Green Globs meets Parabolas (video)

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What are functions?

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Wikipedia.com defines a function in mathematics as an abstract entity that associates an input to a corresponding output according to some rule. A good mental model for this idea is a function machine. My colleague at CIESE (where I worked until last October) Jason Sayres created this nice example which he calls The Mystery Box Game * which is really version of "Guess my Rule". For example, let's say 2 goes through the box and becomes 5 5 goes through the box and becomes 11 9 goes through the box and becomes 19 Can you predict what rule is being used here? (Spoiler alert: click here ) A rule is a mathematical way of explaining a pattern if one exists. Since mathematics can be thought of as the study of patterns, rules or functions are foundational for understanding mathematics. Hopefully, the rest of lesson 1 (Chap 3) will be easier to understand. (Note: Bob has posted the answers for the first lesson of the first edition on our Jacobs group site.) So what'
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The next lesson in chapter 3 of your math book is on graphs, lines, slopes, and equations. Why would you ever want to learn this you might ask? One very good reason is that in the very near future you will be participating in the Great Green Globs Contest and you will need to know all about coordinate geometry. You will learn by playing the Globs game either on the computer* (which is the best way) or with a printout of the score sheet along with a pencil and ruler. Green Globs: The Game (a description) Thirteen randomly scattered “green globs” are displayed on a coordinate grid. The goal is to "explode" all the globs by hitting them with the graphs of equations entered on the keyboard (or paper). The scoring algorithm encourages you to hit as many globs as possible with each equation in order to get the best possible score. The following lesson will help you to learn all you need to know about graphing equations to succeed in Green Globs. Have a good time, but don’t
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Globs Contest meets Parabolas (Lesson 3.4)

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This is the glob array I posted for the contest on 10/31/08 and here is my video that describes one solution using linear equations. In order to get the highest score possible you need to to maximize the number of globs you hit with each equation.   I was hoping to get 5 globs with one and I came close with my first shot, but it missed (1,-1). The same was true for shot #2. Here’s all 5 of my shots. (I made the figure below with Geometer’s Sketchpad.) So I managed to get 37 points. I’m not sure that I can do better using linear equations. But what if could use parabolas? Mr. Jacobs introduces us to parabolas in lesson 4. He defines it as the graph of y equals x squared. In set 1 he explores the patterns that help us to move the parabola along the y axis and also how to draw it “upside down” by making the coefficient of the x-squared term negative. In the next video clip I describe more about graphing parabolas.    Here’s the three parabolas I used to get my score of 61 point
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