Stand up and wave!

I’ve had the skeleton of a lesson on standing waves in physics floating around in my brain for a long time. In the past, I’ve used just a white board and drawn some diagrams, and have occasionally shown some animations from the web to help students visualize the relationship between a vibrating string and the waves moving around on it.

Here was an attempt scanned from my first lesson back in 2006:

And more:

Nothing to write home about. My lesson had a bunch of information in it. Now in my defense, much of this I was eliciting from students. It was a very carefully designed progression of thoughts, and my students did end up understanding it well enough to then apply it to problems. (Better, in fact, than I was when I first tried to learn this in high school. This was one thing in physics that didn’t make sense to me, and I was committed to having my students get it.)

There wasn’t much for the students to grab onto or see – nothing dynamic or moving to engage them. The following year, I added some interactive applets, such as the one here. I didn’t have a class set of laptops, so it was still a demo at the front of the room, but here it was possible to show much more clearly what was actually going on and students themselves could play around with the applet and get some intuition for how it worked.

Fast forward to the present. I’ve been teaching a non-AP level physics class and have been able to write my own curriculum along the way. As a result, we’ve been able to slow down and play with models, do experiments and play with data. Waves are inherently hard to visualize given the frequency of the sound waves with which we typically interact (not to mention their invisibility) so I wanted to try something different. Kate Nowak’s recent post about using technology to do lessons that would be impossible also got me thinking how I could use the technology I have available to do this lesson in a fundamentally different way.

Here is what we did today:

First we watched this video on YouTube showing Tuvan throat singers in action. The kids had never seen this, and it shocked me the first time I heard it too back in high school during a class in electronic music. We also watched parts of the video of Daniel Palacios’ standing waves art piece (Thanks to John Burk, @occam98 for sharing this find yesterday!)

We talked a bit about how the singers might be doing this, but there was no consensus. So I brought them to the back of the room where we had a spring waiting for us on the floor.

I challenged two students to create a single wave with the spring bouncing up and down. This quickly devolved into attempts to make two, three, and eventually six loops in the spring. Hands-on discovery of the relationship between the number of loops and frequency? Check. I had students identify points of maximum amplitude and stationary points, but no vocabulary yet.

We then made our way to another station where I had set up a strobe light and some tuning forks. We played around with the tuning forks and the strobe frequency to be able to visualize the movement of the tines of the fork during vibration. I showed a similar demonstration with a student plucking a guitar string. Objects vibrating up and down in the same way as the spring create sound. Simple idea.

None of this is really revolutionary, I admit it. The good stuff is coming, I promise.

During a workshop I attended with Nick Jackiw on Geometer’s Sketchpad, I learned of a feature that I didn’t know about. With periodic functions, you have the option is to add an ‘action button’ to your sketch that allows you to ‘Play’ the function as a sound. In the previous class when we investigated superposition, we learned how waves added together, even those of different frequency.  I created a simple GSP sketch that added three waves together in which the amplitudes and frequencies could all be changed.

So…wave superposition – they could see it happening in front of them mathematically. Now they could hear the effects of superposition, changing amplitudes and frequencies, and understand how different frequency waves could add together to create the sounds that we hear. Standing waves are things bouncing up and down–> bouncing things up and down make sound waves –> we can also make sound waves by combining waves of different frequencies together.

The real key to making the final connection comes from applying some technology on loan from National Instruments. I have been working with their myDAQ student data collection device in finding ways to adapt it for use in math and science classes. The device has a number of analog and digital inputs and outputs, which are nice, but they pale in comparison with the Audio-In port in terms of ease of getting data into the computer. That data can then be analyzed very nicely by LabVIEW to create a frequency spectrum of the audio data in. I downloaded the software from an NI support site describing a beer bottle music project.

Now to lead up to the climax of the class  – I had a separate function generator create a 880 Hz sine wave which we played over the speaker. We had just seen a graph of this function, so students knew what it looked like graphically, and then what it sounded like. I had them whistle to imitate the frequency, and  now I showed them the graph of the amplitude vs. frequency plot in Labview, which looked like this:

 

I could change the pitch of my whistle and students could see how the peak moved left or right as the frequency went up or down. This was an easy introduction to the concept of frequency space and what it could be used for.

As with most of my classes, I’m the only one usually willing to sing in front of everybody, so I gave it a go:

A plastic bottle I had left over from the day before (though I kept this part to myself during class) was just asking to be played, and that gave some interesting results blowing lightly across it when it was empty…

…partly empty:

…and more full than empty: (Look at those odd harmonics shine!)

What I liked the most about this activity is that we could pretty much put any sound the students wanted to make, and the frequencies would pop out as they did above. They could see how singing a note and changing the shape of one’s mouth changed the amplitude of the various harmonics associated with the sound, and that was what we were really hearing as different sounds.

This all  breaks down to what I think is the most challenging concept for students to really understand about standing waves: If the sound waves we hear are made up of all of these different frequencies (as shown by these graphs we generated in real time), then it means the objects generating those sounds (bottles, voices, strings, tuning forks) are somehow able to vibrate at a whole bunch of frequencies at the same time.

At what various frequencies does it then vibrate?

WHAT ARE THOSE FREQUENCIES IT CAN VIBRATE AT, YOU ASK? 

Now we get to look at those diagrams relating length, wavelength, and frequency that used to show up at the beginning of my lesson before. Fifteen minutes at the end of class to find the frequencies of vibration for a string and a tube closed at one end – they figured it out pretty easily without much prodding.

This was NOT what I used to think a good physics lesson (or any lesson) needed to be: a series of presentations of content that made logical sense. Then we drill like crazy through problems to use the ideas.

Instead, we had a series of experiences that let the students interact in a visceral way with the material. Students could see waves and feel them. They could hear them. They could see what it sounded like when they were added together. They could observe how this new special graph took sounds in and spit back the frequencies of the individual waves that made them. I told them briefly about Fourier and showed them what happens when you play a triangle wave or sawtooth wave into the spectral analyzer – they easily saw the patterns of harmonics for each and (I think) had some intuition for what that meant. No need to have a bunch of theory first before seeing it in action.

I could not have done this with just a whiteboard. Or just a projector. The ease with which I was able to generate a frequency spectrum, create audible sine waves, model super position in a visible and tangible way – these were because of the convergence of technology in front of me. This is how technology is changing things as we speak – the tools to do these things are getting more and more easy to obtain by anyone. We have to get these tools in the hands of our students – they will run with it and get more out of it than we might even understand. We need to be there to guide them in the right direction – we need to help them decide from which parts of the fire-hose to drink.

This reminds me of two years ago when my dad was doing some data collection of sound levels near wind turbines in North-Eastern Ohio. He learned that there were already some applications available to do basic audio analysis on the iPhone that was similar to what the more expensive scientific equipment could do. That was two years ago. Now you can do this (from Faber Acoustical):

Fundamentals are still important. My students wouldn’t understand any of what we did today if they didn’t know what a sine wave is and what it means for a wave to have a frequency.

My argument is that we can bounce back and forth between theory and application. Let’s meet at a happy point in the middle where we can all do a big dance, capture it on video, and send it to YouTube. The whole “they need to understand the basics first” or “do their time” with the boring stuff argument is a hard one to maintain with these amazing tools available that used to only be accessible to universities and companies with deep pockets. Kids take to technology like ducks to water – using it to enable learning in an engaging and relevant way is one of the most powerful aspects of its presence in our classroom.

This lesson felt great today. I’ve always wanted to teach standing waves and show in depth what they are all about, but have never had the right tools at my disposal. This  is exactly the sort of style I want for all my classes – exploration and play that leads to a lesson that motivates itself. Ultimately I hope this lesson leads to a stronger connection and conceptual understanding of the material – they got it a whole lot faster than I did.

I hope that says more about them than it does about me….

Reflections on EARCOS Teachers Conference 2012 – Friday

I decided to use a few digital tools to record my thoughts at the EARCOS conference. At other workshops, I tend to take notes on paper, leave them in a folder, and possibly go back to them when inspiration hits, if I remember I have them. Since I am on my computer so much of the time (and NOT digging around in a filing cabinet to see what is in there) I think this will keep the ideas from this conference fresh and nearby.

I attended a few fantastic workshops Friday and tweeted extensively about each one as important ideas came up. The #earcos12 archive and search function will be really useful for going back and reminding myself of the ideas that came to mind during those workshops.

Workshop 1 – The Geometer’s Sketchpad Workshop: Beyond Geometry with Nicholas Jackiw

It was really a treat hearing the person that defined dynamic geometry talk about the philosophy of his software that implements the model. Having learned mathematics using GSP back in 9th grade, I’ve always seen the dynamic geometry as a natural lens through which geometric concepts can be viewed. Nick mentioned that mathematicians initially had a problem with the concept because two triangles with vertices A,B, and C that aren’t congruent are not the same. Since dynamic geometry defines triangles in terms of the relationships of vertices, two triangles with the same vertices connected in the same way represent the same geometric object. This means that any triangle ABC can be turned into any other triangle ABC just by dragging vertices around the screen.

We went through the basics of plotting points, lines, and measuring slope using the tools of Geometer’s Sketchpad. I hadn’t used it for a while, but it still remains a great program. Nick is a genuine guy with a love for mathematics and what his software can do for students learning concepts. He has a solid grasp and had some great activities that could be used for students to actively learn concepts through exploration rather than listening to a teacher go through a list of boring definitions.

I had the pleasure of sharing with Nick that I used Geometer’s Sketchpad to use geometry in ninth grade and that I still had print outs of the assignments I did using the software. Back then we printed out computer assignments and turned them in, much different from today when turning things in electronically is quite easy. I was a little star-struck talking to him, but as with most good teachers I meet, he was really friendly and appreciative of my comments.

Workshop 2 – The Harkness Method: The Best Class You Never Taught – Alexis Wiggins

One of the things I want help doing is improving the quality of classroom discussions. The shelf life of the discussions we have isn’t much longer than the class period itself. I have been able to extend that a bit having students create wiki pages, interact onor create videos describing their understanding of problems.

I think this workshop provided a real possibility for restructuring my class to do this far more effectively.

Alexis shared how the Harkness method (originated at Exeter Academy) has transformed her classroom and itneraction with students. Students spend class time discussing, arguing, and critiquing arguments. In the process, they learn extensively how to be good community members, be constructive in their criticism, and communicate their ideas. She shrewdly hooked us math/science teachers at the beginning (why are we always the cynics?) by sharing that Exeter does this in their math department. Alexis also shared that she does need to do direct instruction once in a while – her ratio is around 60% discussion, 40% other methods. She also does not do this for the entire class period, particularly for the younger (9th grade) students. Modeling the process and explicitly teaching students skills that make this successful in her room is a key part of her process. She made clear that it takes time to get them to be good at it.

Alexis posted her materials at https://alexiswigginsharknessmethod.pbworks.com.

Workshop 3 – Rules of Engagement – Using Technologies to Motivate Rather than Distract – Doug Johnson

We are constantly having discussions at our school (which is 1:1 Macbooks) about how to maximize student time on task during class – I think this is something almost everyone in schools is currently battling. The presence of technology has so many potential positive applications for learning. It is easy, however, to fixate on the negative aspects almost entirely and stall the process of making these potential benefits available to students in the classroom.

Along with having one of the most useful handouts I’ve ever received at a workshop, Doug Johnson made a number of fantastically relevant points about how school communities can think about the issue. The question he posed at the beginning was “How do teachers compete w/ tablets, smart phones, netbooks, mp3 players, portable games, etc?” What I found most interesting throughout was that he showed how it didn’t need to be a competition. Instead teachers can capitalize on the opportunity

His emphasis on the distinction between entertainment and engagement really resonated with me, as I always wonder if the activities I do with my students are actually helping them learn or not. We then worked together to identify ways to make technology an active part of classroom activities, including a lot of modeling using gosoapbox.com and references to other similar sites such as socrative.com.

Doug’s presentations can all be found at https://dougjohnson.wikispaces.com/engage

Workshop 4 – Digital Citizenship: The Forgotten Fundamental Kim Cofino

This workshop from the excellent Kim Cofino was a perfect pairing with Doug’s workshop and a good ending point for the day. She clearly described her process at the Yokohama International School of rolling out (all at once, which she said was the best idea ever) her 1:1 laptop program with students.

The most important takeaway was how much deliberate planning and community collaboration went into not only creating the acceptable use policy but actively sharing that philosophy with the students, teachers, and parents. The school year started with two days of 1:1 boot camp activities – students discussing and debating different aspects of the policy. She also mentioned that the students will soon repeat some elements of this training and discussion now that the community has been through several months of living out the policy.

An important element of this is that students are explicitly taught and engaged in activities that teach them digital citizenship. She made clear that this does not happen by accident, or by hoping that students will know how to act when they are suddenly given the power afforded them by technology. This is one of the key things I will be taking back with me to Hangzhou.

Her presentation and resources can be found at http://dctff.wikispaces.com/overview

Comments:

This has been a really fantastic experience being at the conference this year – I am learning so much at the workshops and through meeting the incredible collection of teachers here. I appreciate that everyone has been so positive and open in sharing their work and ideas with me. I admit it – I’m addicted to this conference atmosphere. Thankfully, I’ll be able to keep in touch with the people I have met here, and continue learning from them well after I have left Bangkok.

EARCOS 2012 Presentation – Using Geogebra for Skill Development

After a late night getting into Bangkok and a couple hours of sleep (though I suppose few good stories start “I had a good long night of sleep when I first arrived in Thailand) I made it to the start of the EARCOS 2012 Teachers’ conference yesterday morning. I’ll have more to say about the details of the conference later on, but I wanted to post briefly about the presentation I gave on Geogebra in the afternoon.

The room was packed with teachers and coaches armed with laptops and interested in seeing how the program works. My focus was on giving feedback, with Geogebra as the medium for that feedback. I did not intend it to be a beginner’s tutorial on Geogebra for a few reasons:

  • There is so much fantastic material out there already that shows how to use the software.
  • I wanted to specifically focus on the philosophy of using software to provide instant feedback to students on mathematical tasks.
  • Nick Jankiw from Geometers Sketchpad was doing a series of workshops on GSP and I didn’t want to engage in the Geometers Sketchpad vs. Geogebra debate. I see them both as excellent pieces of software. I choose to use Geogebra for a number of reasons that I mentioned in my presentation. The truth is that Geometers Sketchpad defined the field of dynamic geometry, and I do think it’s important to acknowledge that fact.
That said, anyone that wants help getting started with Geogebra should feel free to ask me for help. Thanks to a great suggestion from John Burk (@occam98) and Andy Rundquist (@arundquist) I had some screencasts demonstrating more advanced sketches of my own playing on the screen while waiting for the program to download and while figuring out the basics.
I thought the workshop went well  – I wish I had not felt the need to talk so much and had given more time for people to interact with each other. That said, I think there were many that came and left with much more knowledge than when they entered. A few told me that they already plan to use it next week in their classes.

My slides and accompanying notes can be found here: EARCOS presentation – notes pages

The video is below – unfortunately there wasn’t a great place to put the camera to be able to get me and the slides, and the contrast is not great to be able to see what I am doing in the program. I’ll find some time to post some screencasts of the demonstrations I did with the software later on.

[youtube http://www.youtube.com/watch?v=eW4-OxkqbfM&w=560&h=315]

The TacoCopter? – a gimmick for integration review

I received an email sending me to this site yesterday about the TacoCopter, which of course was spot on given my interest in all things robotic. I also had PID control on the brain thanks to my course on driving a robot car from Udacity. Bits of python code were in my head already, and I had a strong need to put it all together. Given that it was also Sunday (a workday for most teachers) I had to plan for classes tomorrow, specifically Calculus and Physics.

All of this was in the context of the beautiful afternoon I spent on the balcony of the apartment looking out at the warmest, bluest Hangzhou skies of the year so far. It put me in the mood to do something a bit different for tomorrow’s Calculus class. The AP students will be reviewing related rates and implicit differentiation, but the regular students…they get to have a bit more fun.

This is the activity we will be looking at tomorrow in class: CW – TacoCopter Project

The full wiki page that students will be following is located here: http://wiki.hischina.org/groups/gealgerobophysiculus/wiki/42712/Calculus_Unit_8__The_TacoCopter.html

Some python code for simulating the TacoCopter rising to altitude, which can be found here at github.

Then Geogebra for plotting the data, which shows the lovely simulated accelerometer data with noise:

I don’t really know how it will go. At least students will have an excuse to grin as they review.

Snacking on Statistics and Variability

One of my goals this year in Algebra 2 has been to include more discrete math, statistics, and probability when I can. I’ve been convinced by all sorts of smart people that as traditional as it may be to have Calculus as the ultimate goal for math students, statistics and probability are the math that people are more likely to need to use. It compels me to include it in my courses as more than a separate unit.

As if I didn’t need another reason, we are also in a spell of reviewing properties of radicals, and it’s refreshing to get my students thinking differently after a period of simplifying, multiplying, and rationalizing.

I gave them the following scenario:

  • Imagine yourself in twenty years – you are, of course, rich and famous. You are hiring someone to fly your personal jet. your last pilot fell asleep on the job, though he was luckily parked at the gate when it happened.

Two pilots have applied for the position, both equally qualified as pilots. In order to help you make your decision (and avoid the previous situation), you have asked them to keep track of how many hours of sleep they get over a two week period before the interview.

Two weeks later, they return to you with the following data:

  • What differences do you notice about the two pilots?
  • What calculations would you make to describe any quantitative differences between them?
  • Which one would you hire? Why?

Note: This data is completely made up. My new semi-obsession is in using normal distributions to mess up clean functions and force my students (in physics and math) to deal with messy data.

The students almost immediately started calculating means – exactly what I would have expected them to do given what they have been taught to do when faced with a table of data like this. Some did so manually, others used the Geogebra file that generated the data to make their calculation.

The results were fairly consistent – everyone chose the second pilot. When asked why, they said the pilot gets more sleep on average, and so would be the better choice.

When I asked who was more consistent in their sleep, they were easily able to identify the first pilot. When asked why, many had explanations that correctly danced around how most of the data was closer to the average. No students really brought up this fact before I asked though, which leads me to believe they observed one of two things:

  • The importance of the  consistency doesn’t really matter given the difference in the means for how much sleep the pilots got.
  • They didn’t think to look at consistency at all.

Some other interesting tidbits:

  • None of the students thought to construct a histogram to look at the data. When asked, about half of the class said they knew how to construct a histogram. I didn’t dig any deeper to flesh this out. I was going to throw one together in Geogebra, but decided that might be something we should look at with more time available.
  • Half of the class that is taking AP Psychology didn’t think about finding standard deviation. Again, I didn’t dig any deeper to find if this was because they didn’t know that it might apply here, or because they thought the values of the means were more important.

There is plenty here to generate discussion, but the one thing I wonder about is if variation about a mean is a concept that comes naturally to students to consider when given a set of 1-D data. One of my professors mentioned offhand in an experimental design class that any measurement you take is a distribution, a point which I have never forgotten. Up to that moment, I had never really thought much about it either.

Sure, I had collected data in my biology, chemistry, and physics classes before and knew I had to take multiple data points. All I knew then was that doing so made my data “better”. More data makes things better. Get it? My understanding in high school science was also that you never measure the same quantity at the exact same value ten times in a row because someone in your lab group is always messing it up or doing it wrong. Averaging things together smooths that out. I don’t recall ever discussing in either math or science class that the true beauty of statistics comes from managing, communicating, and understanding variability in data that will never really go away. I have always shuddered when students write lab report conclusions that discuss how “the data are/is wrong because” rather than focusing on what the data reveals about an experiment.  We definitely want to work to minimize experimental error, but sometimes the variation in the data is an important characteristic of what is being measured.

Maybe this is something that needs to be explicitly taught in the way we present statistics to our students. It seems like something that needs to be drawn out over time, rather than in one big statistics unit of a course that focuses on other things. I think using  technology to handle the mechanics of calculating statistical quantities allows students to focus more on what the statistics say and develop their intuition about it. We risk letting the important ideas of variation and statistics collect dust and stagnate as  another box of content for students to throw in the closet of their busy, distracted brains.

Bringing robotic cars and Udacity to my classroom

I was really excited to learn about Udacity, a new online education system that premiered two courses on February 20th. That a course on programming a robotic car would appeal to me is probably not surprising to anyone that knows me. I also love having yet one more excuse to continue learning Python, especially one that gets me working with an expert in the field such as Professor Sebastian Thrun. I recall reading about him shortly before his team’s successful bid at the DARPA Grand Challenge, and have since seen his name repeated at many key moments along my development as a robotics enthusiast.

The course is structured really well, with short videos introducing concepts, quizzes and programming tasks (with solutions) along the way to check comprehension, and homework assignments. The students love that I have homework.

I am busy, but this was too cool to pass up.

I also have a pretty hard time hiding the things I’m enthusiastic about in my classroom, so the content of the class has been something I’ve mentioned and shared with students at the start or end of planned activities. The whole classroom gasped at this video from the 25th second onward:
[youtube http://www.youtube.com/watch?v=bdCnb0EFAzk?feature=player_embedded&w=640&h=360]

Based on that reaction, I really wanted to give them a sense for the things I was learning to do. The first week centered on learning about localization – a process that uses probability calculations to estimate the location of the car using sensor readings and a map of the surroundings. I did a quick overview of what this meant as a filler activity to break up work during class, but wanted to find a way to do much more.

Today’s Algebra 2 class was going to be missing a couple students that are attending a Model UN conference, so I figured it would be a good time to try something different.

We started with the following warm-up problems:

Mr. Weinberg tells you we are guaranteed to have a quiz one of the days between Monday and Friday. He tells you that the probabilities of the quiz happening Monday through Thursday are 0.1, 3/8, 1/16, and 36%. What is the probability that the quiz will be on Friday? On which day is the quiz most likely to occur?

This helped review the total probability principle which is key to understanding the localization algorithm. We also did a review of finding the probability of compound independent events, first with a tree diagram, and then using multiplication and the counting principle.

We then went through the following activity for the rest of the period:
Robot Localization activity

I adapted parts of the course material provided by Udacity, primarily simplifying language, cleaning up diagrams, and adjusting the activities for my students who do not have any programming ability. We did have a Python activity back in October, but installing and running Python was a hassle on the 1-1 Macbooks with OSX since I was trying to do it with Python 3 and IDLE. It was only shortly afterward that I learned that an earlier version of Python was automatically installed. Oops. For this activity, we used http://repl.it/ to do the programming. This worked fantastically well.

The students seemed to do really well with the introductory material and filling things in, and modifying the basic programming went smoothly. They ran into some trouble around problem 7, which I half expected – that was the first part of the activity when I told them to do something without any rationale behind it. Most were generally able to implement the procedure and get to problem 9, but at this point at the end of the day on a Friday afternoon, fatigue started to take over. This was after around 45 minutes of working on the activity.

I added a section on motion for possible use in another class, as I ultimately would like them to be able to throw my own homework solution code into a simulator provided by Udacity user Anna Chiara. I did not deal with any of the sensor probability or move probability. The intuition for understanding how those apply in the algorithm is a bit subtle for the background of my students, and would take more of an investment of time than I think my students have the patience for at this state. I think it would be easier to talk about how these issues exist, and then have them observe what they mean by looking at the output of the program.

All in all, it was a cool, low-key way to share my own learning with students after an exhausting week. I think we all needed a bit of a change.

Volumes of Revolution & 3D Modeling

I had a conversation with a colleague a few years ago about volumes of revolution in Calculus. We were both a few years removed from our own Calculus courses in high school and college, and we were talking about how we thought about the concept visually.

For those that need a refresher, here is the idea behind a volume of revolution. Imagine you have a solid object that can be lined up with the x-axis so that its cross section looks like the image below. The object would have a pointy end at the origin (0,0) and a circular face located at x = 1. The closest real world object that fits this description is a Hershey’s Kiss.

The object is axially symmetric about the x-axis. If you were to cut the object with a knife so that the cut passes through the pointy end and the center of the flat face, the image at left would always be the cross section.

A volume of revolution is usually defined by an even simpler idea. Take a region of a graph and rotate it in a circle around some axis. The region at left is defined by rotating the area under the graph of y = x 2 around the x-axis.

My colleague’s way of visualizing this idea started with the solid itself. Cut it into a series of discs, each of width dx , and then analyze a single differential disc to come up with an integral expression for the entire volume. This requires being able to visualize the entire solid first, and then see how it can be cut into discs.

I didn’t see it this way. I could visualize the solid usually, but to then mentally cut the solid into discs, and then construct a differential volume seemed to have one too many steps to make it simple.  I focused on the step that made conceptual sense to me: start with a defined region and rotate it around an axis to create a solid. The differential strip of area we had been making underneath the graph since the first introduction of the definite integral was what I always visualized during integration. I could visualize taking that strip and rotating it around to form a disc, and using that concept for the differential volume. Then add up these discs through an integral to find the volume.

When I taught volume of revolution for the first time, I wanted to introduce it in a way that would emphasize how I had come to understand the concept. Granted, this assumes my way will work for the students, but so far it seems to be doing so pretty well.

Three dimensional computer modeling programs (Blender, Pro-E, Autodesk Inventor, etc) all have a function called ‘Revolve’ which is, by definition, how volumes of revolution are created. The idea is that you define a region, pick an axis, and then the software will create a 3D solid and display it. Having a copy of Pro-E from our FIRST Tech Challenge team, I was able to introduce the process with a series of demonstrations live with the software. Some examples:

The students immediately saw what was going on, and didn’t think much of the process. I could quickly make a sketch, revolve it, and then rotate the object around for students to see what it would look like if actually in front of them. We then proceeded to revolve strips under and between graphs to generate discs and washers. Writing the integrals was then a fairly simple process.

I think the difficulty that might come up with this type of problem is the visualizing step. Students must visualize the 3D shape in order to solve problems related to its volume. I think having this sort of tool available has made a big difference in my students seeing what it means to create a volume of revolution, which then leads to an easier time conceptualizing how to then find its volume using Calculus.

Party games & geometry definitions

Today’s geometry class started with a new random arrangement of student seats. It never fails to amaze me how the dynamics of the whole room change with a shuffle of student locations.

The lesson today was the first of our quadrilateral unit. Normally after tests, I don’t tend to have homework assignments, but I decided to make an exception with a simple assignment:

Create a single Geogebra file in which you construct and label all of the quadrilaterals given in the textbook: parallelogram, rhombus, square, kite, rectangle, trapezoid, and isosceles trapezoid.

This appealed to me because I really dislike lessons in which we go through definitions slowly as a group. I also knew that giving the students some independence in reviewing or learning the definitions of these quadrilaterals was a good thing. Sometimes they are a bit to reliant on me to give them all the information they need. For this assignment, students would need to understand the definitions of quadrilaterals in order to construct them, and that was a good enough for walking into class today.

The warm-up activity involved looking at unlabeled diagrams of quadrilaterals, naming them, and writing any characteristics they noticed about them from the diagrams:

Some had trouble with the term ‘characteristics’, but a peek down at the chart just below on the paper helped them figure it out:

Based on what they knew from the definitions before class, I had them complete this chart while talking to their new partner. There was lots of good conversation and careful use of language for each listed characteristic.

This led to the next thing that often serves as an important (though often boring) exercise: new vocabulary. I used one of my favorite activities that gets students focused on little details – each student received one of the following four charts. The chart is originally from p. 380 of the AMSCO Geometry textbook, and was digitally ruined using GIMP.

The students had a good time filling in the missing information and conferring with each other to make sure they had it all. We then came up with some examples of consecutive vertices, angles, diagonals, and opposite sides.

 

From their work with the chart and using the new vocabulary whenever possible, we then did the following:

What information would you need in order to prove that a quadrilateral is… (use as much of the new vocabulary as possible!)

  • a square?

  • a rhombus?

  • a parallelogram?

  • a rectangle?

  • a trapezoid? (an isosceles trapezoid?)

  • a kite?

I was really pleased with how they did with this exercise – they really seemed to be interacting with the definitions and vocabulary well.

Finally, we arrived at the part that was the most fun. You know that annoying ice-breaker you sometimes are forced to do at professional development sessions where you wear something on your head and have to get the other attendees to tell you who you are?

I hate that activity. That usually means it’s perfect for my students:

Here are the quadrilaterals:
Quadrilaterals – Who-Am-I activity

The students were all smiles during the ten minutes or so we spent going through it – yes, I had one too! They were using the vocabulary we had developed during the day and were pretty creative in getting each other to guess the dog names as well.

In the end, I feel pretty good about how today’s set of activities went. The engagement level was pretty high and everyone did a good job of interacting with the definitions in a way that will hopefully lead to understanding as we start proving their properties in coming classes.

Turning random facts into logistics curves – ODE per day series continued.

I previously wrote about making sure that every class during our unit on differential equations starts with some differential equation they can see or feel in a concrete way.

During the last class, we investigated a draining tank using the video posted by Dan Meyer at his blog.

Today we did something different. I told them that I was doing an experiment with a simple task. They all needed to find the answers to some  questions as quickly as possible:

When they found the answers, I wanted them to quickly throw a hand in the air to let me know. I told them to be honest – they didn’t know what I was doing with the information yet, so there really wasn’t a chance to skew it.

I then showed them the slide with the questions:

I also simultaneously started the following Python program. (UPDATE: Code is posted here.) This let me easily record any time a student raised his/her hand.

I then pasted the data directly into a Geogebra spreadsheet and graphed the data…

…and then fit a logistics curve to the data:

They had seen and heard the concept of learning/performance curves before, but it was really great to be able to develop one on the spot with the class. I was impressed with how good the data turned out. It was then neat to be able to show the differential equation that describes this type of phenomenon and solve it to get this type of function.

As is probably obvious, I only have ten students in this group. It would be really cool to try something like this with a bigger group and see if the data fits as nicely.

Building meaning for momentum from discussions, definitions, and data.

Today we started our next unit in physics with a ‘next time question’ from Paul Hewitt:

My reason for giving this was specifically because of the fact that we haven’t learned anything about it. I wanted the students to speak purely from their intuition. I asked them the following:

We aren’t quite ready to answer this by calculation, but I do want you to make a guess.

Will they move together faster than, slower than, or with the same speed as the ball?

Would your answer change if the ball bounced off Jocko instead of him catching it?

Student responses included:

  • We need to know if he bends backwards when he catches it, because that will affect it.
  • No matter how he does catch it, he will move slower. The larger mass will result in a smaller acceleration.
  • The clown has a non-conservative force, so the kinetic energy will decrease.

Interesting responses. We talked a bit about collisions and throws and catches of objects and what they ‘felt’ doing this with different objects. I introduced the idea that it might be nice to have a physics quantity that contains the direction and rate information of velocity, as well as the mass.  I told them that physicists did, in fact, have such a quantity called momentum. They responded with a few non-physics related ways they had heard the term and described what it meant.

To figure things out about how momentum relates to collisions, I then had them analyze the three air track collision videos from the Doane Physics video library using Tracker. Their tasks were as follows:

  • Find the momentum of each cart before and after the collision for the video you are assigned. Calibration information is contained in the first frame of each video.
  • Find the total momentum of the system before and after the collision.
  • Find the total kinetic energy of the system before and after the collision.
  • What is thechange of the momentum of the system during the collision?
  • What is the change of the kinetic energy of the system during the collision?
  • Talk to your classmates and compare your answers for the three different videos.

It was pretty cool to see them jump in with Tracker and know how to analyze things without too much trouble. Fairly soon afterwards, we had some initial velocities and final velocities, and changes in momentum to compare.

I was, of course, leading them toward something with the change calculations.
We calculated the changes in momentum, which were non-zero. Were the magnitudes significant? A student suggested looking at the percent change compared to the initial momentum. For the first two videos, the loss was less than 1%, though for the third it was around 20%.
A student proposed the possibility that the change should be zero if no momentum is lost during the transfer. Comments were made about how that made sense in the context of our previous unit on energy – things feeling right when all of a quantity can be accounted for.
I then did a little pushing (since we were almost out of time) about what this might mean about total initial momentum and total final momentum.  I also gave them definitions for elastic and inelastic collisions. I then assigned them a couple simple questions that I wanted them to figure out if we can say that the change in total momentum before and after is zero:
Then it was time for Calculus.
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I don’t usually like giving students information. I don’t like giving it away without some sense of where it comes from. I also like when students can discover quantities without equation definitions. Sometimes though, the simplicity of an idea like momentum and its power can come from taking the calculation itself as a tool that can be used to analyze a situation.
In previous classes, I have given the definition, shown situations in which momentum is conserved, and then asked students to use this idea of momentum conservation with their math skills to find unknown quantities. I really liked this alternate approach today of using momentum itself to analyze a situation and then have the idea of conservation come out of discussion. I think its potential for ‘stickiness’ in the minds of students is much greater this way.
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