Hmm... Finally something to exercise my amateur radio side of the brain.
Ok, There are a few tricks you can use to get your PITS board the maximum chance of being heard. First, the balloon payload (your pi radio) will perform best if the antenna is oriented as nearly to horizontal as possible. A misconception about single element antennas is that you can increase signal by pointing the tip of the antenna at your receiver. The opposite is actually true. You can see this for yourself if you look at pictures of multi-element beam antennas used for pinpointing a radio signal. While the boom of the multi-element antennas would be pointed at the target, the actual elements of the antenna are perpendicular to the boom.
A little study of antenna theory will reveal that an antenna element radiates its signal outward from, and perpendicular to the radiator. Additionally, the antenna can get increased gain if there is an appropriately sized reflector behind it and parallel with the radiator. The reflector could be something as simple as an aluminum foil coated cardboard disk of the properly calculated diameter. Since this seems to be a class project, I will not tell you what size to make your antenna or your reflector, however the quarter wavelength for both is the perfect starting point.
Next, the antenna for the software defined radio dongle. This is where you can make the most difference in your signal work. The balloon payload is naturally limited by the weight and wind resistance as to how much antenna you can use. So, the best bet is to magnify the signal received at the dongle radio. Now here we have several factors to consider.
First, you will want to make a beam antenna tuned to your balloon transmitter frequency, and attache it to the dongle with a coaxial type cable. The cable needs to be as short as possible and yet be as close as possible to an odd multiple of quarter wavelengths (1x, 3x, 5x,etc.). This odd number tunes the timing of the signal received to reduce signal loss in the antenna system. But beware, there is another factor to consider here as well. High frequencies move through a cable slower than they do in open air. The difference between those speeds is expressed in a term called "velocity factor". For example, RG58u, the most common type of cable for simple transmitters, has an impedence of 50 ohms and a velocity factor of 0.66.
This means that to calculate the proper quarter wavelength of a cable, you need to multiply the original formula result by 0.66 to arrive at the actual cable length. Remember, the fewer odd multiples of 1/4 wave the better the signal on the wire.
Now for the antenna, I would suggest googling the term "fox hunt antenna project". It will point you to many instructables on building a multi-element beam antenna. Then have the class design a portable mount for the antenna (maybe a camera tripod) and have it near the laptop. Just point the antenna in the general direction of the target and you should be able to receive the signal. The further away the target gets the easier is to point the antenna. Additional study of how the signal lobes of a beam antenna would look (if you could see them in the air) will certainly provide a few days of fun calculations and even visibly show how all of this works. After the first draft of your antenna is ready to test, set up your classroom testing experiment again and report back your progress. I think you will find the signal difference to be huge if you make everything up well. Don't be afraid of the calculations and most importantly, when everything else "looks" right but you still have issues, TRUST your math! If you followed the formulas properly, then the problem will most likely be elsewhere. Second guessing your antenna after it is built is a fools errand. Just do the math backwards and forwards before you build and trust your instincts on this. everything else is just connectors, radios, and batteries
A word of caution. Do not be tempted to make the antenna on the balloon payload to anything more than the single element radiator and an appropriate reflector. In this configuration you will reduce the antenna signal toward space (up) and increase the signal toward earth (down). Any additional elements on this antenna would make it too directional and therefore nearly impossible to track as the signal would be pointed which ever direction the wind wanted. With the suggested configuration above, the higher the balloon flies, the wider the cone of the signal would be pointed down to the earth.
I have played with this stuff before with low power radios (cheaper than the RPi setup) and had lots of fun. But then again my brain craves the mathematics involved in making it right.
Have a blast with this and send pictures later. That would be a treat.
BKM888