Judah L Schwartz, Apr 11, 2016

[b]This is a series of applets drawn from the website mathMINDhabits [[url]sites.google.com/site/mathmindhabits[/url]] that is designed for teachers of mathematics who want to deepen their understanding of the mathematics they teach and that their students are expected to learn.[/b] In trying to understand the nature of motion we are confronted with the problem of reconciling two profoundly different representations of reality. One representation is the path (trajectory: [b][i]f(x,y,z)=c [/i][/b] ) taken by the object in motion as it moves. The eye perceives that path and the brain "records" it. Unfortunately, the representation of the path traversed is inadequate for understanding the motion - it does not tell us where the object was at any given time nor does it tell us how fast the object was moving at any given time. We need to complement the representing the motion by a trajectory with images that supply the missing information, i.e. graphs of position and velocity as function(s) of time [[b][i]x(t), y(t), z(t), x'(t), y'(t), z'(t)[/i][/b] ]. These images are not images that are eyes ever perceive - they are, however, needed [i][b]images for the mind's eye[/b].[/i] Many of us have observed youngsters interpreting a rising line in a distance-time graph as "the girl climbed the hill" and a descending line on the same graph as "the girl came down the hill". College students often confuse the parabolic trajectory of a thrown football [[b][i]an image for the eye[/i][/b]] with the parabolic form of the graph of the football's height as a function of time [[b][i]an image for the mind's eye[/i][/b]]. The applets in this collection all deal with motion and are an attempt to help the user to become nimble is moving between [b][i]images for the eye[/i][/b] and [b][i]images for the mind's eye[/i][/b].

We normally think of the axes of a plane as being a pair of number lines at right angles to one another and whose zeroes are coincident. Suppose, on the other hand, we are interested in the motion of a car down a straight road. As the car moves (or doesn’t move) there is a position along the road that corresponds to each time. Normally, algebra students and teachers, as well as scientists plot such data on a planar piece of paper with time along one axis and position along the other. Both the time axis and the distance axes are quantity lines and not number lines. The plane in which this graph is plotted is not the same sort of plane in which we normally explore the geometry of triangles, circles and rectangles. This means that transformations in the plane must be approached very carefully. Although there are exceptions, for the most part the transformations of translation, dilation, rotation and reflection of functions that map number lines to number lines have little to do with the ways in which scientists use these transformations when dealing with functions that map quantity lines to quantity lines. Let’s consider a specific example which illustrates how translation has to be approached differently. Suppose we have a point in a time-distance plane whose coordinates are t,d [to make the example more concrete let’s say that t = 15 seconds and d = 60 meters]. We can assume for example, that time t is plotted along the horizontal axis, and distance is plotted along the vertical axis and the units of d are meters. Now let’s consider another point in this plane, one whose time coordinate is 20 seconds and whose distance coordinate is 70 meters. Seen as numbers these two points are (15, 60) and (20,70). Seen as quantities these two points are (15 seconds, 60 meters) and (20 seconds, 70 meters) Does the following question make sense? What is the distance between the two points? Our definition of distance between two point (x1, y1) and (x2,y2) based on Pythagoras is that the distance is equal to the square root of the sum of square of separation in time (5 seconds) and the square of the separation in distance {10 meters). This computation requires us to add seconds2 to meters2- something we can no more do than we can add apples to oranges. On the other hand, we can translate the distance coordinates and the time coordinates separately. We just can't combine them into a single translation with a single set of units. Translation in a plane made of two perpendicular quantity lines differs from translation in a plane made of two perpendicular number lines. How does dilation differ in these two kinds of plane? Once again let us think of a time-distance plane. A dilation of the point t,d means multiplying both t and d by some quantity, say r. Although mathematically, we can regard r as a dimensionless number, we must pay attention to whether r represents a scale change as in meters/meter or seconds/second or a unit change such as 100 centimeters/meter or 1/60 minute/second. For example, if r = 1 minute / 60 seconds then we can multiply the t seconds by r, but if we try to multiply the d meters by r we will get d/60 minute meters /second This highlights the fact that can't use the same quantity r for both coordinates. On the other hand, we can dilate the distance coordinates and the time coordinates separately. We just can't combine them into a single dilation. A reflection of this point (in the origin, say) means transforming the position from beyond a starting point to one before a starting point as well as a time after a starting time to one before the starting time. In many real circumstances, the transformed point has no meaning. The bottom line is that there is often a great deal of confusion among many students and teachers about the difference between a plane formed by two crossed number lines and one formed by two crossed quantity lines if the two kinds of quantity are different. This will happen with time-distance planes, time-temperature planes, volume-mass planes and so on. In geometry classes we deal with planes formed by two crossed distance lines while in doing application or word problems in algebra classes we plot functions in which the target quantity is almost always different than the domain quantity. Both of these sorts of mathematical activity lead to images drawn on paper – a sheet of paper is planar – and the implicit conclusion is often drawn, incorrectly, that they are the same sort of plane. The reader may wish to explore this issue further - please see the applet There are planes & there are planes

• 1. There are planes & there are planes

• 1. Baton twirl
• 2. There are planes & there are planes
• 3. Projectile in parameter space
• 4. Football & Handball in Parameter Space
• 5. Jack & Jill
• 6. The Princess and the Key -

• 1. starting wave at a stop light
• 2. falling dominoes

• 1. Pebble stuck in a wheel -traditional view
• 2. Pebble stuck in a wheel - space-time view

• 1. two clocks in k/n time !
• 2. Lissajous - oscillatory motion in two dimensions
• 3. Two dimensional oscillatory motion in space-time

• 1. Meeting Up - on the road with Red & Blue
• 2. Ship rendezvous
• 3. Monkey drop
• 4. 'Monkey drop' in 3 frames of reference
• 5. monkey drop - space time diagram

• 1. TWO MOTIONS - one path in space