Note: This post details my version of the paradigm experiment for what is Unit 2 (Particle Moving with Constant Velocity) of the Modeling Instruction Mechanics curriculum. My interpretation varies a bit from the official materials, and my classes build this as their first model, starting with the very first day of school.
Here’s a binder. And the handshake. [As they walk in, I'm handing out their binders (with materials, labeled dividers, spare graph paper, their names on the sides) and shaking their hands as we make the transaction. I've seen some discussion in various places online about the appropriateness of shaking students' hands on the first day of school. It seems to vary widely depending on the situation and history of the school, the teacher, the students. Since I've known most of these students for at least a year (and, in fact, was likely saying "goodnight" and turning off lights for some of them once or twice a week when they were freshmen), it makes sense for me. In a way, it's a small signal that our relationship has to be a bit more formal inside the classroom than it has been (and might continue to be) outside the classroom. We're doing some business, here. You would likely know whether that would be weird or normal at your own school and for your own situation. And also, this handshake business really has nothing to do with the constant velocity particle model, so I'll stop discussing it here.]
Hey guys, I want to show you something cool. Let’s go next door.
Come around this table in the front. Okay. Are you ready? Mentally, physically? Metaphysically? It’s going to be pretty cool. (We’re ready.) Great. So in a moment, I’m going to say “start”, and I’m going to say “stop”. Some things might happen before I say “start”, and some things might happen after I say “stop”, but we want to focus on observing just what happens in between the “start” and “stop”. Got it? (Yep.)
Cool. [I flip the switch on the cart. 1] Start. [The cart moves pretty slowly across the table in front of me. I watch it carefully.] Stop. [I turn off the cart. I look up at them.] Whoa. You want to see that again?
Of course. Let’s watch. [I repeat the same thing.]
Now, for this next part, let’s talk about our observations. There’s not anything in particular that I’m looking for. I just want to get your brains going. Anything that observed between the “start” and “stop”. Go for it. [A lot of responses will happen. Someone will mention the cart being turned on and off even though it happened outside the boundaries. That's fine. I don't admonish them for it (though their classmates might, depending on the group). Someone will mention that it made a noise. Someone will say, "The cart is black." Usually, too, I'll get some really helpful observations. It was going in a straight line. The speed was the same the whole time. On those, there will often be a student (or several) who observed that it didn't go in a straight line or that it didn't have a constant speed. We'll turn the cart back on to watch for that specifically. I'll ask them, "How will we know whether it is [the thing we're looking for] or not? What will we look to see? What will it look like if it isn’t?”]
(It had a constant speed.) Oh, that’s interesting. How do you know it is constant? [The conversation here varies a lot with each group. Essentially, though, we're looking for a verbal definition of constant velocity. We usually come around to something like, "it went the same distance in each chunk of time". Some variation on that. I told them I wasn't looking for anything in particular, but that was sort of a lie. I don't care about what their other observations are (unless they are inaccurate, but then we'll just challenge them). But I do care that we get to that verbal model of constant velocity.]
Huh. So, we’ve gotten a pretty good description of this motion in words. I think a great next step in understanding this cart would be to get a more quantitative description. For that, we’d need to do an experiment. So.
What can we measure?
What can we measure about the motion of the cart? But also, what tool would we use to measure it? We don’t have a lot of time today, and this is our first experiment, so ideally we will use tools that are easily found in the lab and that we are mostly familiar with already.
There’s a space to write this down in your packet on the inside of the front page. Don’t worry about the sketch just yet. You’ll have time to do that later.
The Potential Measurements
(Distance, ruler.) Great. Has everyone used a ruler before? And we have a bunch of those, so we can definitely use that tool.
(Time, stopwatch.) Sure. Has everyone used a stopwatch before? And we have a drawer full of them right over here.
(Speed.) Cool. What tool would you use? (The ruler and stopwatch, then you just divi—) That sounds like it’s going to be more of a calculation than a measurement. What tool could we use to measure it directly? (Uh, a radar gun?) That would be so cool! I wish we had one of those. We don’t, so is it okay if I leave speed off the list for now? (Uh, okay.)
(Direction, compass.) Fair enough, but I actually don’t have compasses either. [I've had a student say they carry one in their backpack before...! If you do leave it on the list, it's easy to eliminate later because it doesn't seem to depend on anything else, and the other things don't seem to depend on it.]
(Weight, scale.) Okay, and I do have a scale. [This one doesn't always come up, but it is a pretty frequently raised idea.]
Culling the List
[So now we have something like: distance, time, direction, weight. To do the experiment, we need to figure out what relationship we are trying to find. Usually the relationship is between just two quantities, so we need to narrow down our ideas, here.]
Which of these things can we choose values for? (Definitely distance. And time. And direction. And weight.) Hold on. how would you choose the values for weight? (Oh. Actually, good point. So not weight.) Okay, so if the value for weight can’t change, it probably won’t be a dynamic part of our experiment, huh? So should I cross that one off the list? (Fair enough.)
Now. Which of the things on our list depend on the things we can choose values for? (Um, distance depends on time. Or time depends on distance.) How about direction? Does it depend on the others? (Well, it always goes in the same direction, so I guess not.) Okay, so if it doesn’t depend on these other things, it doesn’t seem so relevant for today’s experiment, right? (Yeah.) So should I cross it off the list? (Sure.)
Okay. We’re getting close, now! There is one thing I have a suggestion about.
So, I’m thinking about how in a few minutes, we’re going to split up and do experiments in different places. After, we’ll want to compare our data to each other. So I was thinking, maybe instead of just using distance, we should do a measurement that we could compare to each other. Like, if we all agreed on the same place that was zero, even if we didn’t start there. But it would be the same zero for everyone. That way we could compare where each cart was in a more meaningful way.
So instead of distance, it would be more like where you were on a number line. [Draw a number line on the board.] We would just need a common zero. An, uh, what do you call the zero on the number line in math? (Oh, um, the origin.) Yeah, a common origin.
Oh, hey! There’s an origin already taped across the room that we could use.
Wow, that’s convenient! So I guess, besides just an origin, we also need to say which was is positive and which way is negative, right? So let’s say toward physics is positive, toward chemistry (the chemistry lab is next door) is negative. For obvious reasons.
So is it fair for us to measure position instead of distance? (Sure.) It’s pretty similar. I think it still keeps the spirit of what you meant by distance, but just allows us to compare data more easily in the end.
So now we’re down to position (ruler) and time (stopwatch) for our measurements, right?
What relationship are we trying to determine?
What is our objective for this experiment? To find a relationship between what and what? (Distance and time. I mean, position and time.) Okay. So we’re finding a relationship between position and time for the motorized cart. You can write the objective down on the experiment page in the packet.
What should we do with the data?
So what are you going to measure, again? (Position and time.) Right. And so you’re going to take a bunch of data points, then where will you put them all? (In a data table?) Good idea. So what will go in the data table? (Uh, position… and time…) Okay. So let’s choose symbols for those things that will help us remember which is which. (x and y.) I think we can be even more specific.
Let’s take time, for example. What symbol would you use to represent time that would help you remember it was for time? (Seconds.) [I write (s) at the top of the column I've drawn on the board, leaving a space for the symbol in front of it.] I think seconds for the units makes sense. What about the symbol, though? (Oh, ok. t?) That’s a good idea. Now how about for position? (p?) That would make sense, but physicists have claimed that for something else already. [Moving back over toward that number line I sketched before.] So position is the spot on the number line. What symbol would you use to label the number line in math? (x?) Sure. If it were a vertical number line? (Oh, y.) So x makes sense for position to me. [Filling it in on the sample data table on the board.] How about the units? (Inches? No, centimeters?) Okay, fight! [I'd let them use whatever the decided on together, but usually centimeters wins "because it's science class" and for no other reason.]
Once you have a table full of data, what will you do with it? (Graph it?) Yeah, my favorite! [Drawing sample axes on the board.] So what will go on our axes? (Position and time?) Of course. So which one where? [Now the real fight begins. I try to coach them in the direction of saying "vertical" and "horizontal" instead of x and y for the axes. The big conflict is that they really want to put position on the horizontal because it is x, but they also really want to put time on the horizontal because it is "independent" (whatever they think that means from math classes of old). In the end, I might just end the debate by telling them "what physicists usually do" and having us agree to draw x-t graphs so that we can all compare our results more easily (after all, that's why we decided on position instead of distance).2 I might throw out the line—So we'll all have a slight moment of panic when we put x on the vertical axis. But we'll get through it together.—which tends to calm them down and let us move forward.]
Off you go!
Okay, so now you know what you’re doing, right? Ready to go do it? [Very awkward pause ensues. This whole thing has moved very quickly, and the students aren't yet sure what to do. I mean, it's the first day of classes and everything. Am I actually expecting them to be ready to do an experiment? What is with this class, anyway? So I'll throw them a rope of comfort instead of just chucking them into the deep end.] Well, okay, should we try out one method together before we break into groups? [Relieved sigh of, "YES!"] Now, this is only one of many good solutions to taking data for this experiment. You absolutely do not have to use this same method. Since it’s the first day of school and your brains aren’t totally into physics mode yet, I just want to give you one example. If you have other ideas about how to take this data, please do that! I’ve seen several good ways, but I’m also always looking for new ones that are even better. [All that being said, a lot, even most, of the Honors Physics students will all do exactly what I am about to show them (down to the number of seconds between each breadcrumb). It's the first day of Honors Physics, and they.are.scared. The regular students will usually have at least one or two groups that will try their own way.]
I call this the “breadcrumb” method. I need someone who is a pro at using a stopwatch, and I need someone to grab one of those meter sticks. Just a one-meter stick, not a two-meter stick. Thanks! So basically, you’ll start the stopwatch and I will turn on the cart at the same time. Then, every, say, 2 seconds, you’ll say “now” and I will drop a breadcrumb at the new position of the cart. I’m going to put the breadcrumbs at the back of the cart because… [I mimic putting a breadcrumb in front of the cart and they can see the problem of my fingers being roadkill immediately.] We can measure the positions after that. Got it? Great. Let’s try it out. [There will sometimes be a quick discussion about whether to put the back or the front of the cart on the Start line at this point. If there isn't, they will figure that out in their groups easily enough.]
Okay, let’s try out the measuring technique. Important note! Everyone put your pencils down, now. Do not record this data. This is not your data. You will get to take your own data in just a couple of minutes. Don’t write this down! [I point to a breadcrumb.] What was the time when the cart was here? (Two seconds.) [Glance at the clock.] Two seconds after the start of time…? (No, from when we started the stopwatch.) Okay, so it’s more like a change in time, not an absolute time, right? We should come back and talk more about that later. Anyway, how about when the cart was here? [Point to the next breadcrumb.] (Four seconds after the stopwatch started.)
Where was the cart at 0 seconds, then? (Here. [Student(s) point to the Start line.]) That’s our first data point, right? So let’s measure that position. [Student who retrieved the meterstick], can you measure that position for us? [Usually, there's a discussion now about whether that position was 0 centimeters or not. The "nots" eventually win, since the Start line was not on our origin line, and the origin line is our 0 cm line. We get the student to measure from the origin line to the Start line in centimeters (and not in inches).] Is it on the positive side or the negative side of the origin? (Uh… the negative side.) Okay, so negative [measurement]. How about this breadcrumb? Can you measure the position here? Stop writing this down. [Now, almost without fail, the measurer will try to go from Start to the breadcrumb instead of from the origin to the breadcrumb, despite the discussion just seconds before. He will get corrected by his classmates, though, and we'll see that likely mistake so that they will recognize it in a few minutes when they make it again.]
Okay, I think you all have an idea now of one way to take this data, right? In a moment, not yet, let’s do groups of two or three. The carts, tape, and stopwatches are over here. The metersticks are over by the hood. Remember not to change the speed dial on the cart, even though it is incredibly tempting to do so. There might be a group in another class that isn’t finished taking data, and you don’t want to ruin their experiment.
Off you go!
[That all seemed to take quite a long time in writing, but it's not so exceptionally long in person. No more than 20 minutes, for sure, but more likely close to 10 or 15 minutes, depending on how picky the students are deciding to be in their answers on the first day of classes.]
[The following bits are said to individual groups when appropriate as they are graphing. I don't make these as general announcements at all.] This graph is the only one for an experiment that I will ask you to plot by hand this year, so let’s do a really careful job with it. Make sure you are using a ruler to draw a single best fit line. Don’t connect the dots. Don’t just connect the first and last data points. Try to line up the ruler so that it gets as close to every data point as possible. There are more precise ways to do this work, but the eyeballed best fit line will be good enough, here, I think.
Once you have the line drawn, imagine that your data points have disappeared so that the only thing on your graph is the line. Find points on the line (not your data points) to find the slope. Use the variables on the graph (x and t) for your equation. Don’t default to x and y. There is no y-axis on the graph you’ve drawn.
Be sure to put units on the numbers in your equation, but not on the variables. That is, the slope and intercept should both have units. Go back and put units in your work to see what they should be.
When you are finished making your graph, go ahead and clean everything up. Then, grab a whiteboard and make a sketch of the shape of your graph. Sketch, not plot. Don’t make tick marks. No rulers or metersticks allowed. Just a really quick sketch, then write the equation of your line. When we are all finished, we’ll get together and have a board meeting (ha ha).
Okay, go ahead and sit on top of the tables in the circle. We will show everyone’s boards at once, and we’ll look to see what’s the same and what’s different. After a few experiments, you won’t really need me to lead this anymore because you will be able to have the conversation yourselves, but I’ll help a bit this time. In general, you want to talk to each other, not to me. So. What’s the same about everyone’s boards? [At this point, the responses can be all over the map. Eventually, you'll get to some more substantial answers, like, "they are all lines."] So everyone got a linear relationship? (Yeah.) Oh! That seems pretty significant. We were looking for the relationship between position and time for the cart, and everyone found that relationship to be linear, even though they were using different carts. Cool.
So what’s different about everyone’s boards? [Hopefully they will talk to each other a bit about how the slopes are different, the signs of the slopes are different, and the intercepts are different. Then we can dig into what each of those differences represents. Asking, "Who had the fastest cart? Who had the slowest cart?" can help a lot. Eventually, we get to the idea of the slope telling both the speed and the direction of the motion. When we're there, we wish for a word that meant both parts of that info, and we get velocity.] Sweet. It seems like we’ve defined this relationship pretty well at this point. Let’s try using it in new problems!
Next, we might have a really brief discussion that pulls together the pieces of our model (the verbal piece from our observations, the features of the evenly-spaced “breadcrumb” motion map on the table, the position-vs-time graph features). Maybe 2 or 3 minutes, there. Then on to working some problems in the packet. I let them encounter the velocity-vs-time graphs on their own. They get confused and nervous for a small time, but they can figure it out pretty easily (especially if they are using the collaborative work correctly and arguing at their tables). Within a day or two, we’re Mistake Gaming the problems with whiteboards.
And in general, we are also wrapping up the unit as quickly as possible. We’ll come back to everything in this model when we build the constant acceleration model in a couple of weeks. We want to move into forces as quickly as possible and start explaining, not just describing, with our nascent physics powers.
1. The cart in my photo is the crazy-expensive PASCO version. There is a cheaper (but less straight-line-going, I hear) cart that has been oft-discussed on Modeling Instruction listservs.
2. I might also just let each group decide on their own. I might do a hybrid, making a suggestion of what we usually do, but giving them the freedom to make their own choices. In general, I haven’t found that giving them the choice here is worth the time unless it’s a group that I am pretty sure is going to be able to deal with the graphs being different (and therefore the slopes and intercepts meaning different things) during the discussion really efficiently and easily. (A group this year decided to go rogue on the graph, but quickly realized that you could compare it to the others if you rotated the board, then pretended to look at it through the back of the not-transparent board—not every class is going there on day 1 though!) Even though it’s the first day of class, knowing my students pretty well before they get here (small boarding school + working on freshman dorm) can help me make that call in the moment. It just varies from class to class.