Category Archives: physics

2012-2013 Year In Review – Learning Standards


This is the second post reflecting on this past year and I what I did with my students.

My first post is located here. I wrote about this year being the first time I went with standards based grading. One of the most important aspects of this process was creating the learning standards that focused the work of each unit.

What did I do?

I set out to create learning standards for each unit of my courses: Geometry, Advanced Algebra (not my title – this was an Algebra 2 sans trig), Calculus, and Physics. While I wanted to be able to do this for the entire semester at the beginning of the semester, I ended up doing it unit by unit due to time constraints. The content of my courses didn’t change relative to what I had done in previous years though, so it was more of a matter of deciding what themes existed in the content that could be distilled into standards. This involved some combination of concepts into one to prevent the situation of having too many. In some ways, this was a neat exercise to see that two separate concepts really weren’t that different. For example, seeing absolute value equations and inequalities as the same standard led to both a presentation and an assessment process that emphasized the common application of the absolute value definition to both situations.

What worked:

  • The most powerful payoff in creating the standards came at the end of the semester. Students were used to referring to the standards and knew that they were the first place to look for what they needed to study. Students would often ask for a review sheet for the entire semester. Having the standards document available made it easy to ask the students to find problems relating to each standard. This enabled them to then make their own review sheet and ask directed questions related to the standards they did not understand.
  • The standards focus on what students should be able to do. I tried to keep this focus so that students could simultaneously recognize the connection between the content (definitions, theorems, problem types) and what I would ask them to do with that content. My courses don’t involve much recall of facts and instead focus on applying concepts in a number of different situations. The standards helped me show that I valued this application.
  • Writing problems and assessing students was always in the context of the standards. I could give big picture, open-ended problems that required a bit more synthesis on the part of students than before. I could require that students write, read, and look up information needed for a problem and be creative in their presentation as they felt was appropriate. My focus was on seeing how well their work presented and demonstrated proficiency on these standards. They got experience and got feedback on their work (misspelling words in student videos was one) but my focus was on their understanding.
  • The number standards per unit was limited to 4-6 each…eventually. I quickly realized that 7 was on the edge of being too many, but had trouble cutting them down in some cases. In particular, I had trouble doing this with the differentiation unit in Calculus. To make it so that the unit wasn’t any more important than the others, each standard for that unit was weighted 80%, a fact that turned out not to be very important to students.

What needs work:

  • The vocabulary of the standards needs to be more precise and clearly communicated. I tried (and didn’t always succeed) to make it possible for a student to read a standard and understand what they had to be able to do. I realize now, looking back over them all, that I use certain words over and over again but have never specifically said what it means. What does it mean to ‘apply’ a concept? What about ‘relate’ a definition? These explanations don’t need to be in the standards themselves, but it is important that they be somewhere and be explained in some way so students can better understand them.
  • Example problems and references for each standard would be helpful in communicating their content. I wrote about this in my last post. Students generally understood the standards, but wanted specific problems that they were sure related to a particular standard.
  • Some of the specific content needs to be adjusted. This was my first year being much more deliberate in following the Modeling Physics curriculum. I haven’t, unfortunately, been able to attend a training workshop that would probably help me understand how to implement the curriculum more effectively. The unbalanced force unit was crammed in at the end of the first semester and worked through in a fairly superficial way. Not good, Weinberg.
  • Standards for non-content related skills need to be worked in to the scheme. I wanted to have some standards for year or semester long skills standards. For example, unit 5 in Geometry included a standard (not listed in my document below) on creating a presenting a multimedia proof. This was to provide students opportunities to learn to create a video in which they clearly communicate the steps and content of a geometric proof. They could create their video, submit it to me, and get feedback to make it better over time. I also would love to include some programming or computational thinking standards as well that students can work on long term. These standards need to be communicated and cultivated over a long period of time. They will otherwise be just like the others in terms of the rush at the end of the semester. I’ll think about these this summer.

You can see my standards in this Google document:
2012-2013 – Learning Standards

I’d love to hear your comments on these standards or on the post – comment away please!

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Speed of sound lab, 21st century version


I love the standard lab used to measure the speed of sound using standing waves. I love the fact that it’s possible to measure physical quantities that are too fast to really visualize effectively.

This image from the 1995 Physics B exam describes the basic set-up:
Screen Shot 2013-05-16 at 3.43.30 PM

The general procedure involves holding a tuning fork at the opening of the top of the tube and then raising and lowering the tube in the graduated cylinder of water until the tube ‘sings’ at the frequency of the tuning fork. The shortest height at which this occurs is the fundamental frequency of vibration of the air in the tube, and this can be used to find the speed of sound waves in the air.

The problem is in the execution. A quick Google search for speed of sound labs for high school and university settings all use tuning forks as the frequency source. I have always found the same problems come up every time I have tried to do this experiment with tuning forks:

  • Not having enough tuning forks for the whole group. Sharing tuning forks is fine, but raises the lower limit required for the whole group to complete the experiment.
  • Not enough tuning forks at different frequencies for each group to measure. At one of my schools, we had tuning forks of four different frequencies available. My current school has five. Five data points for making a measurement is not the ideal, particularly for showing a linear (or other functional) relationship.
  • The challenge of simultaneously keeping the tuning fork vibrating, raising and lowering the tube, and making height measurements is frustrating. This (together with sharing tuning forks) is why this lab can take so long just to get five data points. I’m all for giving students the realistic experience of the frustration of real world data collection, but this is made arbitrarily difficult by the equipment.

So what’s the solution? Obviously we don’t all have access to a lab quality function generator, let alone one for every group in the classroom. I have noticed an abundance of earphones in the pockets of students during the day. Earphones that can easily play a whole bunch of frequencies through them, if only a 3.5 millimeter jack could somehow be configured to play a specific frequency waveform. Where might we get a device that has the capacity to play specific (and known) frequencies of sound?

I visited this website and generated a bunch of WAV files, which I then converted into MP3s. Here is the bundle of sound files we used:
SpeedOfSoundFrequencies

I showed the students the basics of the lab and was holding the earphone close to the top of the tube with one hand while raising the tube with the other. After getting started on their own, the students quickly found an additional improvement to the technique by using the hook shape of their earphones:
Screen Shot 2013-05-16 at 4.03.13 PM

Data collection took around 20 minutes for all students, not counting students retaking data for some of the cases at the extremes. The frequencies I used kept the heights of the tubes measurable given the rulers we had around to measure the heights. This is the plot of our data, linearized as frequency vs. 1/4L with an length correction factor of 0.4*diameter added on to the student data:
Screen Shot 2013-05-16 at 4.14.22 PM

The slope of this line is approximately 300 m/s with the best fit line allowed to have any intercept it wants, and would have a slightly higher value if the regression is constrained to pass through the origin. I’m less concerned with that, and more excited with how smoothly data collection was to make this lab much less of a headache than it has been in the past.

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Visualizing the invisible – standing waves


I wrote a post more than a year ago on a standing waves lesson I did. Today I repeated that lesson with a few tweaks to maximize time spent looking at frequency space of different sounds. The Tuvan throat singers, a function generator, and a software frequency generator (linked here) again all made an appearance.

We focused on the visceral experience of listening to pure, single frequency sound and what it meant. We listened for the resonant frequencies of the classroom while doing a sweep of the audible spectrum. We looked at the frequency spectrum of noises that sounded smooth (sine wave) compared to grating (sawtooth). We looked at frequencies of tuning forks that all made the same note, but at different octaves, and a student had the idea of looking at ratios. That was the golden idea that led to interesting conclusions while staring at the frequency spectrum.

Here is a whistle:
Screen Shot 2013-05-13 at 3.10.40 PM
…a triangle wave (horizontal axis measured in Hz):

Screen Shot 2013-05-13 at 3.09.45 PM

…a guitar string (bonus points if you identify which string it was:
Screen Shot 2013-05-13 at 3.12.14 PM

…and blowing across the rim of a water bottle:
Screen Shot 2013-05-13 at 3.14.04 PM

The ratios of frequencies for the guitar string are integer multiples of the fundamental – this is easily derived using a diagram and an equation relating a wave’s speed, frequency, and wavelength. It’s also easily seen in the spectrum image – all harmonics equally spaced with each other and with the origin. The bottle, closely modeled by a tube closed at one end, has odd multiples of the fundamental. Again, this is totally visible in the image above of the spectrum.

I’m just going to say it here: if you are teaching standing waves and are NOT using any kind of frequency analyzer of some sort to show your students what it means to vibrate at multiple frequencies at once, you are at best missing out, and at worst, doing it plain wrong.

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Computational modeling & projectile motion, EPISODE IV


I’ve always wondered how I might assess student understanding of projectile motion separately from the algebra. I’ve tried in the past to do this, but since my presentation always started with algebra, it was really hard to separate the two. In my last three posts about this, I’ve detailed my computational approach this time. A review:

    • We used Tracker to manually follow a ball tossed in the air. It generated graphs of position vs. time for both x and y components of position. We recognized these models as constant velocity (horizontal) and constant acceleration particle models (vertical).
    • We matched graphical models to a given projectile motion problem and visually identified solutions. We saw the limitations of this method – a major one being the difficulty finding the final answer accurately from a graph. This included a standards quiz on adapting a Geogebra model to solve a traditional projectile motion problem.
    • We looked at how to create a table of values using the algebraic models. We identified key points in the motion of the projectile (maximum height, range of the projectile) directly from the tables or graphs of position and velocity versus time. This was followed with the following assessment
    • We looked at using goal seek in the spreadsheet to find these values more accurately than was possible from reading the tables.

After this, I gave a quiz to assess their abilities – the same set of questions, but asked first using a table…
Screen Shot 2013-03-08 at 6.55.40 PM
… and then using a graph:
Screen Shot 2013-03-08 at 6.57.23 PM

The following data describes a can of soup thrown from a window of a building.

  • How long is the can in the air?
  • What is the maximum height of the can?
  • How high above the ground is the window?
  • Is the can thrown horizontally? Explain your answer.
  • How far from the base of the building does the can hit the ground?
  • What is the speed of the can just before it hits the ground?</li

I was really happy with the results class wide. They really understood what they were looking at and answered the questions correctly. They have also been pretty good at using goal seek to find these values fairly easily.

I did a lesson that last day on solving the problems algebraically. It felt really strange going through the process – students already knew how to set up a problem solution in the spreadsheet, and there really wasn’t much that we gained from obtaining an algebraic solution by hand, at least in my presentation. Admittedly, I could have swung too far in the opposite direction selling the computational methods and not enough driving the need for algebra.

The real need for algebra, however, comes from exploring general cases and identifying the existence of solutions to a problem. I realized that these really deep questions are not typical of high school physics treatments of projectile motion. This is part of the reason physics gets the reputation of a subject full of ‘plug and chug’ problems and equations that need to be memorized – there aren’t enough problems that demand students match their understanding of how the equations describe real objects that move around to actual objects that are moving around.

I’m not giving a unit assessment this time – the students are demonstrating their proficiency at the standards for this unit by answering the questions in this handout:
Projectile Motion – Assessment Questions

These are problems that are not pulled directly out of the textbook – they all require the students to figure out what information they need for building and adapting their computer models to solve them. Today they got to work going outside, making measurements, and helping each other start the modeling process. This is the sort of problem solving I’ve always wanted students to see as a natural application of learning, but it has never happened so easily as it did today. I will have to see how it turns out, of course, when they submit their responses, but I am really looking forward to getting a chance to do so.

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A computational approach to modeling projectile motion, part 3.


I’ve been really excited about how this progression is going with my physics class – today the information really started to click, and I think they are seeing the power of letting the computer do the work.

Here’s what we did last time:

In a fit of rage, Mr. Weinberg throws a Physics textbook while standing in the sand box outside the classroom. By coincidence, the book enters the classroom window exactly when it reaches its maximum height and starts to fall back down.


  • Is it appropriate to neglect air resistance in analyzing this situation? Justify your answer.
  • We want to use this problem to estimate the height of the classroom window above the ground. Identify any measurements you would take in order to solve this problem. (No, you may not measure the height of the classroom window above the ground.)
  • Use your spreadsheet to find the height of the window as accurately as you can.

Note: This activity got the students using the spreadsheet they put together last time to figure out the maximum height of the object. They immediately recognized that they needed some combination of dimensions, an angle, and a launch speed of the book.


These tables of values are easy to read, but we want to come up with a more efficient way to get the information we need to solve a problem.

The table below represents a particular projectile. Identify as much about its movement as you can. How high does it go? How far does it go? When does it get there? That’s the kind of thing we’re interested in here.

Screen Shot 2013-03-04 at 5.02.17 PM

Note that at this point the students are spending time staring at tables of equations. This is clearly not an efficient way to solve a problem, but it’s one that they understand, even the weakest students. They can estimate the maximum height by looking at the table of y-values, but the tedium of doing so is annoying, and this is what I want. I try to model this table of values with the spreadsheet they put together with them telling me what to do. Every time I change a value for initial speed or initial height, the location of the maximum changes. It’s never in the same place.

Eventually, someone notices the key to finding the maximum isn’t with the y-position function. It’s with the vertical velocity. When does the y-component equal zero?

This is where the true power of doing this on the spreadsheet comes alive. We look at the table of values, but quickly see that we don’t need a whole table. We go from this:
Screen Shot 2013-03-04 at 5.18.48 PM

…to this:
Screen Shot 2013-03-04 at 5.20.52 PM

Clearly this t-value is wrong. Students can adjust the value of the time in that cell until the velocity in the cell below is zero. A weak student will get how to do this – they are involved in the process. The tedium of doing this will prompt the question – is there a better way? Is this when we finally switch to an algebraic approach? No, not yet. This is where we introduce the Goal Seek tool.

Screen Shot 2013-03-04 at 5.23.12 PM

The spreadsheet will do the adjustment process for us and find the answer we are looking for. With this answer in hand, we can then move on to posing other questions, and using goal seek to find the values we are looking for.

The process of answering a projectile motion question (how far does it go? how high does it go?) through a spreadsheet then becomes a process of posing the right questions:
Screen Shot 2013-03-04 at 5.26.42 PM

This is the type of reasoning we want the students to understand within the projectile motion model. Whether your tool of choice for answering these questions is the graph, equations, or a table of values, posing these questions is the meat and potatoes of this entire unit in my opinion.

The next step is to then introduce algebraic manipulation as an even more general way to answer these questions, including in cases where we don’t have numbers, but are seeking general expressions.


Today I had a student answer the following questions using the goal seek method with the numerical models I’ve described above:

A ball is thrown horizontally from a window at 5 m/s. It lands on the ground 2.5 seconds later. How far does the ball travel before hitting the ground? How high is the window?

He solved it before anyone else. This is a student that has struggled to do any sort of algebraic manipulation all year. There’s something to this, folks. This is the opening to the fourth class of this unit, and we are now solving the same level questions as the non-AP students did a year ago with an algebraic approach and roughly the same amount of instruction time. Some things to keep in mind:

  • My students are consistently using units in all of their answers. It is always like pulling teeth trying to get them to include units – not so much at the moment.
  • They are spending their time figuring out the right questions to ask, not which equation to ‘plug’ into to get an answer.
  • They immediately see what information is missing in their model at the beginning of a problem. They read the questions carefully to see what they need.
  • The table of values gives them an estimate they can use for the problem. They have an idea of what the number should be from the table, and then goal seek improves the accuracy of the number.
  • At the end of the problem, students have all of the initial information filled out to describe all of the parts of the problem. They can check that the horizontal range, maximum height, and other waypoints of the path match the given constraints of the problem. This step of checking the answer is a built-in feature to the process of matching a model – not an extra step that I have to demand at the end. If it doesn’t match all of the given constraints, it is obvious.

I am looking for push back – is there anything I am missing in this approach? I get that deriving formulas is not going to come easily this way, but I think with a computer algebra system, it’s not far away.

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A computational approach to modeling projectile motion, continued.


Here is the activity I am looking at for tomorrow in Physics. The focus is on applying the ideas of projectile motion (constant velocity model in x, constant acceleration model in y) to a numerical model, and using that model to answer a question. In my last post, I detailed how I showed my students how to use a Geogebra model to solve projectile motion.

Let me know what I’m missing, or if something offends you.


A student is at one end of a basketball court. He wants to throw a basketball into the hoop at the opposite end.

  • What information do you need to model this situation using the Geogebra model? Write down [______] = on your paper for any values you need to know to solve it using the model, and Mr. Weinberg will give you any information he has.
  • Find a possible model in Geogebra that works for solving this problem.
  • At what minimum speed he could throw the ball in order to get the ball into the hoop?

We are going to start the process today of constructing our model for projectile motion in the absence of air resistance. We discussed the following in the last class:

  • Velocity is constant in the horizontal direction. (Constant velocity model)
  • x(t) = x_{0} + v t

  • Acceleration is constant in the vertical direction (Constant acceleration model)
  • v(t) = v_{0} + a t
    x(t)=x_{0}+v t +\frac{1}{2}a t^2

  • The magnitude of the acceleration is the acceleration due to gravity. The direction is downwards.

Consider the following situation of a ball rolling off of a 10.0 meter high platform. We are neglecting air resistance in order for our models to work.
Screen Shot 2013-02-25 at 6.15.15 PM

Some questions:

  • At what point will the ball’s movement follow the models we described above?
  • Let’s set x=0 and y = 0 at the point at the bottom of the platform. What will be the y coordinate of the ball when the ball hits the ground? What are the components of velocity at the moment the ball becomes a projectile?
  • How long do you think it will take for the ball to hit the ground? Make a guess that is too high, and a guess that is too low. Use units in your answer.
  • How far do you think the ball will travel horizontally before it hits the ground? Again, make high and low guesses.

Let’s model this information in a spreadsheet. The table of values is nothing more than repeated calculations of the algebraic models from the previous page. You will construct this yourself in a bit. NBD.
Screen Shot 2013-02-25 at 6.39.23 PM

  • Estimate the time when the ball hits the ground. What information from the table did you use?
  • Find the maximum horizontal distance the ball travels before hitting the ground.

Here are the four sets of position/velocity graphs for the above situation. I’ll let you figure out which is which. Confirm your answer from above using the graphs. Let me know if any of your numbers change after looking at the graphs.

Screen Shot 2013-02-25 at 6.42.35 PM

Now I want you to recreate my template. Work to follow the guidelines for description and labels as I have in mine. All the tables should use the information in the top rows of the table to make all calculations.

Once your table is generating the values above, use your table to find the maximum height, the total time in the air, and the distance in the x-direction for a soccer ball kicked from the ground at 30° above the horizontal.

I’ll be circulating to help you get there, but I’m not giving you my spreadsheet. You can piece this together using what you know.


Next steps (not for this lesson):

  • The table of values really isn’t necessary – it’s more for us to get our bearings. A single cell can hold the algebraic model and calculate position/velocity from a single value for time. Goal seek is our friend for getting better solutions here.
  • With goal seek, we are really solving an equation. We can see how the equation comes from the model itself when we ask for one value under different conditions. The usefulness of the equation is that we CAN get a more exact solution and perhaps have a more general solution, but this last part is a hazy one. So far, our computer solution works for many cases.

My point is motivating the algebra as a more efficient way to solve certain kinds of problems, but not all of them. I think there needs to be more on the ‘demand’ side of choosing an algebraic approach. Tradition is not a satisfying reason to choose one, though there are many – providing a need for algebra, and then feeding that need seems more natural than starting from algebra for a more arbitrary reason.

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Struggling (and succeeding) with models in physics


Today we moved into exploring projectile motion in my non-AP physics class. Exhibit A:
Screen Shot 2013-02-22 at 4.29.27 PM

I launched a single marble and asked them to tell me what angle for a given setting of the launched would lead to a maximum distance. They came up with a few possibilities, and we tried them all. The maximum ended up around 35 degrees. (Those that know the actual answer from theory with no air resistance might find this curious. I certainly did.)

I had the students load the latest version of Tracker on their computers. While this was going on, I showed them how to use the program to step frame-by-frame through one of the included videos of a ball being thrown in front of a black background:
Screen Shot 2013-02-22 at 4.34.45 PM
Students called out that the x-position vs. t graph was a straight line with constant slope – perfect for the constant velocity model. When we looked at the y-position vs t, they again recognized this as a possible constant acceleration situation. Not much of a stretch here at all. I demonstrated (quickly) how the dynamic particle model in Tracker lets you simulate a particle on top of the video based on the mass and forces acting on it. I asked them to tell me how to match the particle – they shouted out different values for position and velocity components until eventually they matched. We then stepped through the frames of the video to watch the actual ball and the simulated ball move in sync with each other.

I did one more demo and added an air resistance force to the dynamic model and asked how it would change the simulated ball. They were right on describing it, even giving me an ‘ooh!’ when the model changed on screen as they expected.
Screen Shot 2013-02-22 at 5.04.31 PM

I then gave them my Projectile Motion Simulator in Geogebra. I told them that it had the characteristics they described from the graphs – constant velocity in x, constant acceleration of gravity in y. Their task was to answer the following question by adjusting the model:

A soccer ball is kicked from the ground at 25 degrees from the horizontal. How far and how high does the ball travel? How long is it in the air?

They quickly figured out how it works and identified that information was missing. Once I gave them the speed of the ball, they answered the three questions and checked with each other on the answers.

I then asked them to use the Geogebra model to simulate the launcher and the marble from the beginning of the class. I asked them to match the computer model to what the launcher actually did. My favorite part of the lesson was that they started asking for measuring devices themselves. One asked for a stopwatch, but ended up not needing it. They worked together to figure out unknown information, and then got the model to do a pretty good job of predicting the landing location. I then changed the angle of the launcher and asked them to predict where the marble would land. Here is the result:

Nothing in this lesson is particularly noteworthy. I probably talked a bit too much, and could have had them go through the steps of creating the model in Tracker. That’s something I will do in future classes. When I do things on the computer with students, the issues of getting programs installed always takes longer than I want it to, and it gets away from the fundamental process that I wanted them to see and have a part of – experiencing the creation of a computer model, and then actually matching that model to something in the real world.

My assertions:

  • Matching a model (mathematical, physical, numerical, graphical, algebraic) to observations is a challenge that is understood with minimal explanation. Make a look like b using tool c.
  • The hand waving involved in getting students to experiment with a computer model is minimized when that model is being made to match actual observations or data. While I can make a computer model do all sorts of unrealistic things, a model that is unrealistic wont match anything that students actually see or measure.
  • Students in this activity realized what values and measurements they need, and then went and made them. This is the real power of having these computer tools available.
  • While the focus in the final modeling activity was not an algebraic analysis of how projectile motion works mathematically, it did require them to recognize which factors are at play. It required them to look at their computed answer and see how it compared with observations. These two steps (identifying given information, checking answer) are the ones I have always had the most difficulty getting students to be explicit about. Using the computer model focuses the problem on these two tasks in a way that hand calculations have never really pushed students to do. That’s certainly my failure, but it’s hard to deny how engaged and naturally this evolved during today’s lesson.

The homework assignment after the class was to solve a number of projectile motion problems using the Geogebra model to focus them on the last bullet point. If they know the answers based on a model they have applied in a few different situations, it will hopefully make more intuitive sense later on when we do apply more abstract algebraic models.

Algebra is very much not dead. It just doesn’t make sense anymore to treat algebraic methods as the most rigorous way to solve a problem, or as a simple way to introduce a topic. It has to start somewhere real and concrete. Computers have a lot of potential for developing the intuition for how a concept works without the high bar for entry (and uphill battle for engagement) that algebra often carries as baggage.

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When things just work – starting with computers


Today’s lesson on objects in orbit went fantastically well, and I want to note down exactly what I did.

Scare the students:

Screen Shot 2013-02-05 at 3.23.59 PMhttp://neo.jpl.nasa.gov/news/news177.html

Push to (my) question – how close is that?

Connect to previous work:

The homework for today was to use a spreadsheet to calculate some things about an orbit. Based on what they did, I started with a blank sheet toward the beginning of class and filled in what they told me should be there.
orbit calculationsScreen Shot 2013-02-05 at 3.30.08 PM
Some students needed some gentle nudging at this stage, but nothing that felt forced. I hate when I make it feel forced.

Play with the results

Pose the question about the altitude needed to have a satellite orbit once every twenty four hours. Teach about the Goal Seek function in the spreadsheet to automatically find this. Ask what use such a satellite would serve, and grin when students look out the window, see a satellite dish, and make the connection.

Introduce the term ‘geosynchronous’. Show asteroid picture again. Wait for reaction.

Screen Shot 2013-02-05 at 3.23.59 PM

See what happens when the mass of the satellite changes. Notice that the calculations for orbital speed don’t change. Wonder why.

See what happens with the algebra.

Screen Shot 2013-02-05 at 3.36.51 PM

See that this confirms what we found. Feel good about ourselves.

Wonder if student looked at the lesson plan in advance because the question asked immediately after is curiously perfect.

Student asks how the size of that orbit looks next to the Earth. I point out that I’ve created a Python simulation to help simulate the path of an object moving only under the influence of gravity. We can then put the position data generated from the simulation into a Geogebra visualization to see what it looks like.

Simulate & Visualize

Introduce how to use the simulation
Use the output of the spreadsheet to provide input data for the program. Have them figure out how to relate the speed and altitude information to what the simulation expects so that the output is a visualization of the orbit of the geosynchronous satellite.

Screen Shot 2013-02-05 at 3.55.37 PM

Not everybody got all the way to this point, but most were at least at this final step at the end.


I’ve previously done this entire sequence starting first with the algebra. I always would show something related to the International Space Station and ask them ‘how fast do you think it is going?’ but they had no connection or investment in it, often because their thinking was still likely fixed on the fact that there is a space station orbiting the earth right now . Then we’d get to the stage of saying ‘well, I guess we should probably draw a free body diagram, and then apply Newton’s 2nd law, and derive a formula.’

I’ve had students tell me that I overuse the computer. That sometimes what we do seems too free form, and that it would be better to just get all of the notes on the board for the theory, do example problems, and then have practice for homework.

What is challenging me right now, professionally, is the idea that we must do algebra first. The general notion that the ‘see what the algebra tells us’ step should come first after a hook activity to get them interested since algebraic manipulation is the ultimate goal in solving problems.

There is something to be said for the power of the computer here to keep the calculations organized and drive the need for the algebra though. I look at the calculations in the spreadsheet, and it’s obvious to me why mass of the satellite shouldn’t matter. There’s also something powerful to be said for a situation like this where students put together a calculator from scratch, use it to play around and get a sense for the numbers, and then see that this model they created themselves for speed of an object in orbit does not depend on satellite mass. This was a social activity – students were talking to each other, comparing the results of their calculations, and figuring out what was wrong, if anything. The computer made it possible for them to successfully figure out an answer to my original question in a way that felt great as a teacher. Exploring the answer algebraically (read: having students follow me in a lecture) would not have felt nearly as good, during or afterwards.

I don’t believe algebra is dead. Students needed a bit of algebra in order to generate some of the calculations of cells in the table. Understanding the concept of a variable and having intuitive understanding of what it can be used to do is very important.

I’m just spending a lot of time these days wondering what happens to the math or science classroom if students building models on the computer is the common starting point to instruction, rather than what they should do just at the end of a problem to check their algebra. I know that for centuries mathematicians have stared at a blank paper when they begin their work. We, as math teachers, might start with a cool problem, but ultimately start the ‘real’ work with students on paper, a chalkboard, or some other vertical writing surface.

Our students don’t spend their time staring at sheets of paper anywhere but at school, and when they are doing work for school. The rest of the time, they look at screens. This is where they play, it’s where they communicate. Maybe we should be starting our work there. I am not recommending in any way that this means instruction should be on the computer – I’ve already commented plenty on previous posts on why I do not believe that. I am just curious what happens when the computer as a tool to organize, calculate, and iterate becomes as regular in the classroom as graphing calculators are right now.

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Who’s gone overboard modeling w/ Python? Part II – Gravitation


I was working on orbits and gravitation with my AP Physics B students, and as has always been the case (including with me in high school), they were having trouble visualizing exactly what it meant for something to be in orbit. They did well calculating orbital speeds and periods as I asked them to do for solving problems, but they weren’t able to understand exactly what it meant for something to be in orbit. What happens when it speeds up from the speed they calculated? Slowed down? How would it actually get into orbit in the first place?

Last year I made a Geogebra simulation that used Euler’s method  to generate the trajectory of a projectile using Newton’s Law of Gravitation. While they were working on these problems, I was having trouble opening the simulation, and I realized it would be a simple task to write the simulation again using the Python knowledge I had developed since. I also used this to-scale diagram of the Earth-Moon system in Geogebra to help visualize the trajectory.

I quickly showed them what the trajectory looked like close to the surface of the Earth and then increased the launch velocity to show what would happen. I also showed them the line in the program that represented Newton’s 2nd law – no big deal from their reaction, though my use of the directional cosines did take a bit of explanation as to why they needed to be there.

I offered to let students show their proficiency on my orbital characteristics standard by using the program to generate an orbit with a period or altitude of my choice. I insist that they derive the formulae for orbital velocity or period from Newton’s 2nd law every time, but I really like how adding the simulation as an option turns this into an exercise requiring a much higher level of understanding. That said, no students gave it a shot until this afternoon. A student had correctly calculated the orbital speed for a circular orbit, but was having trouble configuring the initial components of velocity and position to make this happen. The student realized that the speed he calculated through Newton’s 2nd had to be vertical if the initial position was to the right of Earth, or horizontal if it was above it. Otherwise, the projectile would go in a straight line, reach a maximum position, and then crash right back into Earth.

The other part of why this numerical model served an interesting purpose in my class was as inspired by Shawn Cornally’s post about misconceptions surrounding gravitational potential and our friend mgh. I had also just watched an NBC Time Capsule episode about the moon landing and was wondering about the specifics of launching a rocket to the moon. I asked students how they thought it was done, and they really had no idea. They were working on another assignment during class, but while floating around looking at their work, I was also adjusting the initial conditions of my program to try to get an object that starts close to Earth to arrive in a lunar orbit.

Thinking about Shawn’s post, I knew that getting an object out of Earth’s orbit would require the object reaching escape velocity, and that this would certainly be too fast to work for a circular orbit around the moon. Getting the students to see this theoretically was not going to happen, particularly since we hadn’t discussed gravitational potential energy among the regular physics students, not to mention they had no intuition about things moving in orbit anyway.

I showed them the closest I could get without crashing:

One student immediately noticed that this did seem to be a case of moving too quickly. So we reduced the initial velocity in the x-direction by a bit. This resulted in this:

We talked about what this showed – the object was now moving too slowly and was falling back to Earth. After getting the object to dance just between the point of making it all the way to the moon (and then falling right past it) and slowing down before it ever got there, a student asked a key question:

Could you get it really close to the moon and then slow it down?

Bingo. I didn’t get to adjust the model during the class period to do this, but by the next class, I had implemented a simple orbital insertion burn opposite to the object’s velocity. You can see and try the code here at Github. The result? My first Earth – lunar orbit design. My mom was so proud.

The real power here is how quickly students developed intuition for some orbital mechanics concepts by seeing me play with this. Even better, they could play with the simulation themselves. They also saw that I was experimenting myself with this model and enjoying what I was figuring out along the way.

I think the idea that a program I design myself could result in surprising or unexpected output is a bit of a foreign concept to those that do not program. I think this helps establish for students that computation is a tool for modeling. It is a means to reaching a better understanding of our observations or ideas. It still requires a great amount of thought to interpret the results and to construct the model, and does not eliminate the need for theoretical work. I could guess and check my way to a circular orbit around Earth. With some insight on how gravity and circular motion function though, I can get the orbit right on the first try. Computation does not take away the opportunity for deep thinking. It is not about doing all the work for you. It instead broadens the possibilities for what we can do and explore in the comfort of our homes and classrooms.

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Who’s gone overboard modeling in Physics? This guy, part I.


I’ve been sticking to my plan this year to follow the Modeling Instruction curriculum for my regular physics class. In addition to making use of the fantastic resources made available through the AMTA, I’ve found lots of ways to use Python to help drive the plow through what is new territory for me. I’ve always taught things in a fairly equation driven manner in Physics, but I have really seen the power so far of investing time instead into getting down and dirty with data in tables, graphs, and equations when doing so is necessary. Leaving equations out completely isn’t really what I’m going for, but I am trying to provide opportunities for students to choose the tools that work best for them.

So far, some have embraced graphs. Some like working with a table of data alone or equations. The general observation though is that most are comfortable using one to inform the other, which is the best possible outcome.

Here’s how I started. I gave them the Python code here and asked them to look at the lines that configure the program. I demonstrated how to run the program and how to paste the results of the output file into Geogebra, which created a nice visualization through this applet. Their goal through the activity was to figure out how to adjust the simulation to generate a set of graphs of position and velocity vs. time like this one:

Some used the graph directly and what they remembered from the constant velocity model (yeah, retention!) to figure out velocity and initial position. Others used the table for a start and did a bit of trial and error to make it fit. While I have always thought that trial and error is not an effective way to solve these types of problems, the intuition the students developed through doing came quite naturally, and was nice to see develop.

After working on this, I had them work on using the Python model to match the position data generated by my Geogebra Particle Dynamics Simulator. I had previously asked them to create sets of data where the object was clearly accelerating, so they had some to use for this task. This gave them the chance to not only see how to determine the initial velocity using just the position data, as well as use a spreadsheet intelligently to create a set of velocity vs. time data. I put together this video to show how to do this:

.

It was really gratifying to see the students quickly become comfortable managing a table of data and knowing how to use computational tools  to do repeated calculations – this was one of my goals.

The final step was setting them free to solve some standard  Constant-Acceleration kinematics problems using the Python model. These are problems that I’ve used for a few years now as practice after introducing the full set of constant acceleration equations, and I’ve admittedly grown a bit bored of them.Seeing how the students were attacking them using the model as a guide was a way for me to see them in a whole new light – amazingly focused questions and questions about the relationship between the linear equation for velocity (the only equation we directly discussed after Day 1), the table of velocity data, and what was happening in position vs. time.

One student kept saying she had an answer for problem c based on equations, but that she couldn’t match the Python model to the problem. In previous classes where I had given that problem, getting the answer was the end of the story, but to see her struggling to match her answer to what was happening in her model was beautiful. I initially couldn’t do it myself either until I really thought about what was happening, and she almost scooped me on figuring it out. This was awesome.

They worked on these problems for homework and during the beginning of the next class. Again, some really great comments and questions came from students that were previously quiet during class discussions. Today we had a learning standard quiz on constant acceleration model questions, and then decided last night during planning was to go on to just extending the constant acceleration model to objects in free fall.

Then I realized I was falling back into old patterns just telling them that all objects in free fall near Earth’s surface accelerate downward at roughly 9.81 m/s^2. Why not give them another model to play with and figure this out? Here’s what I put together in Python.

The big plus to doing it this way was that students could decide whether air resistance was a factor or not. The first graph I showed them was the one at right – I asked whether they thought it could represent the position versus time graph for an object with constant acceleration. There was some inconsistency in their thinking, but they quickly decided as a group after discussing the graph that it wasn’t. I gave them marble launchers, one with a ping-pong ball, and another with a marble, and asked them to model the launch of their projectiles with the simulation. They decided what they wanted to measure and got right to it. I’m also having them solve some free fall problems using the gravity simulation first without directly telling them that acceleration is constant and equal to g. They already decided that they would probably turn off air resistance for these problems – this instead of telling them that we always do, even though air resistance is such a real phenomenon to manage in the real world.

A bit of justification here – why am I being so reliant on the computer and simulation rather than hands on lab work? Why not have them get out with stopwatches, rulers, Tracker, ultrasonic detectors, air tracks, etc?

The main reason is that I have yet to figure out how to get data that is reliable enough that the students can see what they have learned to look for in position and velocity data. I spent an hour working to get a cart on an inclined air track to generate reasonable data for students to use in the incline lab in the modeling materials from AMTA on constant acceleration, and gave up after realizing that the students would lose track of the overall goal while struggling to get the mere 1 – 2 seconds of data that my 1.5 meter long air track can provide. The lab in which one student runs and other students stand in a line stopping their stopwatches when the runner passes doesn’t work when you have a small class as I do. The discussions that ensue in these situations can be good, but I have always wished that we had more data to have a richer investigation into what the numbers really represent. The best part of lab work is not taking data. It’s not making repetitive calculations. Instead, it’s focusing on learning what the data tells you about the situation being measured or modeled. This is the point of spending so much time staring and playing with sets of data in physics.

I also find that continuing to show students that I can create a virtual laboratory using several simple lines of code demonstrates the power of models. I could very easily (and plan to) also introduce some random error so the data isn’t quite so smooth, but that’s something to do when we’ve already understood some of the fundamental issues. We dealt with this during the constant velocity model unit, but when things are a bit messier (and with straight lines not telling the whole picture) when acceleration comes into play, I’m perfectly comfortable with smooth data to start. Until I can generate data as masterfully as Kelly does here using my own equipment, I’m comfortable with the computer creating it, especially since they can do so at home when they think nobody is looking.

Most of all, I find I am excited myself to put together these models and play with the data to model what I see. Having answered the same kinematics questions many times myself, being able to look at them in a new way is awesome. Finding opportunities for students to figure out instead of parrot responses after learning lists of correct answers is the best part of teaching, and if simulations are the way to do this, I’m all for it. In the future, my hope is to have them do the programming, but for now I’m happy with how this experiment has unfolded thus far.

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