It was the week before Xmas and the Upper Sixth had just returned after a study leave period. The next topic in the scheme of work was continuous distributions, discrete ones having been explored for several weeks previously. The school had just acquired an OHP TI82 screen for use with the set of TI82's. The following describes what turned into an excellent set of lessons which I will certainly repeat.
Firstly we needed some data. Having done some work many years ago working on a project modelling military manoeuvres I came up with the idea of parachute drops. The class made their own parachutes out of carrier bags (a circle cut round a large dinner plate), string and a weighted cardboard cut-out Santa Claus (chocolate ones would have been better it was suggested!). A point was marked on the ceiling and a plumb line dropped to give another mark directly below. The parachutes were then held up to the ceiling mark and released and their distance from the 'drop zone' measured. This was repeated 50 times. The following was one set of data - all the groups however got remarkably good bell shaped distributions although the parameters varied according to various design elements in the parachute.
The following lesson this set was used as an example. Everyone put the data in their TI82 (the first use any had made of these) and after adjusting the scaling under WINDOW first a Box & Whisker and then a histogram was drawn. Having the screen of my calculator displayed on the white board was very reassuring for the students in knowing they had done the right thing - a few minor problems were sorted where awkward MODE's had been left in.
The class were then told that the purpose of the lesson was to try and find a mathematical model to fit this data. 'Inverse quadratic' came up fairly quickly and after some discussion the graph of y = - x² was sketched on the whiteboard in position on the histogram axis displayed by the OHP. After several suggestions and trials by the students the quadratic was shifted, enlarged and stretched sideways to give a curve that the students agreed was probably the best they were going to get from a quadratic:-

One student suggested a split function with the inverse quadratic up to say x = 35 and then a positive quadratic to give a nice tail at the end. All agreed that this was a good idea but a bit clumsy. " The trouble with all of these", said one student, "is that they will all cut the x-axis at some point but in reality they need to got to infinity, " What if the parachute had caught a draught.?"
" You need something like a sine curve", said another. At this we proceeded to fit a suitable sine function, although it was pointed out that this still cut the x-axis. At this point I gave some direction by looking at curves which tailed off to infinity, 1/x was suggested by the class We switched off theSTATPLOT at this point and I suggested they look for a function which gave a positive and negative infinite tail. After arriving at e^x and e^-x and discussing how we could combine them into one function I suggested that Gauss faced with the same problem had come up with exp[-x²] and there was an almost universal hum of approval as they saw the bell shaped curve it produced. We then returned to the STATPLOT histogram and with very little prompting they were able to add various constants to produce the following model which they all agreed was an eminently good fit.

Several students commented at the end how they had seen that day the purpose of 'all that graph plotting they had done in the Lower Sixth'. By the end all were confidently using the TI82's which were previously unfamiliar to them.
In the new year it is planned to spend a further lesson to finally bring this round to p.d.f.'s. By looking at the area using the integral function (Option 7: under CALC) the total area under the curve can be found. Functions can then be divided by this area as a constant that will give a probability function. Once done the integral can be used to work out areas that will give the probability of a value lying in a particular range.

All such modelling exercises should end up with a simulation. Using random numbers to generate distances from the spot ( perhaps a rectangular distribution to give a bearing?) a series of spots can be placed an a scaled diagrams to give,( hopefully) a realistic picture of what went on in practice.

Many thanks to my colleagues Bryan Dye, and Crisp Wilmington for their technical assistance in setting up the lessons.