Lerner Physics

So, after a day of learning that the slope of the momentum-time graph is the force, and another day when we figured out the area of the force-time graph gives you the change in the momentum, we were ready to put this model to use.

I know that lots of people use momentum bar charts, but I'm not 100% sold on them. I could be convinced, and I know that energy bar charts are A-MA-ZING. I wanted to play around with momentum bar charts that made the focus more on the motion of the objects and less on the blocks. 

So, following in the footsteps of the LOL diagram, I use the SOS diagram. The name will make more sense a little later. 

Remember, my students have done balanced forces yet. They only know that a force is a push or a pull and is the slope of the momentum-time graph. I start by introducing interaction diagrams.

First, we make our first interaction diagram together as a class. We use the setup in lab as our situation. (I took a picture of the setup and posted it at the beginning of this post.) We start by listing all the objects we care about and putting them in bubbles. Then we connect the objects that interact with lines. Usually, to interact with an object in a physics sense, that is, to put a force on it, you have to be touching it. There seems to be only one exception we have right now, and that would be the Earth. It seems to push everything around, even when we're not touching the Earth. I mean, that's why babies are FASCINATED by dropping objects from high chairs. "Look!" they seem to scream. "This object is being pushed around and NOTHING IS TOUCHING IT." We aren't usually as impressed nowadays that we're older, but we should be. Anyway, here's what the interaction diagram looks like:

At this point, what interaction or interactions are changing the motion of the object do we care about? Really only the spring on the cart. That's the one that changes the car's momentum. Wait! This diagram shows what changes a car's momentum! Cool! So I can draw the picture before that interaction and after that interaction and put a (simplified version of the) interaction diagram in the middle:

The first diagram represents pᵢ, the second diagram represents ∆p, and the third diagram represents the final momentum (there's no Unicode for subscript f. That makes me mad.)

So, we have our equation here, in picture form: Initial momentum + change in momentum = final momentum. Or, since change in momentum is the area of the force-time graph, and the simplest way to calculate that area would be a rectangle where the base was the time and the height was the average force for the situation, we can write it as...

Initial momentum + average force * change in time = Final momentum.

So why is it an SOS diagram? The O is for the system schema in the middle (which I call an interaction diagram, because 'system' is used differently by the AP, and I don't want to confuse my students), and the two S's are for sketch. It also is a cry for help when you're using the momentum model.

If you've been following me and read this tweet, you know that I'm going to be defining momentum in a different way this year. We're defining force as the slope of the momentum-time graph. This means we have to define momentum too, which is the hand-wavy part of this unit. Why even come up with this concept? Luckily, we define momentum as "how hard it is to stop something" and "mass times velocity" before anyone was too awake.

Then we went to lab and had a cart bounce off a force probe. We changed the mass of the cart and the spring on the force probe. We graphed momentum-time and force-time graphs. Here's what students noticed in lab:

• The momentum of the cart stayed constant except when it is touching the force probe.
• The force was negative because it pushed in the opposite direction of the initial motion of the cart.
• The momentum of the cart started positive and went negative.
• The force was the largest when the momentum of the cart was zero.
• When we used the springs, the momentum after hitting the spring was the opposite of the momentum before hitting the spring. The two momenta had the same magnitudes.
• When we used the clay, the momentum after hitting the spring was less than the momentum before hitting the spring.
• We could never get the final momentum to be larger than the initial momentum except that one time when the plunger exploded and hit the force probe, but we don't think that counts.

We had great discussions today about different situations where objects are accelerating. They were deep and interesting and in each class, students brought up some great ideas.

I took no pictures, though. I feel like a #teach180 rookie.

So, instead, I'm going to focus on how hard it is to be wrong. Students don't want to put up wrong answers. I had a student today who told me he was worried that other students would judge him for not understanding the material. I talk about how important it is to discuss different ideas. I train students to ask questions rather than just point out flaws.

But I feel like I'm still caught up in the right/wrong paradigm. I don't think I'm pushing the conversation from answer-finding (which is all about right and wrong) to meaning-making (which is all about clarity/connections and confusion/disconnections). I need to work on communicating that to students. Instead of, "what do you like or not like about these boards?" (which is my go-to question about whiteboards and is way too vague), I'm going to try other approaches. My ideas so far:

• What connections between the ideas on the board to you see? (Like, how do the position-time and the velocity-time graphs relate?)
• What connections do you see between this problem and the problem where...?
• What makes the most sense to you on this board?
• What did you learn by doing this question? (Or what did you learn from the discussion of this question?)

I'm still thinking about this.

Here are the issues my students thought about when they used the motion detectors to look at carts speeding up and slowing down on ramps:

• Is the position-time graph for an object going down a ramp starting at rest an exponential or a parabola? 
• How can the slope of the velocity-time graph be negative when all the velocities we measured were positive?
• So, when the velocity is negative, the acceleration is always negative, right?
• So, when the object is moving in the negative direction, the acceleration is always negative, right?
• So, if the object is slowing down, the acceleration is always negative, right?
• So, if the velocity is decreasing, the acceleration is always negative, right?
• So, if velocity is a slope and acceleration is a slope, they have the same shape, right?

We took our first quiz today about CVPM. But our short lesson today was about the particle in constant velocity particle model.

Mr. Kaar, Mr. Engels, and Ms. Pollack, and I created a unit to explore what "particle" means in CVPM, and how it relates to conservations of quantities in systems. This lesson is the first part of this.

We start with a video of two air pucks connected by balsa wood sliding along the ground. We make sure we know what those air pucks are, and how they seem to move at CVPM. We then learn how to track one air puck. (See awesome gif above. Thanks, gifmaker.me!) Cool, a motion map! That motion map doesn't look like CVPM at all! We look at the position-time graph, and that doesn't look right either. It's way more complicated than a straight line. What should we track? Students usually quickly realize the middle is what we should track. We then do the same thing for three air pucks attached into an equilateral triangle. What should you track now? That's more difficult, because the middle isn't really part of anything. But it still works!

We then give them the two pucks again but this time, when they track the center, it doesn't look right. Why? Because we secretly filmed it on the ramp outside. So that's why the dots on that motion map aren't equally spaced. It seems to be speeding up, which is what we'd expect on a ramp. Guess we'll have to test things on ramps with the motion detectors on Monday...

• We will get a chance (literally in the next unit) to deepen our understanding of these representations.
• It takes a lot of effort to get the last bit of refining. Do I care what they do on motion maps with the last dot? Well, no, not really. If a student decides to draw the position-time graph one more second to what the arrow says will happen next, I'm OK with that. I'm sure other physics teachers have strong opinions about this, but I don't, and more importantly...
• I don't know what we'd get out of having the best constant velocity model. Constant velocity isn't the cornerstone of physics. The next model is. I don't care if there's a small mismatch between my model of constant velocity and theirs. 

Still, even with this leeway, we still have some important discussions:

• Where does a position-time graph? Does it always start on the vertical axis? Does it always start with the first dot?
• What happens between one region of constant velocity and another region? How can the velocity change and no time pass?
• What do the numbers on the motion map diagram represent? 

I teach two sections of AP Physics 1. They're both in the morning. They both have the exact same number of kids. They both had the exact same amount of time to do a lab with motion detectors. As two classes, though, the vibe couldn't be different.

One period talked about all the different situations with the motion detector pretty quickly. All the students basically agreed, and the questions seemed mostly limited to questions about the conventions of the representations. (Do the axes have arrows on them? When do the arrows go on the dots in the motion map?) When I was done, I was uneasy that they had not yet made their intuitions explicit. So we made a class consensus focused around two questions: how we show the direction of motion and how we show if the object is moving fast or slow in these different representations.

The other period, when going over the different situations, got into it. Some students seemed to want to represent what they actually saw in lab, while others were more willing to idealize it (use their "physics goggles"). Students could reason between the representations and give arguments for their answers. Not all of them, though, saw the situation the same way. It got a little hot. All the answers they whiteboarded were reasoned and not intuited. Even if their reasoning was not the same reasoning I would use, it was consistent and intelligent. By the time we did the class consensus, which I felt I should do, partially because I did it in the previous class and partially because I wanted to make sure we all ended up in the same place, every student was articulate about how to answer the two questions. The arguments were productive, but a bit too critical.

When we moved on to the next activity with the second class, I knew it was time to pull out Kelly O'Shea's Fun-Time Super-Cool Mistake Game™. (That's not what she calls it, but that what I call it in my head.) The mistake game is great; I learn more every time I read Kelly's post. I'm introducing it here because students were passionate about their answers, which is good, but were phrasing everything in terms of "right" and "wrong." I contributed to that; they started to echo my language about what I like and don't like by exclusively talked about what they don't like, meaning what they thought the whiteboard got wrong. This is a trap I fall in all the time, and Kelly O'Shea's Fun-Time Super-Cool Mistake Game helps me get out of that trap. I need to shift the conversation to questions like "According to the motion map, where does your object start? How does your motion map show that?"

After just quickly introducing the four ways to represent constant velocity motion of a particle (see my last post), I sent the students out to try to walk various position-time graphs, and then use the computer to find the velocity-time graph, and then represent that motion as a motion map and as words.

Some questions my students were thinking about today:

• Is going towards the detector the positive direction or is going away from the detector the positive direction? (Student tries it.) Wait, really? Why is going away positive?
• Do I ignore that first bit where I'm not moving yet, or do I draw it on my velocity-time graph? 
• Can I just call where I start x = 0? (As in, if I start where I normally start a little ways away from the detector and don't move, can't I call that point zero?)
• How does the velocity change in an instant? Doesn't it take time?
• How can I tell which way to draw the arrows on my motion map?
• How does the slope of the position-time graph relate to the velocity of the object? (One student came up with a great rule for this. I don't want to put his or her name in here because I didn't ask for permission, but every time I came back to that group, I always went back to that rule, naming it after the student who came up with it.)

What does summer work look like for AP Physics 1? I don't want them to start doing any physics, because they don't know how the class works yet. But summer work is a good way, I think, to have students check in with their math skills and see if they know some of the mathematical moves they'll use in the class. I've posted two pages from the seven pages of the summer work above. The other pages are about simple trigonometry and literal equations.

We whiteboarded this problems today. I was happy to see that classes were ready to ask each other questions, without putting all the questions through me. 

After that, it was time to sketch out, quickly, CVPM (the Constant Velocity Particle Model). Many units in this class are about the four representations we use to describe them:

• Diagrams: I introduce the motion map by doing an example. Sometimes I'll talk about a car with a leaky fluid. Other times, I refer back to what students did in lab to measure the speed of the buggy. I don't spend too much time on it. I focus on doing one quick motion map, asking if students can tell where are walked faster/slower/stopped, when I was walking forward/backwards, and move on.
• Algebra: Another quick one. At this point, I don't want to show all the equations we could use. Some students, as soon as they see an equation, use that representation above all others. I understand why; they take years of classes in the math wing where the equation is the best representation. So I try to deemphasize it as much as possible. I ask students what equation they used in math class to solve questions about speed. Here, I usually hear s = d/t; at other schools, I've heard D = RT. I write down either. I ignore it for now. (Later, when we've developed a good equation for velocity, I'll add it to the notes here, but not yet. I'm just keeping this representation open as a possibility.)
• Words: What words will we use for to describe this motion. Starting point, velocity, fast/slow usually come up quickly. I often have to give an example until forward/backward comes out. I don't like forward/backward because in this model, points don't have a front. I will sometimes do an example where I walk both forwards and backwards from the left side to the right side of the room. My belly button does the same motion, even though one is forward and one is backward. So I tell them, since I'm mathematical, I prefer positive direction/negative direction.
• Graphs: We found from doing the buggy lab that graphs are a great way to show your results. So a straight line on the position-time graph means constant velocity. We then answer these two questions: What does the vertical intercept mean in this graph? What does the slope mean in this graph? They are very useful, and that fact allows me to reiterate how important graphs are to physicists.

I don't always teach these four representations in this order, but I wrote them in this order because it's how I remember to do all four. I use the acronym DAWG. It's very 1990s, but it works for me.

Students whiteboarded their results for the buggy lab, and it was a lot of small handwriting, long tables, and half-hearted conclusions. No one, in words of Brian Carpenter, was willing to bet me a burrito on their results. We did come up with a list of things we wanted to see in whiteboards in the future: large, legible handwriting, multiple trials, and visuals would be nice. On the way home last night, I was thinking what would happen if we lined up all the whiteboards together. That's where the picture came from today, with many students timing the buggy for 9 meters every meter. I didn't like how teacher-directed it was, but my students and I did like how convincing the results were: