Quadratic Function. Parabola. Second Degree Polynomial: Parabolas in Football and Everyday Life

Quadratic Function. Parabola. Second Degree Polynomial. GA2 and TPC

Quadratic functions. Parabolas. Second Degree Polynomials.  There's so many names, you just know this has to be important.  For your blog, you'll find one aspect of quadratics to focus on and to develop further.  Choose something that's a little outside what we did in class but still includes quadratic functions.  My blog, hopefully, will send you to a bunch of different places where you can check out two categories of information about these functions.  The first category is the cool mathematics of these things.  The second category is the application of quadratics to the "real world". Of course, if you want to expand this and discuss something else relevant, then that's ok, too. If you have questions about whether your idea is ok, then just ask me.

As an intro, here's a site that has a slider (you'll have to scroll down to see what I mean) that shows you how the transformations to the quadratic function are paralleled with an athlete kicking a soccer ball. mathisfun. 

Please go to that site now and use the slider. Scroll down. See if there's something else on that page that catches your eye. There's several good ideas for your study there.

How many different mathematical ways can you think about a quadratic function?

1. Geometrically, as a conic section (for GA2 kids, if you don't know what that means, googleconic section).
2. Geometrically, as a set of points equidistant from a single point and a line. Here's an example of this. Again, you'll have to scroll down a little to see the circles and line. (Artsy folks: you might like the first image on this page; surely a child of the 1960's came up with this idea.)
3. Using algebra to plot points on a Cartesian plane using a quadratic equation..lo and behold, you get the parabola. (Sorry, no link for this one; you've all done this hundreds of times before.)
4. Use algebra to express the function in a variety of helpful forms: vertex form, descending form, factored form. (Again, we did this in class.)
5. Using creative forms of construction: folding paper  or using the circles paper shown in #2 link above. (There's some interesting youtube videos that show paper folding to create other conic sections; that can be an interesting exploration for you.)

So...what about interesting applications?

1.  A parabolic reflector.  (Someone's trying to make money with this one...)
2. A parabolic dish for a telescope. 
3. A football punt. (NFL and NSF got together for this video.) The idea applies, as you'll see in the video, to any projectile.

Find something that strikes your fancy. Choose ONE idea and develop it.  Make it more than 8 sentences; it must be a single full idea.  Cut and paste.  And, of course, submit through canvas. 

                                                                                                                                            

I found it interesting that I can find an infinite amount of parabolas in real life. For example, in architecture, rollercoasters, and nature. The McDonald's arches even is made of two parabolas. A dolphin's jump out of the water is a parabola. The Axis of Symmetry becomes very important in each case. This is so we understand the what the other half of the parabola is going to be like. I found this image below to understand the engineering in rollercoasters. I didn't realize it'd be so complicated. 

Also, there are parabolas in football. I remember the video we watched in class and found it interesting that they needed a mathematician to help the punter improve. I re-watched the video at home and found it interesting how the trajectory is a parabola. it follows this pathway due to gravity and the decrease of the vertical vector. The last thing I found interesting about the video is the closing statement where it it said "but takes NFL punters like Craig Heinrich, to turn science into a an art form". The distance between points of a parabola mean so much and can be studied to hit perfect spots.

           

Tessellations

A New Pi. For GA2 and TPC

I must credit Trevor Kann and his most recent post with this idea.   Click on his blog.  I'll summarize below, but I recommend you see his blog as the "original source." He also gives, in his blog, a lot of the answers to the questions I ask in my blog.   He explored the ratio of the circumference to the diameter of a circle (otherwise known as pi) in "Taxicab Geometry."  We had discussed this as a tangent in class, but being a good thinker and good blogger, he started asking himself, "What if...."

To set this up, in class we talked about a new way to look at  "point," "line," and "plane," after all, they are undefined terms.  What if we defined these things such that a plane was a grid, much like city streets on a NS EW grid. Points would be lattice points, or intersections of the "streets" on the grid, and we could travel only on the grid its self, much like a taxicab would drive.  A circle is still the set of all points equidistant to a single point.

In the grid to the right, the center point of a "circle" is shown with a circle of radius 2 (diameter = 4).  Remembering that you can only travel on the lines and points are only places where the lines intersect, find the circumference. Then take the ratio of the circumference to the diameter.  This ratio is known as pi.  (On a Euclidean plane, like we have been using, the ratio is about 3.14159....). Verify that this calculation of the new "pi" is indeed 4. Make your own grid; identify a circle with a different radius.  Is the ratio still 4? 

The above is our "tangent" in class last week.

So what did he do that was so cool?  Trevor asked, What if we have the same rules about lattice points, traveling only on the lines, but we set up the plane to be equilateral triangles.  What would pi be? (Actually, he said, "This got me thinking....") To the right is a snip of his new grid.  One unit is the length of the side of an equilateral triangle; points are only vertices of the triangle. You can see the center and a circle shown. The radius of this circle is 2.  Find the circumference.  Find the value that is the ratio of the circumference to the diameter ("pi"). It's not 3.14159.....  and it's not 4.


Then he thought some more.  What if the grid was tessellating hexagons.  In the process of thinking, he figured out that the only tessellating regular polygons are squares, triangles, and hexagons.   To the right is his image of the tessellating hexagons. Of course, he used a radius of 2 and found a value for the circumference, diameter and the ratio of the circumference to the diameter.  Go ahead, find that ratio. (Answer is in his blog :) )



So ok.  In this next blog, I know Trevor is still thinking about this (DO keep thinking, Trevor!).

For your blog, you can expand on this idea.  OR better yet, find something like, "Why are there only three regular polygons that tessellate?" or "What is the relationship between regular tessellations and regular polyhedra?" or "What if we put a circle on a soccer ball?" (Trevor's question, lay off that one, I think he may be writing about that one.....) BTW, a soccer ball is also a"truncated icosahedron."  OR even better, think of a tangent we took in class (or maybe your brain just took in class without us being with you on that mental adventure....) As yourself, "What if..." or be like Trevor and just start thinking....

                                                                                                                                              

Well, I have noticed that there are tessellations almost everywhere. If you look at your kitchen tile you notice that it too tessellates so that it fits together. Another example, is a fence. The diamond shape is able to figure together so that it can build into a tall structure. I think Trevor O'Brien said this is comment on Ms. Mariner's blog post, but the beehive built out of hexagons is too important. It is similar to the way glass domes are built. In Epcot the large dome is made out of  tessellating triangles. I think the idea of tessellation works well because it fits itself together like a puzzle, but a puzzle isn't really tessellating because all the small pieces aren't the same. Overall, I think the idea of tessellations is very interesting because it can be used in everyday life and also to transform graphs just like in the one above.