Today's lesson on objects in orbit went fantastically well, and I want to note down exactly what I did.
Scare the students:
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.
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.
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.
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.
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.