This would make a great project for polynomials. Students can either take or find pictures on the internet, overlay a coordinate grid and the curve and then write the function that would produce that curve. Look at more examples here.
Polynomial Pictures
Quadratic Equations and Projectile Motion
“Can you think of something that can be modeled with the quadratic equation defining a projectile’s motion?” From SFSU, here are 9 activities with handouts and rubrics centered around this question.
Graphing Calculator 3D
This is a wonderful free download: Runiter’s Graphing Calculator 3D. (The “try online” version is not very good).
In middle school, there’s little need for 3D graphing, but sometimes it’s good to show students where they’re going and not simply where they are. It also has 2D capability which would be more useful in regards to graphing linear equations or quadratic equations for Algebra.
It took less than a minute to install, and if you’re familiar with graphing calculators at all, it’s fairly self explanatory. For the Algebra 1 classroom, the basic functions you would need to know are:
1) Input your equation(s) in the “y=” on the top left.
2) Click on the coordinate plane icon at the bottom left if you want to change the default 5×5 grid to 10×10.
3) The table icon near the top middle will hide/show data.
It doesn’t do too much more than the TI software your school might have except that it’s more colorful and requires no set-up. But one school I taught at had very little technology resources at all – I wish I had known about this back then.
There are a few bugs, nothing that will affect your computer and nothing really important. For instance, the “example” button will show you various graphs. Each time you click this example button, a new graph appears. There’s only 10 or so that it cycles through but if you switch from 2D examples to 3D, then it’ll just stop graphing anything even though the data still appears to change. Well, like I said, this isn’t important.
Also, with this free version your capabilities are limited. Many of the functions are disabled such as the ability to save, etc. The most annoying part about that is that if you accidentally click on an icon that has been disabled, it will ask you if you want to upgrade (pay about $50 for the real deal). Which I understand since this is a free version, and they’d prefer to sell their product. But I think this free version is all that you would need for a middle school class, though maybe it would be worth purchasing for Calculus or Statistics (they have an advanced Stats calculator).
Quadratic Equations Project: Catapults
I have always wanted to incorporate catapults into the quadratic unit, but frankly, the catapult designs I’ve always come across seemed complicated and daunting. So I was very excited to come across this design made out of craft sticks and rubber bands.
Here’s the link for video instructions:
Craft Stick Catapults
I don’t plan on buying the kits, as craft sticks and rubber bands are easy enough to get a hold of. As for the basket, I’m sure that could be made either with masking tape, or by simply gluing down a washer or a nut. As far as a good projectile, I’m thinking Skittles.
After building the catapults, students would then have to measure how far down they pull their basket (the height of the basket from the ground), the distance and time in the air of their projectile, and then model the quadratic equation.
The other thing that I really like about this design is that it makes it easy to adjust the settings. For example, since the catapults in the video are 9 craft sticks thick, students could easily change it to 8 or 10 craft sticks thick, and model those equations too. And then maybe come to some conclusion as to how many craft sticks they would need to have the projectile go x distance.
**11/10 update: Actually, I don’t think there would be a significant difference between 8, 9 or 10 craft sticks. Maybe just compare between 3 and 9 or something like that.**
Just mulling over ideas…
**11/18 update: I think this would have to be done outside to keep the skittles from bouncing around so much. Also, instead of changing the number of craft sticks, students could change the initial height of the catapult by first placing it on the ground, and then placing it on something higher, like a picnic table (there happens to be some picnic tables in the courtyard of our school).
Graphing Quadratic Equations: Pendulums
It’s best if students get to experience collecting actual real world data and seeing that data actually forming into the quadratic model. A worksheet filled with numbers that claims to be “from a science experiment” or “real world data” is not. To me, real world is something students can gather themselves or experiment with.
One experiment that you could use is the pendulum experiment. This is a classic physics lab that demonstrates how length affects the period of a pendulum. There are 3 options:
1) The best option is an actual, hands on experiment. Go here for tips on how to set up the experiment. All you need is some string (sturdy-ish), a weighted object (unbreakable, preferably), something heavy (like a stack of books, or someone could hold/sit on it) and a stopwatch. The gym should have stopwatches, or there are online timers. Your science department might even have actual pendulums.
2) A second option is to use a computer simulation of a swinging pendulum. The students can collect their data in a computer lab or with class laptops. The worksheet on Algebrafunsheets is a worksheet for this online version.
3) A final option is that the teacher could perform either the hands on or the computer model experiment in front of the class, soliciting student participation, and gather the data together as a class. Probably the least effective method.
What students should discover is that their data will curve in the typical quadratic model (you’ll need to set the x-axis as time and y-axis as length if you want to keep the “u” shape Algebra 1 students are typically asked to graph).
A great way to finish the graphing quadratic section is with this brief film “Parabola” by Radiolab. 4 minutes long. (Most are not true parabolas, but I love the into).
Quadratic equations – Population Project
I just made a worksheet in which students estimate the U.S. and world population for various years, an application that’s (very) roughly modeled with a quadratic equation, and came up with an idea for a project.
Students create a time line (poster board, computer, etc.) of the United States, and using the quadratic equation that models U.S. population growth, have them select 20-25 (or however many they have time for) important dates (Civil War, World Wars, etc) and calculate and graph the approximate U.S. population for that year. So the y-axis would be U.S. population and on the x-axis, time. This way, students are graphing a relevant, meaningful quadratic equation instead of just a random equation with arbitrary values. And since they can relate a growing population with the growth of a quadratic curve, it adds meaning to graphing quadratic equations.
Of course, they should elaborate each coordinate with a picture to represent the events they’ve selected.
Just a thought.
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Concordia University
