Symbiosis, Math Style

Last week, a conversation with the Algebra 1 (Mr. Steele) teacher went about like this:

Mr. Steele:  Hey, I’m thinking of doing a project about quadratics, and using projectiles.  Any advice?

Mr. Gant:  Well, don’t use those water-propelled 2-liter bottle rockets – they don’t follow projectile motion very well, because rockets aren’t under free-fall.

Mr. Steele:  Ok – what will?

Mr. Gant:  Ooh!  Water Winger!   Water balloons!   You can launch them, and boy are those true projectiles.

Mr. Steele:  (He would say “Ooh!” to follow mine, if were much less restrained)   Ah.  You know, that could tie into something like Angry Birds.

Mr. Gant: (not restrained)  Ooh!  I bet I could help out!   I need to address normal distributions!   MY students’ work could inform YOUR students’ work!   

Thus was born, in a period of about 3 days, the following entry document.   Comments/questions encouraged.  Suggestions for next year especially welcome.


Mr. Steele teaches Algebra 1, and he is about to start teaching quadratic equations.   As you might recall from physics, the pathway of a projectile follows a parabolic (parabolic = like a parabola = the graph of a quadratic equation)  path.   

So creating a project about projectiles seems like an obvious choice, and there is a great potential for fun.  In our conversations, Mr. Steele has mentioned the notion of a project where students create a real-life Angry Birds sort of affair, where students engage in some target practice as the end of the project.

Of course, you have to launch the projectile with some sort of slingshot thingy if you are going to emulate Angry Birds.    It turns out that as a former Physics teacher, I happen to be in possession of several “Water Wingers”.    These are contraptions that are meant to launch water balloons quite far.   Below, you can see a photo of one in use (with a pink water balloon in flight), and you might even be able to tell that it requires three people – two holders, and one person who pulls back and releases.  

Mr. Steele is thinking about using these devices, but he needs to communicate to his students how accurate they can expect their shots to be.    

This is where you come in.   Because every launch will have a little variation in it, we can expect that the landing spots will vary.   However, if we keep the launches as consistent as possible, then we can expect the variation in landing spots to be fairly predictable.  
In fact, I predict that the variation in the landing spots, along the direction of travel, will follow a Normal Distribution, but that is only a hypothesis.  

Mr. Steele needs you to discover that distribution, so that you can advise him on how accurate he can expect the water wingers to work.  That way, he will know how to best assess his students for his project.

Of course, that means that you will need to generate a means of consistently launching the balloons, and a means of measuring their landing positions, and then you will need to analyze their landing spot distribution.

As well, your group will need to present your findings to Mr. Steele, me, and perhaps some Sophomores in Mr. Steele’s classes.   During the presentation, based upon your analysis, you should be able to tell us z-scores and the probability of hitting a target with the water winger, depending upon the target’s size.    

Additionally, given a launch angle and distance of stretch, you should be able to give Mr. Steele a range where you think that if the students do their work right, they should land a balloon.   If the balloon lands outside of that range, then you can be confident that the algebra 1 students made an error.

Lastly, in addition to the presentation, every individual will need to create a written explication of your data-taking methods, results, and a complete analysis of the distribution of landing spots, including calculations of probabilities.