.”  This was very explicit and gave them about 30 seconds to move 20 meters.  ”Don’t miss the time!” I called as they were in transit.

To Group 2, I said, “Group 2, please move as a group to the 18-meter mark, and be there at 140 seconds!”  This gave them much less time to essentially do the exact same thing that Group 1 had been done.

At this point I have 2 groups on the line and 3 groups in a big clump near 0, still.  I whip out my portable dry erase board, and draw the graph with the two points in space time that groups 1 and 2 had hit.  I made sure to leave plenty of room for the next five minutes on the graph.

“Group 3!” I commanded.  ”Please move to the point 10 meters, 200 seconds!  Please move as a group!”  This was a slightly different format of instruction, but it was close enough to the earlier instructions that they got it.

“Group 4!”  I’m at the top of my voice now, trying to invoke the spartan leader from “300.”  ”Please move as a group to 5 meters, 215 seconds!!”  They have to run to get there, but it’s a short distance so that I won’t embarrass anyone who still doesn’t quite get it.

Now there are 4 groups on the line.  There has just been running, and there is still giggling.  I call for everyone’s attention and bring my voice back down so that they bring their voices back down.  ”Here are the two points that groups 3 and 4 just hit,” I said, graphing them carefully because I’m going to reference it later.  ”I have a special instruction for you, group five.”  Everyone is paying attention now – this is suspense.  ”When I say go, please move to the point 7 meters, 260 seconds.  Once you have hit that point, please say ‘here we are!’  This is a new part of the instruction.”  They move to (7m, 260s), and say “here we are,” but softly.  I say, “everyone, when I count to 3 please say ‘here we are’ as loud as you can! One, two, three!” and about half of them say “here we are.”  I say, “one, two, three!” and more of them say it a little louder.  ”One, two, three!” and most of them say it really loudly.

Here’s the most memorable part of the activity.  After the last loud “here we are!” I call on group 5 again.  ”Group 5, please move to 18 meters, 300 seconds,” making them hurry but not run.  ”When you get there please yell ‘here we are’ as loud as you can.”  While they were in transit, I moved to group 4 and said, “Group 4, please move to 23 meters, 310 seconds, and when you get there please yell ‘here we are!’ as loud as you can!”  Moving quickly down the line, and before group 4 was done, I addressed group 3, and said, “group 3, please move to 40 meters, 310 seconds.  You don’t have much time!” and they were off.

I kept moving down the line until each group had gotten a couple of instructions.  My last instruction to each group was “please move to the classroom, 500 seconds.  Please sit as a group.”

So I just took 20 minutes to run around on the sidewalk (getting out there and synchronizing the watches takes a while).  Here are questions the students can answer fluently now:

• What is it like moving from (5m, 200s) to (20m, 210s)?
• Which is harder, moving from (10m, 150s) to (15m, 155s), or moving from (10m, 150s) to (20m, 155s)?

These questions are relevant to them because they’re still breathing hard from running.  They’re interested because they still remember giggling when I asked the first group to run and they still remember scoffing when I asked them to move 20 meters in 5 seconds.

I do the teacher-at-the-front-of-the-room thing to practice graphing a couple of these points, and reading from the graphs.  If I were omnipotent I would have had worksheets so that the groups could practice them themselves.  But after we practice graphing points for a few minutes, we get to the whole point of the lesson.

There are two sets of two points on the board, one in red and one in blue.  One is labeled group 1, and the other group 2.  The kids now understand intrinsically what is being represented.  I haven’t put any numbers on this particular graph, but I ask anyway: “which group had to run faster?” and it’s freaking obvious to them.  I’m not saying they couldn’t have figured it out before we spent an entire class period on the setup, but they don’t have to figure this out.  They see those points and they know.  I say, innocently, “what’s the slope between these two points?” and we figure it out pretty quickly, having dredged up the slope formula the day before.  I get in a couple of words about units, blah blah blah, and then I have them calculate the average speed for 10 different situations.  We run up against the bell, so I let them go before we get to compare answers, but they got it.  Everyone.

Everyone got it, and what’s interesting is that it’s not because it was useful or relevant to anything outside of class.  They were immersed in it because it was fun, and because I made them move, and because they were involved in the exercise quite literally with their entire being.  We talk about pseudocontext sometimes as if problems with genuine context can’t also be crap.  Here’s a situation with no context – we created our own context – that was more valuable than a hundred interesting and relevant word problems.

This post is part of a series of lessons about immersing students in an environment so that they can ask questions of their “surroundings” instead of their teacher.  See my other posts and an introduction.

The night before I wanted to teach trigonometry to my students I gave them the following diagram.

This diagram took up an entire side of a piece of paper.  On the other side of the sheet I gave a huge, 4-column, 37-row chart.  The first four rows are below.

Angle	x-coordinate	y-coordinate	ratio of y/x
0 degrees
10 degrees
20 degrees

and so on, down to 360 degrees.

I included the instructions below.

“This is a diagram showing a circle with a radius of one.  There is a 40-degree angle drawn on top of it.  Please check now to see that the angle ends at (approximately) the point (0.76, 0.64).  We can approximate to the hundredths place from this diagram.

“On the opposite side of this sheet you will see a table with rows for every angle between 0 and 360 degrees, in increments of ten degrees.  In the row for 40 degrees, please enter 0.76 under the x-coordinate, because 0.76 is the x-coordinate of the end of the 40-degree angle on the diagram.  Please enter 0.64 under the y-coordinate, because 0.64 is the y-coordinate of the end of the 40-degree angle.  Finally, please enter 0.84 under “ratio of y/x,” because 0.64/0.76 is approximately 0.84.

“Your homework is to fill out this chart completely, for all 35 of the other angles listed.  The coordinates you get will vary as you choose other angles.  You will need a protractor to draw angles – please actually draw the angles you need to measure, and do not attempt to estimate angles without a protractor.  If you find a logical shortcut, you may use it.”

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Of course, the “x-coordinate” column is what we call , the “y-coordinate” column is , and the “ratio” column is , but the students don’t need those names yet, so I didn’t introduce them.  I wanted to meet the students at their own fluency level, and they are perfectly comfortable with x- and y-coordinates, protractors, angles, and ratios (this is a precalculus class).

Why would I start with “there’s a function called sin(x) that gives you the y-coordinate of a point at the end of an angle inside a unit circle?”  We focused on the calculation of cosine and sine before we had the names, and calculated it many times.  In our discussion of our values, the students were getting frustrated with saying “the x-coordinate of the point at the end of the angle where it intersects the circle,” and so I innocently mentioned that this is what mathematicians call “cos(x).”  Pretty shorter, huh?  They dug it.  Some of said, “oh, this is how you calculate cosines?”

The work they did at home varied.  The clever idea here is that you can actually fill out the entire chart with only nine measurements by using various symmetries. I would say that most students made 18 measurements – the x- and y-coordinates in the first quadrants – and then filled out the rest of the chart.  Some students came in with fully half of the measurements made but their chart incomplete, giving variations of “I can’t be bothered to do this kind of grunt work” (and I don’t grade homework).  Some students made all 70 measurements.  All of my students made some attempt at the homework – I think that somehow it’s kind of a fun activity!

This simple circle, with its highly-structured instructions and almost no student initiative, has the benefits of an immersive physical environment.  The students measure values directly off of the circle, and when they see the symmetries involved they can check them for themselves.  During class, when we’re talking about various properties of cosine and sine, they can check values directly.  When I eventually ask about 45 degrees and 15 degrees, they are fluent in this calculation and can easily adjust what they are doing.  When I ask about circles with different radii, they have intuitive guesses (some right and some wrong: great!) about what will happen to the x-coordinates and y-coordinates.  They are so comfortable with the measurements at this point that they can see the logic in using instead of making a new cosine function for a circle of radius two.

Learning happens when you’re comfortable enough with a situation that you can experiment.  You have to be able to change the initial conditions a little bit, see what happens, and use that to find a pattern, to form a generalization in your mind.  This circle gives students something they can experiment with in a way that triangles don’t.  While they could certainly construct right triangles, it would be too hard for them, and take too long – they would be able to quickly try different angles, or to see all of the angles at once like they can with the circle.  The symmetry is hidden with the triangles, but it’s glaring with the circle.  This circle, even though it’s just a circle on a piece of paper, is a whole environment which the students can explore.  And, importantly, they are already good at the skills they need to explore it.  They can find new information with old skills.  That’s what this series is about!

Update 2/1/2011: Changed evaluation of sin(40) from 0.72 to 0.64.  I’m that powerful.

This post is part of a series of lessons about immersing students in an environment so that they can ask questions of their “surroundings” instead of their teacher.  See my other posts and an introduction.

I wanted to teach data modeling.  The students already knew the basic shapes of lines, hyperbolas, parabolas, cubic functions, square roots, etc, and how to graph them, but we hadn’t talked about what to do with real data.

The tempting approach is to give out 25 points and say “find an equation based on the functions you know that goes close to these points.”  In fact, that’s what I did in my first year.

This is terrible.

In my second year, still burning from the bored stares of the first year, I convinced our facilities manager to climb something like nine meters up a terrifying ladder to hang a couple of ropes from the roof of our gym.

Later, I brought my class to the gym with bunch of weights between 2 and 10 pounds.  I described a couple properties of pendulums, did the whole omg-its-going-to-smash-his-nose-oh-whew-he’s-fine trick, and finally defined “period,” complete with a tutorial of how to measure it (give all the kids stop watches, have everyone time it at once, and then give an official measurement so they can see if they were too slow or too fast or what).

I made sure NOT to say that the period of a simple pendulum is determined by its length alone, and that weights have no effect.  They do not know at this point that period vs. length is a square-rooty function.

Then, I said, “please make your pendulum swing so that its period is within a tenth of a second of 4.5 seconds.”

This is great.

Here’s why: the students are conversationally fluent in this environment.  They know that pendulums swing back and forth, they know how to tie knots, lift weights, push things, measure time and distance, and they consider themselves experts in this area.  In fact, they are experts – they have been practicing physical interactions every day for the last 16 or 17 years!  But they lack the technical precision to describe it well enough to predict specific measurements.  When they realize they actually don’t know how to achieve a specific period, they will be intrigued.  Some will think they do know how, but then be unable to do it, and they will be even more intrigued.

This is why you spend hours to create a real environment.  They are good at reality, and feel comfortable in it.  They have vocabulary to talk to each other about it.  Compare this with discussing square root regression in class – they’ve learned about root functions and practiced for, what, 2 hours total (against 75,000+ hours of practice with physical motion and perception).  They don’t feel comfortable talking about what they’ve learned yet, and they certainly don’t have the vocab to experiment with what they’ve learned by tweaking it in small ways.  In the real world, they can easily say “let’s try pushing harder” or “add more weight!” and see results instantly.

Anyway, the students spend the next 40 minutes trying things, different weights, different initial velocities, different angles, and different pendulum lengths.  Only the lengths make a difference.  They naturally figure out which direction makes the period longer and which makes it shorter, and I jump on that opportunity to point out that they’re already fitting a pattern to what they’re seeing (aka translating reality into math).

They try a bunch of different lengths, but never get that close to 4.5 seconds.

“Why not?” I ask innocently.  Some say they don’t know, others say they can’t reach high enough.  I ask, “Well, how high will you have to go?,” again innocently enough.  ”I could bring a ladder to the next class I suppose, but I don’t want to go to the trouble of bringing the wrong length.  Those things are really awkward and weigh a ton.”

Guys, they are hooked right now.  My question was so natural, so authentic that they want to find me an answer just as a matter of course.  I obviously need that information to help them out, not just to be an obnoxiously unhelpful math teacher.  As homework they look at their data, figure out how they have to organize it to get me an answer more precise than “it needs to be higher,” and bring that answer to me at the next class period.  We go to the facilities office, get the appropriate ladder, and try out their answer, which turns out to be astonishingly close to 4.5 seconds (easily within our measuring precision).  A cheer goes up!

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Now, I’m happy, but not that happy, because every group used a line to model the data.  It turns out that a line is a pretty good model for pendulums as long as you stay in a pretty small range.  So I asked the kids, “how high would you have to be to get the period to be one second?”  Their linear models told them something like 16 meters, and it was obvious (after converting to feet) that the ceiling was nowhere near 16 meters tall.  We spent maybe 20 minutes at this problem, talking about domains of models, practicing using the models to predict various values, etc.  A ton of normal curriculum at the board fell in to this discussion, and they got a lot of personal practice time (please always mix practice in to your lessons).

I suggest that we use a smaller pendulum to get more data.  This required changing our models to be functions of the length of the rope instead of the height off the ground (if all of my kids graduated knowing only how to change the parameters of a function I would consider myself a success).  I bring out a few pendulums of the 1-, 2-, and 3- meter variety (I brought some twine and a pair of scissors to class) and we get some more data points.  CLEARLY not linear now.  We get MORE practice fitting the same data with a different type of graph.  Now we can compare models, learn about different kinds of errors, answer fun questions like “so how high do you think the gym ceiling is?” etc.

I hope I’m making the benefits of an immersive environment clear.  When you are working with new mathematical concepts, students cannot play with them or experiment with them because they don’t understand them well enough.  If you give them a framework with which they are comfortable, they can feel confident enough to experiment, think, discuss, and ultimately learn.  Additionally, when students are asking questions of the environment instead of you, they are responsible for their own questions and data.  You won’t accidentally give away the lesson by providing only and all of the pertinent data.

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If you do this pendulum thing, I recommend you measure while you can reach the whole rope.  I forgot until just after the facilities manager had climbed down from the ceiling, and I wasn’t about to ask him to go back up there.  It needs to be a thin, light, sturdy rope that doesn’t stretch too much, and it can’t rub too much at the top when it swings.  Please emphasize, organize, and enforce safety procedures.  You can hurt someone with 15 pounds on an 8-meter pendulum (attached with knots by kids with Velcro shoes).