The Setup

One time I made my students measure the heights of buildings around our school with an engineer’s transit. They had to use trigonometry.  I was a pretty great teacher so I made them take all of their measurements multiple times.  Man, they loved it.  It was a great activity because I give out points for each successful thing they do, which really motivates them. Everything is on a scale of five to ten (I have a progressive attitude about grades), so when a team gives an answer I give them some points, and at the end of the week we average them to get final grades. (side note: this is extra great because I can give students feedback pretty quickly)

Anyway, when they were done we did some serious analysis on the board. One team measured the gym, and they took four separate measurements. They found that the gym building was 40, 39, 42, and 39 feet tall. To really make the significance of that stick I made a spark graph.

I gave that team three 9/10s, one for each incorrect measurement, and one 10/10, for the correct measurement of 40 feet.

The other team measured the science building, and I guess they were slow or something because they only measured it twice – 19 and 21 feet. In the end it worked out because we could still make a spark graph. Two 9/10s.

I was starting to think that the spark graphs weren’t that useful, but luckily this was on a smart board so I could have one of the students come up and drag them around.

We figured out that the science building was about 20 feet tall and the gym was about 40 feet tall.

Here’s what I couldn’t believe, though.  I asked them what the average height was, and here’s what they did:

$\frac {(19 + 21) + (40+39+42+39)} {6} = 33.33$

Whoa guys! I know the scale is supposed to start at 5 but I’ve got to give a zero for that.  Very disappointing.  Are you sure you’re in the right class?

The Problem

The students couldn’t understand that the numbers they collected shouldn’t just be added up and averaged together.  I mean, you can add up numbers and average them, but you have to understand what you’re doing.  Averaging is an algorithm that really only works for equal, independent measurements of the same thing.  Obviously the average height of the buildings is 30 feet. We have to make sure that the numbers we’re sticking into our averaging algorithm are actually compatible.  Even though all six of our measurements were in feet… some are measurements of the science building and some are measurements of the gym.  If you average them you get the average measurement, not the average height!

The Punchline

Overall, I think the activity was a success.  After I added up the points everyone got during the day and averaged them together, everyone had over 90%!  Then I added up the points they got on homework and tests, and averaged that in too.  Finally, I added up all the points that everyone had earned in the whole week and averaged those together, and I had a final grade of B+ for the class overall – pretty good!

Afterwards, someone asked me what they needed to work on to improve.  I looked up their grade and saw they had an 85%, so I suggested they try to get more points the next time I asked a question. I love that self-motivation that points systems provide.

But I was the proudest when the director of maintenance heard that our class had been measuring the buildings.  He actually came into the class to ask some advice! He needed to get new ladders so that he could easily repair the roofs of the building, and almost my entire class could easily answer that he should get – you guessed it – the thirty-foot model!

WTF at these graphs

This is a graph of the activity on ActiveGrade for the last 24 hours.

It’s Saturday at around noon right now, so you can see that maybe a couple of people have been entering grades this morning, and a bunch of people were entering grades over the course of Friday – petering out around 8 PM (ugh – I feel for you).  The units here are a little complex – the vertical axis marks “requests per second,” which is proportional to “the number of things that are happening in ActiveGrade per second.”  When the line is at 0.3 requests per second, it means that about 3 things (maybe entering an assessment, looking up grades, changing a grading policy) were happening every ten seconds.  On average, of course.

I don’t think many of my students could really understand this graph.  It’s deceptively complex.  How many requests do you think ActiveGrade got total in the last 24 hours?  How many requests do you think are in a single spike? What does it even mean to be getting a request per second at a particular instant in time – less than a second long? Do you think you could say when the most people were logged on?

Still, it’s easy to see when the program was the busiest.  That’s useful.

But look at this graph, which covers the last 48 hours:

At first glance, it looks like activity levels were lower in the first 24 hours and higher in the second 24 hours.  The requests per second stay around 0.05 for the first bit, and are frequently up over 0.1 on the second day.

How many of you would look at this next graph, of the last four days…

… and guess at the distribution of visits shown below?

This is the same data.  Wednesday had WAY MORE activity than Friday!  Could this be right? Is there a problem with the graphs?

So here are the questions:

• What is going on with these graphs?  Why does Friday, which looks so busy, do so poorly in the final count?
• Is the line graph an appropriate model for reporting this data?
• How many total requests have we gotten here?
• Is the business going to succeed?

How would you structure a lesson to give students the fluency they need to ask, and answer, these questions?  Graphs with these characteristics will probably not appear in your textbook.

If you don’t teach calculus, does this kind of question (“hey wait a minute, wtf at these graphs?”) have a place in your class?

What did you do to the x-axis?!? Using the most relevant context possible

In my Precalc class we were studying average rates of change.  As you do.  Anyway, the formula looks something like $\frac {x_{final} - x_{initial}} {time_{final} - time_{initial}}$.

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.

Bringing the Problem to Physical Reality: Trigonometry

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.”

Of course, the “x-coordinate” column is what we call $cos(\theta)$, the “y-coordinate” column is $sin(\theta)$, and the “ratio” column is $tan(\theta)$, 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 $2\cdot cos(x)$ 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.

Monty Hall, right?

For the last three weeks I’ve been studying probability with my Algebra 2 students, thinking: “too bad there’s no good way to teach this.”  Probability only works on large scales, and then it really only approaches working.  The kids all get the basic stuff, like coin flips having 50/50 odds, they all don’t get the more complicated stuff like standard deviation.  They lack the tools for problems with continuous distribution, and as far as I can tell that leaves us with carnival games and card tricks.  Which only work on average.  By being my interested, lively self during class I managed to interest half of them for most of the time (what are the odds that you’re interested in this question, Johnny?).  But it was a hard unit for me.

Today, last day of class, the final is over, and I think to myself, “ok, let’s talk about the Monty Hall problem.”  (Knowing what the problem is is necessary to understanding this post.)  I bring the requisite bowls and candy prizes.

“Who wants to play?” I ask, and J raises her hand.  They don’t know the rules yet.  I just say, “One of these three doors (bowls) has a tootsie-roll pop in it.  If you guess which one, you get to keep the candy!”

So she guesses, and gets it right, and I give her the lolly pop, arcing it to her desk.

“Who else wants to play?”  More hands go up, and we play several more times with this simple rule.  ”What’s the probability of winning?” I ask, and they easily respond with some form of one third.

On the fifth round, L is playing, and she makes her choice.  I say, “L, your guess was as good as any other.  But I want to give you another chance.  I will show you that this bowl over here <lifting bowl> is empty!”  She, and the rest of the class, stare at me blankly.  ”Do you want to keep your original guess, or switch!?”  I say this channeling Regis Philbin on the \$500,000 question.  She stays.  ”Is that your final answer?”  She says that it is, and with a flourish I show her that she is right; she’s won the candy!  Congratulations!  Congratulations all around!

We play again.  I stop after showing an empty bowl to ask about the odds of the last two unknown bowls, and the entire class is quite confident that there is a 50% chance that the candy is under bowl 1, and a 50% chance that the candy is under bowl 2.  This is intuitively obvious and not true.

Excellent.

They still don’t see anything fishy when we play another time, and again I ask about the chances of the bowls.  I say, “So, when you chose at first, you had a 1/3 chance of winning.  Now you say that the same bowl you picked has a 1/2 chance of winning.  How could flipping over an empty bowl over here improve the chances that this bowl is a winner?”  The students sort of stammer – they aren’t fluent in this language and don’t have the vocabulary to convince me that it’s true.  A student in the back starts to give an example.

He says, “Imagine if you had fifty bowls,” and I immediately whirl around and draw 50 bowls on the board.  It takes a long time, and I make a spectacle out of it.

50 bowls. One of them has candy in it.

“Ok,” he continues, a little incredulous that I actually just drew fifty marks on the board.  ”And the player chooses one, so -”

“Which one?” I interrupt.

“Uhh…”

“Please come up and circle one!”

Does this bowl have the candy?

I say, “What are the odds that you just chose the right bowl?”

He says, “1/50,” and the rest of the class agrees.

I say, “Ok.  So now, as the host, I’m going to open all the doors but the one you chose and one other.  If you chose right at the beginning, the extra door will be empty.  If you chose wrong, it will have the candy in it.  Here we go!”  And I make a big, slow deal of erasing all but one other mark.

The original choice and one other bowl remain. What are the odds that the candy is in the other bowl?

“Which bowl do you think has the candy?” And they all think that the lower left bowl probably has the candy.  Several students are laughing at this point.

“What are the odds?” I ask, and they falter!

“Aren’t they 50/50?” I ask.  ”There are two things, and one of them has the candy, and we don’t know which one, right?  So what’s the big deal?”

H protests, “but in this example you chose the right one to leave!  Obviously it has the candy!”

“Isn’t that what I did before?”

The rest of the lesson goes on like you’d expect.  We make a tree diagram.  I extend the example: imagine you had to pick the correct blade of grass out on the lawn, and then I went and mowed all the other blades of grass except one.  Do you think you’d pick correctly first, or second?  Etc.  We work out numerical probabilities.

But this lesson is different from the others in probability in that the kids are engaged.  This problem should have been first.  I wish I based the whole subject of probability around it:

• The students can describe the problem well.  They think they are fluent at first but later find they are not, and are intrigued to find better ways to describe what is going on.  No other probability scenario I found had this quality.
• We don’t have to try a million times before the probability gives actual results.  Because of the role of the host, knowing probability actually helps kids win the game in an interesting way (much better than “you should bet on 7 because it comes up 1/6 of the time which is more than other numbers”).
• This game is actually fun to play.  Who knows why.  The switching thing is great.  All of my quarter and die games were flops.  For the last five years.

I thought I would like teaching probability, but I don’t, because it claims to be so practical and is actually so impractical.  Do you have any ways of teaching it that you like?

Bringing the Problem to Physical Reality: Modeling a Pendulum’s Period

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!

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.

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).

A Probability Discussion

There’s a major problem with probability: it doesn’t work very well.  For example, common knowledge tells me that in a fair coin toss I have 50% odds of flipping heads and 50% odds of flipping tails.  But flip a coin once and you’ll either get 100% tails or 100% heads.

Now, I get it.  I mean, don’t worry.  After many flips, blah blah blah.  We can study a fair coin toss for two weeks, counting total possibilities, various games, expected values, special orders with surprising probabilities, etc, but none of this helps me guess the next flip at all.  Even in the next six flips there’s a depressingly low chance of getting three heads and three tails.  We certainly can’t rely on it happening in our math class while we try to convince people of what “50% likely” means!

I believe in probability, but it is indemonstrable.  Once you do something, it 100% happened despite its 50% (or 5%) odds.  The challenge for us, then, is to reconcile these seemingly-disparate pieces of evidence.

I chose to let the students have a 30-minute discussion about the issue and I brought several props.  My goal was to get the students to seriously consider their beliefs about probability – to be wary before they were confident.  By the end of this conversation, I want some students to doubt that heads and tails are equally likely.

I started the conversation by showing them this quarter, through which I have inserted a piece of paper:

After the requisite jokes about how hard it was to get that piece of paper through the quarter, I asked, “What are the odds of flipping tails?”

“50%,” one student immediately asserted.  The other students all agreed (I was surprised that they all agreed).  Experimentally, we showed that the odds are not even – there was a strong correlation between the side that starts up on your thumb and the side that lands up on the table, apparently because the paper drastically reduced the coin’s tendency to flip at all.  It sort of parachuted down instead of flipping and rolling.  One of the students tried cutting the paper (which started as a clean square) to see if she could affect the odds.  The students’ assumptions were incomplete and wrong, which is one of my favorite ways of getting them interested.

Then I asked another set of questions, this time with a plain coin and no paper:

“When I hold this coin way above the table with tails facing up and drop it, what are the odds that tails will land up?”

50%, Riley, [obviously | I think | assuming it bounces].

“Ok… when I hold this coin one inch above the table with tails facing up, what are the odds that tails will land up?”

100%, Riley, it won’t even flip.

But at two inches, the coin bounced a bit, and at three inches it was hard to tell what was going to happen.  ”At what height does it become 50%, class?”

They got my point.  If we can tell what is going to happen after an inch, why can’t we tell what is going to happen in a foot?  Philosophical questions abound here, right?  And they started to doubt themselves about everything.  They started to see that there’s no way to test the probability with perfect accuracy.  They started to see that you might be able to rig a coin flip to turn out the same every time.  They started to see the effects of human bias in selection games (I had a deck of cards too).

By the end of this discussion, my students were using phrases like “assuming that there is enough complexity” and “assuming that no one is cheating” and “assuming that the sides are equally heavy.”  They saw the assumptions that go into a statement like “heads and tails are equally likely,” and saw that we can’t rely on that information in all cases.  I sacrificed 30 minutes of class during which they could have practiced permutations and combinations, but I hope that their increased savvy about statistics will pay off.

The Sound of Music: Experiential Exponential Potential?

For several years I have used the frequencies of sounds to give students practice using exponents.  This year I did things backwards: we’ve been studying exponents in various abstract ways first, and then as a sort of conclusion / practice / experiment / “hands-on” kind of thing brought out the tuner.

For this 1-hour lesson you will need:

• A computer per group of students,
• A microphone per computer,
• G-tune,
• Either the money to buy g-tune, the gall to use the demo version in class, or an alternative piece of software, and
• A worksheet much better than this one (docx).1

Suggested Lesson Activity (50 minutes)

5 minutes: have a student read the first part of the first question aloud.  There is a lot to read here – too much to give to each group to read, I think.  After “the chromatic scale,” have groups turn on g-tune and attempt a chromatic scale.  This will be fun and hilarious for them and perhaps more so for you.  Make sure each group has g-tune working and can identify the notes that it is displaying.  Surprisingly, many groups could do this simultaneously – my groups were less than 10 feet apart and did not suffer too badly from interference from nearby singing.

10 minutes: have a student read the next part of the first question.  Stop after “number to the note,” or whatever you have rewritten it to be (please do rewrite it).  Explain that your goal for the class period is that the students will be able to find a mathematical pattern in the musical scale that will allow them to predict the frequencies of different notes (and the notes at different frequencies if you’re studying logs too).  Then let each group read the discussion questions aloud and decide on their numbering system.  Today I had groups start with 0 at C4 and C3.  Another group started with 0 at G#5 and go backwards, and another group started with 1 at A2 and go up by 0.1 (so C3 was 1.3).  Of course the scale does not matter as long as it’s linear, so whatever will be easiest for them to work with will be best.  Probably the easiest scale would go up by twelfths, but of course you will not tell them that.

10 minutes: students will find the specific frequency of ten different notes.  After they have done this and recorded their data in duplicate or triplicate, you (the teacher) can take their sheets and swap them between groups to increase measurement speed.  You are now done with my worksheet, and luckily have added on to it the last half of this lesson, which I did not have time to do before class.

10 minutes: Ask groups, “do you see any patterns in the data?”  Your rewritten worksheet says something like “look for patterns in the data.”  If they don’t see any, ask how they can look for patterns in numbers.  They should be thinking of strategies like “make a table,” or “make a graph.”  You might be extra-direct and prompt them to look for a relationship between C3, C4, and C5, or F3, F4, and F5, etc.  Today in my class 100% of students noticed that the frequencies approximately double between notes one octave apart.

10 minutes: Ultimately you want to be able to bring this back to exponents and logs.  I had students graph their data on geogebra, and I asked questions like “what would the frequency for C6 be?” and “what note would be at 1000 Hz?”  Inevitably their answers included the exponents and logs (though only one group called their logarithms logarithms).

If you are not a small-group kind of teacher, the closure of this lesson seems weak.  But just wait until you hear the kind of discussion that happens with 3 or 4 kids trying to figure this out, with such a fun activity (singing) and fun tool (cool waves and stuff that respond to your voice) and high skill levels (my kids can already work with exponents and logs relatively comfortably).  One of my four groups actually found the equation that best matched their data, AND its inverse – it was really neat to see it in geogebra.  Every group used exponents to talk about the frequencies, and all but one group started to talk about logarithms too.  This is practice using math in a casual way and feeling the fun and power of it.

This is a fun class full of joyous noise, and the kids were really into measuring precisely and graphing precisely.  Every group eventually made a graph and noticed it looked exponential or logarithmic (depending on what they assigned to which axis).  They got to think about what kind of scale makes sense in the first question, and got to see exponents at work in nature.  I used the last five minutes of class to indulge in a monologue about the philosophy of sound: “is ‘sound’ the only way to interpret vibrations?” and “the computer registers 180 and we hear a certain pitch – which is more useful?” etc.

1. Sometimes when I am self-deprecatory I am actually trying to emphasize how awesome I am.  In the case of this worksheet, however, please be advised that you probably actually want a much better one.  For one thing, the worksheet provided only covers half of the lesson plan (ran out of time!).

Programming with Geogebra

This post is about some of the virtues of programming computers in math class.  I include a long anecdote and a quick geogebra tutorial.  The punchline: teaching kids to program introduces them to an environment that gives instantaneous, continuous, 100% correct, 0% helpful feedback without judgement.  The computer doesn’t say, “you’ve made a mistake here,” it just shows you a result, and it’s up to you to interpret it, decide if it’s a correct result, and find the problem if it’s not.

In my calculus class we’re looking for a way to guess how long it will take a glass of cold water to rise to within a few degrees of room temperature.  We’ve taken a lot of data, and discovered that ${dT}/{dt}=0.01(70-T)$1.  However, no one in class could find a $T$ that satisfied the equation (not even $T=70$).  So I broke it down a little:

If the water is 40 degrees right now, what do you think its temperature will be in 1 minute?

You can imagine where it went from here – lots of guesses, including some really good ones and pretty bad ones.  Instead of helping them write it in math (I didn’t even tell them that this is, like, a method), I took a little time to teach them some geogebra 2 programming techniques so they could flesh out the ideas themselves.

We started with a blank file and created a point, A.  I showed them how they could make a point that was 1 unit right of and 1 unit above A.  Type ( x(A)+1, y(A) + 1) into the input field below to get a taste of this.

Sorry, the GeoGebra Applet could not be started. If you're using a reader, try visiting the post directly.

They liked this; I’m always surprised by how much students like stuff like this.  So this alone has them interested in learning geogebra.  Then I lay the geogebra Tool Manager on them.

In the applet below, open the “Tools” menu and choose “Create New Tool…” and choose B as an output object.  You’ll notice that A has been chosen automatically as an input object.  Choose the name and icon you like from the third tab.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

What follows is a picture to help you – it’s not an actual geogebra applet!  Don’t try to click on it!

After you’ve gone through all the tabs and clicked “Finish,” your new tool will appear in the toolbar, all the way to the right.  Now you have a tool that accepts a point as input and creates a point 1 to the right of it and 1 above it, automatically!  Try clicking around with your tool.  You can even click on the output of the tool to feed it back into the tool, creating a long line.

Well, at this point my students were ready to get back to the temperature thing (we’ve been working on it for maybe 3 hours at this point, over several classes).  With a little nudging from me they make a tool that does Euler’s method (fixed width of 1) on a point, by choosing B = (x(A)+1, y(A)+0.01(70-y(A))) and using the tool repeatedly.  Check this out – try dragging the point A up and down to different temperatures!

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

Note that now I can ask them questions like “how do you make the curve flat?” and “why does the curve change direction?”

With a few simple commands the students have created a complex piece of software, and they can be proud of it.  It has a pleasing organic flow that they like intrinsically.  But the best reason to teach them to program geogebra is that geogebra programs only work if they are mathematically sound.  I can see in an instant whether a student has created the program correctly or not.  When they go on to create other geogebra programs, I can assess whether they understand the concept or not, and more importantly, they can assess their own knowledge.  Geogebra will show students if they understand or not, but won’t give suggestions or hints.  It also doesn’t mind if they are wrong 400 times in a row.

I start teaching geogebra commands to kids right away, in the first month of school.  My algebra 2 kids can transform an arbitrary function, calculate a line or other curve through 2 points, and animate sliders.  The ones that get really interested learn more on their own (one student has all but mastered latex).  The kinds of assignments I can give and the kinds of exploration they can do now are really something else.  Please teach your students to use computer technology well.  It’s not enough that they can use the built in functionality – they have to be able to make their own.

1. The kids came up with many ideas about temperature change and I directed them towards this equation
2. If you’re a math teacher and you don’t know how to program geogebra, I recommend looking into it – it’s fun and extremely helpful for creating interactive diagrams.

What do you mean, “relevant?”

Do you try to make your lessons relevant to your students and their lives?  Do you struggle to find ways your students will actually use your math in the future?  I suggest a reframe: make lessons relevant to your students by immersing them in mathematical situations during class.  Don’t worry too much if you can’t find something in their life outside of class.

My text book refers to bank accounts to make exponential equations relevant, and so does yours, probably.  Some problems with this example are obvious: my account is compounded monthly instead of continuously, I put money in and take money out at weird intervals, etc, so this math actually does not apply without modification.  Some problems are more subtle: you don’t need to know about exponential equations to pick a bank account, you just pick the one with the highest interest rate.  My students have very little money and differences in interest are on the order of pennies per year.  My students don’t have bank accounts.  My students don’t care about bank accounts.

The most subtle problem (that I’ve found, anyway) is that even students who do care about bank accounts aren’t opening bank accounts at the moment I am trying to teach them about exponential functions.  Even if the example is relevant to their outside life, when they are in my class they are not in their outside life.  The best I can do is “imagine you’re choosing a bank account” or maybe “find three bank accounts online.”  What I want is a student faced with a problem that he wants to solve during our class.

Give up on outside relevance in favor of inside relevance.  Pass out super balls.

This idea is from CPM’s Algebra 2: Connections.  Ask students, “how high will your ball bounce?”  They will have a mental crisis as they think of all the reasons this question is unanswerable in its current form.  Already they are thinking about a problem during my class.  Better yet, they are pretty fluent in bouncing already – they have bounced a lot of things in their day, you know?  They have some vocabulary to describe what’s going to happen, and a framework in which to place guesses.

You get the idea: it turns out that the height of bounce x follows a path almost exactly exponential, and you can ask students how high the second bounce will be, and how high the tenth bounce will be.  Please note that this has nothing to do with anything that “matters” in “real life.”  This activity makes exponential functions relevant by showing kids how exponential functions can improve their fluency about bouncing.  Exponential functions give them the power to, with absurd accuracy, guess how high the fifth bounce will be – and everyone likes that.  Increased fluency about reality is relevant to everyone.

So I’ve traded outside relevance for inside relevance, and the biggest difference is that I am teaching during the problem-solving process instead of before the problem-solving process.  The kids who learn about bank accounts won’t use that math until they are opening a bank account, and at that point they can’t get help from me or from classmates.  The kids who learn about bouncy balls use the math at the same time they are learning the math, and so their ideas are immediately challenged and reinforced by results.  It’s better!