### Activities for teaching motion.

In general we'll need a Cartesian grid that we can put icons in. Some of the icons will represent objects (e.g., trees, houses, "clouds") that don't move. Clouds obscure anything that is "behind" them, making it possible to hide things and challenge students to find them, given clues. Other objects (people, bicycles, cars) have the ability to move and represent their velocity either as x and y components or as an arrow denoting the velocity vector, or both. Alternatively, we might want to build something like the old ThinkerTools "datacross" - an array of four "thermometers" pointing up, down, left, and right, that denote the x and y components of the velocity. This has the advantage that the datacross is not attached to the moving object, making it easier to read and perhaps avoiding conveying the impression that the velocity is somehow "attached" to the object. (Note that we can add an arrow to the datacross representing the resultant velocity vector. This will be useful in some circumstances, and not useful in others. If/when we do this, it should be possible to manipulate the arrow and affect the velocity component "thermometers," and vice versa.)

#### Activities and software for helping students represent position in terms of displacement vectors.

Challenge them to move an object from one place to another by defining the displacement vector between the origin and the desired destination. Place one or more obstacles in the direct path so that they can't do it with just one displacement vector, but challenge them to do it with a few displacements as possible. (A "golf" analogy might work here - a "par-3" hole is accessible with 3 displacements, if you use more you "bogey" the hole.)
A variant of this idea is the "treasure hunt" metaphor that we used with MAC - get to the position of the treasure by adding displacement vectors. The point there was to recognize that the displacements "commute" in the sense that the order of their execution is irrelevant.

For activities involving more than one person we could have the kids hide treasure in various places in "their" territory and then send out "sniffers" that can find it if they travel over it. The sniffers are controlled by giving them displacements, so the idea is to maneuver them so that they cover as much of the other person's territory in as few moves as possible. If they accidentally bump into an obstacle (e.g., a tree or a house) or the border of the screen, their sniffer stops and they lose points.

Activities and software for helping students understand constant velocity in 1 and 2 dimensions.
We could give the kids a do-it-yourself distance-rate-time problem construction kit. They can put icons on the screen and command them to move at constant speed - initially in one dimension, later in two. The motion of the objects can be programmed so that at any particular time after the start of the model the object can be given a new velocity. The interface for doing this could be similar to that for iMovie or other animation editing software that kids may be familiar with - we would create a timeline for each object on the screen and let the students drag arrows onto it. (Note: this is significantly different from the "boosters" that we developed for MAC. Those incremented the velocity when anything ran over them. It was harder, I found, to come up with good pedagogical uses for them.)

#### One-dimensional motion

The standard real-world situation for 1-dimensional motion used to be a train, but a car or bus on a single highway might be better (most kids have probably never seen a train!). The standard activity would be to challenge students to create a situation (e.g., "Start this express train out from Chicago just enough later than the slower train so that the fast train catches up with the slower one just as both of them are passing the Cleveland station."). At first they do it purely by experimentation, later on they're contrained to "getting it right" in at most two trials, say, so that they have to make predictions before blindly trying things out. And we track the number of trials, whether successive trials get closer to the goal, and so forth.
When it comes to average velocity over a trip (in a situation where the velocity changes during the trip, obviously) we can set up "round trip" situations (e.g., the train travels from Chicago to Cleveland at 60 mph. How fast does it have to go on the return trip in order that the average velocity for the round trip be 80 mph?). We can also set up challenges where the goal is to make the train go 40 mph, 60 mph, and 80 mph for variable lengths of time that the students can program, in such a way that the average speed is 50 mph, say. (Equivalently, we require that the train arrive at the destination by such-and-such a time.)

#### Two-dimensional motion

As the ThinkerTools project demonstrated lo these many years ago, it's a Good Idea to start with rectangular coordinates and design activities that demonstrate the independence of motion along orthogonal coordinates. A generic activity of this kind involves steering a vehicle (or mass particle) around a particular trajectory, either by delivering fixed-size dollops of horizontal or vertical velocity or by programming such dollops by dragging velocity "boosters" onto a timeline. (The real time version also makes for a great collaborative game -- you give one kid control over the x velocity and other controls the y velocity and together they have to guide their "dot" over the desired trajectory. You get great conversations out of this, as the students try to make sense of what's going on.) Both the real time and the programming version of this activity will  provide great data for diagnosing a student's  understanding of motion in two dimensions. For the real time version we can simply report how far the kid got before crashing into the wall -- and look to see if that distance increases steadily from one trial to the next. For the "delayed reaction" programming version, the number of boosters used (we can put a premium on doing the job with as few boosters as possible, or limit the number of boosters available), the nature of the boosters (i.e., what direction and magnitude delta-v they produce), and where they are placed along the time line will tell us a lot. Also, how those measures vary from one trial to the next -- in other words, are the students able to diagnose their errors and fix them after observing the motion?

### Activities for teaching about forces.

The way we approached this in ThinkerTools still seems to be a good one. We started out, as described above, by giving students the ability to exert a fixed-size impulse in one of four directions (to the right, to the left, up, and down). Each impulse resulted in an instantaneous change of velocity; because the objects involved all had the same mass, each change of velocity had the same magnitude. In real time mode we allowed students to apply multiple impulses to the object, but we restricted the number of impulses that could be imparted per unit time and we also restricted the total Vx and Vy that could be applied. In this world, the concept of force really doesn't appear -- what we have is impulses: infinite forces applied over infinitesimal times such that the product of the force and the time interval (which equals the change in momentum caused by the impulse) is a constant number.

We then approached the study of continually applied forces by introducing a limit process: we progressively halved the time interval between impulses and simultaneously halved their magnitude. This had the effect of maintaining constant the average applied force, but as the process continued the time between applications became small enough that students could ignore it and the effect was indistinguishable from that of a continuously applied, constant force. Later on, we introduced changes in the mass of the object upon which the force was applied, and worked on activities that would force kids to take the mass into account.