California Science Standards and LOOPS Activities

Initial brainstorming and commentary by Paul Horwitz.

Motion: The velocity of an object is the rate of change of its position.

Students should know:

1. Position is defined in relation to some choice of a standard reference point and a set of reference directions.
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   Look at Dynamica Treasure Hunt activity and Hanging with Friends. Both are too long but might serve as a starting point. Modified versions of "Battleship" game? Find the target given rectangular or polar coordinates, possibly in more than one step.

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2. Average speed is the total distance traveled divided by the total time elapsed. The speed of an object can vary.
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   Object traveling around a track at some constant speed. What speed does it need to go on the next n rounds in order to increase its average speed by x? Predict and then try. Score on number of tries.

   Two ways to get somewhere: freeway all the way (where you travel at v1) or take a shortcut where you have to travel at v2 < v1 but the distance is shorter by x. Which one to choose? Race.

   Take a look at the Simcalc activities - Distance equals integral under a velocity-time curve. Jim Kaput's famous paradox: two different ways of describing the same thing: (1) changing velocity and resulting distance vs. (2) changing salary and resulting total earnings. (People instinctively understand the problem in the second case, but not in the first.)

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3. How to solve problems involving distance, time, and average speed.
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   The difference between these activities and the ones just before them might just be that in this case we ask the kids to solve a problem and plug numbers in before seeing the simulation, rather than having them figure out the answer by experimenting. (But of course, this is a continuous scale - depending on what premium we associate with experimenting "efficiently.")

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4. The velocity of an object must be described by specifying both the direction and the speed.
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   Not clear whether we should be teaching polar, rectangular coordinates, or both. Assume both. Should we teach the "parallelogram method" for finding the resultant of two vectors? Probably not.

   Velocities will initially be constant and the vector designating them can simply be an arrow pointing to the location of an object some short-but-perceptible time (like half a second, say) later. The new position is separated from the old one by a displacement vector that is characterized by a magnitude and a direction (or by horizontal and vertical components).

   Start with orthogonal (horizontal and vertical) velocity vectors: what happens if we apply both at once? Or single vector: what happens to the x and y components if we elongate it in the same direction? If we rotate it without changing its length? Activities involve translating from one system to the other: given the x and y coordinates, draw the vector; given the vector, draw the x and y coordinates.

   Note: this kind of thing is much easier to do with displacement vectors than with velocity vectors (think "Manhattan" coordinates vs., navigating at sea). We need to connect the two via movement the velocity is just the displacement the thing will experience in one (short) unit of time, so think in terms of where you're going to end up, rather than how fast you're going to go in order to get there.

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5. Changes in velocity may be due to changes in speed, direction, or both.
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   We need to separate the two kinds of changes so that the students see them as separate categories. We can do this by giving the students a vehicle that moves on a frictionless surface and to which they can apply forces by pressing the up, down, right, and left arrow keys, respectively. The directions up, down, right, and left are relative to the orientation of the vehicle, and the vehicle always points in the direction in which it is moving. So the up and down keys change the vehicle's speed and the left and right keys vary the direction of its velocity, keeping its speed constant. Display the velocity and the applied force with differently styled arrows and have the vehicle leave dots every half-second or so, designating its position at successive instants. We decree that they can't press two keys at the same time, so they can only change the speed or the direction, but not both at once. Have them play with this vehicle by racing it around variously shaped tracks (which should be pretty simple), then give them problems like: Look at the dots the vehicle left and draw the arrows representing the force (if any) that was applied at various positions. Or: have them produce a simple sequence of commands (e.g., by dragging force arrows onto a time axis) that will get their vehicle to traverse a given path. (A "paper and pencil" - actually, online - version of this could just be to match up a set of such command sequences with a set of trajectories.)

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6. How to interpret graphs of position and speed versus time for one-dimensional motion.
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   There's a good Dynamica activity that does this. Introduce a graph and a vehicle that travels on a one-dimensional track. Kids play around with it and then we show them a graph and try to get them to match it by moving the vehicle. Or we hide the vehicle (it goes into a "foggy" patch and disappears) but show the graph and ask the kid to point to where he thinks the vehicle is. Or we allow them to control the vehicle by manipulating the graph and accomplish something (e.g., arriving at the train station on time, or matching the velocity of some other vehicle. Or two trains running on parallel tracks and our hero has to jump from one to another - where should he jump? Indicate where on the graph and then run the simulation to see.)

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Scoring Stategies

Force

Unbalanced forces cause changes in velocity. Students should know:

1. A force has both direction and magnitude.
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   Actually this one is tied very closely to the equivalent one for velocity and could be taught together with that one. We can play the same game that we played with velocities (defining them in terms of the displacements they result in over time) and start with forces that last only for some short-but-perceptible time like half a second but are constant as long as they are on. These forces change the velocity of an object- either in direction or in magnitude or both - by an amount that depends on the magnitude and direction of the force, and the mass of the object. The difference is that it's much harder to perceive the difference between two velocity vectors, particularly when the change is only in the direction. (The notion that a force exerted at right angles to the direction of motion produces a change in direction with no change in speed is not what Andy diSessa would call a "p-prim"!

   Again, the activities would involve setting up a force, or a sequence of forces, that achieve a particular trajectory or goal state - e.g., making the object trace out an octagon by exerting equal-magnitude "puffs" of force at each exterior vertex - and checking to see how close the student comes and how many tries it takes to come that close.

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2. When an object is subject to two or more forces at once the result is the cumulative effect of all the forces.
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   It would be good if students could collaborate on this one. Each student controls one force. Initially pairs, later on greater numbers of students exert forces on the same object, which behaves accordingly. The purpose is to get the object to move in a certain way - e.g., to get it through a goal on the screen. Each student controls only one aspect of his force - e.g., its magnitude, but not its direction. For instance, in the two-student case, one student controls a vertical force and the other controls a horizontal force. (Imitating ThinkerTools, we could have arrows pointing to the object, one along the bottom of the screen, the other along the right side of the screen. These then represent the x and y coordinates, respectively, of the object, and one arrow is controlled by each student.)

   In "play" mode the students just fool around and get to know the system; then in intro mode (where there are only two forces, horizontal and vertical, and the students can only change their magnitude) they try to "keep the car on the road" or something. Then we can add a third or fourth force (at arbitrary angles) and eventually we go into "problem solving" mode where the students can't change things in real time but have to decide what they're going to do without (or with a minimum of) experimentation. It's useful to have the forces "snap to a grid" so that the resultant can be calculated easily. (Dynamica does this.) A good problem for kids to solve is: given a set of forces acting on an object, find another force which, when added to the set, "cancels out" all the other forces.

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3. When the forces on an object are balanced the motion of the object does not change.
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   It's relatively easy to figure out that when something isn't moving (over some period of time) the net force exerted on it must be zero. It's much harder to get students to recognize that a moving object that is not accelerating has no net force exerted on it. (Everyone thinks that the motorboat that is moving at a constant speed must have a net force on it "to combat the frictional forces that are trying to slow it down." Didn't TEEMSS 1 find that a majority of middle school science teachers fell into that trap?) We need to produce several isomorphic activities that force this principle down the kids' throats in various situations. We may need a velocity-dependent (e.g., v or v-squared) frictional force that enable kids to discover the concept of "terminal velocity."

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4. How to identify separately the forces acting on a single static object.
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   This concept flows directly from the activity described above in which the student is asked to create a situation where the resultant of all the applied forces is zero. The standard situation is the object supported against a gravitational field - e.g., the book on the desk. We depict them the gravitational force on the object and ask them to draw the constraint force (not hard!). Or we show a team of oxen pulling (at constant speed) a very heavy object and challenge them to draw the frictional force (harder!).

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5. When the forces on an object are unbalanced the object will change its velocity (speed up, slow down, or change direction).
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   Again, this flows the earlier ones - IF you know that a single force causes a change in velocity AND you know that a set of unbalanced forces is equivalent to a single force, then it follows that whenever the forces are unbalanced the velocity is going to change. The key point is to recognize the second "if" - that no matter how many forces are acting on an object they are equivalent to a single force (plus a torque, but we aren't dealing with that!). So "all" we have to do is really pound that point home and then this learning goal becomes a test of whether they really understand it.

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6. The greater the mass of an object the more force is needed to achieve the same rate of change in motion.
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   Clearly, we start by giving them different masses and having them exert the same force and see what happens. At first, the size of the object can be a clue about its mass - later on, we can mix that up and see if they can, for instance, sort objects by mass even though their radii are not correlated with their mass. Or pick out of a set of identical-looking objects the one with the greatest mass. The trick here, and what makes this a bit problematic to teach, is that gravity won't work as a force, but gravity is by far the most common force. So we don't use gravity as our force. (Do we also point out that gravity is different, or does that transcend the standards? God forbid we should teach them something that's not in the standards!)

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7. The role of gravity in forming and maintaining the shapes of planets, stars, and the solar system.
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   This one is quite different from all the others! I wonder how it got into the standard! It's also peculiar in other ways. It asks for the kids to understand three completely different (to them, anyway) things: the shape of planets, the shape of stars (what shape? Stars don't have a "shape" - at least not to the kids!), and the "shape" of the solar system! That last one isn't even well defined, though I suppose they mean the shape of the orbits of the planets (and the comets???). Bizarre!

   OK - enough ranting. We can do something with this one, I imagine. For instance, the shape of the Earth ("round") we having the kids make an "Earth" with MW and letting "gravity" compact all the particles into a more or less circular shape. Note: we'll need a long-range force in MW - do we have one? As for the stars - someone's got to tell them that they have a shape and that it's round, just like the Earth's and for the same reason. End of story. The "shape" of the solar system I suppose means just experimenting with orbits. We can give them a fixed (infinite mass) sun and a finite mass earth and have them experiment with different initial velocities and try to make a circular orbit. Once they have one (within some tolerance - and we need to generate an event when they get there), we give them another planet, or maybe a moon, and have them try to circularize that orbit. (Which won't work very well because the scale will be way off and the three bodies will all interact with each other! Oops! We'll have to try it and see if we can make it work well enough.)

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