This past Monday my cohort and I presented at our GRS Expo. We shared as a cohort our journey this past year in GRS towards becoming teachers, as well as our individual commitments to reform based science teaching. There was one commitment I want to make but did realized I didn’t have time to discuss at expo that I want to talk about now. Something I want to do in my physics teaching practice is for my students to develop the physics equations we use through investigation and analysis, rather than me supplying them.
To showcase this in my station at Expo I of course picked a tricky equation, the center of mass for two particles:
We’re going to make one more simplification to our equation by saying we’re going to leave one of the masses, m2, where it is. Also, instead of measuring where or center of mass is located relative to the end of the meter stick, we’re going to measure it relative to m2. That is, our objective will be to find how far the center of mass is from m2. m2 is zero meters away from itself, so in our equation x2 (how far m2 is away from our starting point) is zero. So our equation now looks like:
(if we leave m2 at the 50 cm mark of our meter stick, and include the mass of the stick in m2, we have also successfully accounted for the mass of the stick)
This equation is looking nicer. While the relationship between the position of the center of mass and the mass of our two objects is still complicated, we do have a linear relationship between the position of the center of mass and the position of m1 (bonus points: the y intercept is 0). Everyone loves a linear relationship! It’s not too difficult to determine an equation for a linear relationship from a graph by finding the slope of the line (a process we would have practiced earlier in the year, perhaps when determining F = ma). For this equation, the slope would end up being m1 / (m1 + m2).
Our initial challenge for our students can then be something along the lines of “what is the mathematical relationship between the location of the center of mass and the position of m1 for the given setup” (the setup being the meter stick + binder clips). I would give each team the same mass for m2, but a different m1. The reason for this will be come apparent in a second.
Once each team has determined the relationship for their masses, they can then pool their findings together. One way to do this would be for each team to add their best fit line and equation to a graph on the Smartboard.
In this made up example data, m2 = 1 g.
Notice the pattern? the numerator of the slope for each equation is equal to the mass of m1, while the denominator is equal to the mass of m1 plus 1. We could write our current understanding of this relationship then as:
This is starting to look very similar to the equation we’re aiming for, it’s just missing the dependence on m2 (though students may point out that m2 was equal to 1 g, and a 1 shows up in our denominator). If students conduct the same investigation (with their own tweaks based on their experiences the first time round), except this time with m2 = 2g and we share our data once more…
We end up with a similar pattern, except this time our equation looks like this:
Based upon these two equations, we could then infer that the number in our denominator is in fact the value of m2, and our final equation demonstrating the relationship between the position of the center of mass and these variables is then:
The inference we made to arrive at this final equation was perhaps a bit of a stretch, so maybe we should do one more value for m2 just to be safe. We can first predict what we should find if our equation is correct: slopes of 2/5, 3/6, 4/7, and 5/8, and then check if it works out in reality (spoiler: it does).
Woohoo, we did it!
Bonus investigation: What happens to my equation when I add a 3rd mass? This investigation would actually end with formulating the original center of mass equation, before any of our simplifications.
Phew, that was a lot of work. Anyways, that’s my thought process for how we could develop a physics equation through experimentation. No way I would have been able to do all that in my 10 minute expo presentation. Especially if I wanted to be able to discuss my vexations. Those being:
- There isn’t much opportunity for students to really lead this investigation. The setup and overall procedure is predetermined. I left the process by which students actually come up with their equations (what data they take, how they take it, how much they take) open, but in the end they need to come up with the linear equation so that when we combine our findings we can see the pattern. I’m worried about this experience being hands on without minds on.
- What will the students get out of doing this? My goal in determining these equations through investigation is for students to have a context what the equation is representing. In my placement I found my students understood if you push on something with a greater force it will accelerate faster, and that something with more mass won’t accelerate as fast as something with less mass, but they did not understand that F = ma represented that very same concept. Will building up the equation themselves help students’ conceptual understanding of an equation?
I realize I gave the bare overview of the process we would use to come up with this equation and didn’t talk much about how I would scaffold the whole thing, but I’m curious as to anyone’s thoughts on these issues.