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## Tim Chartier

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# Frustrated With Math? Try Angry Birds!

Posted: 06/11/2012 5:56 pm

All over the world youth moan and groan when their math calculations are off and celebrate, often accentuated with a fist pump, when they're right. It's happening right now. In a class? Possibly, but much more frequently on a bus, subway, car, or in doctor's office. It's a craze. Children, youth, and adults wear Halloween costumes, winter hats, and T-shirts related to this application of math. Sound impossible? Not at all. Simply play the popular video game Angry Birds, and every time you prepare to launch an incensed bird through the air, you are using math.

On a smartphone or tablet, or through your browser, let's play Angry Birds. Each launch of a bird from the slingshot with the intent of a collision with a pig or fortified structure requires you to estimate a path of flight. Let's play with a bit of a different goal, in part, so you look at the game in a new way, especially if you are an avid player.

Looking at the screenshot above, how would you launch the bird so that it lands in front of the tower without hitting anything but the ground? Keep in mind that every time you change the launch angle for the bird, you are predicting its path. Now, launch the bird. Did you foresee the bounce? How close was the actual trajectory to your predicted one? Note the puffs of smoke that mark the path; they follow a nice smooth curve. Physicist Dr. Rhett Allain studied the game and verified in his article The Physics of Angry Birds that our feathered friends travel without air resistance. So, a bird's path is a parabola.

Mathematically, parabolas can be written as y = ax² + bx + c. Choosing specific values for a, b, and c defines a curve. In Angry Birds, if you find the right a, b, and c, the birds enact their revenge on the pigs and their fortresses.

Can our knowledge of parabolas yield other information about this popular game? To begin, we must define a scale that can be consistent from screen to screen. We'll use a scale of 1 slingshot and adapt Dr. Allian's notation and write it as 1 AB, as seen below.

A parabola, in the absence of air resistance, is also defined by an initial velocity and angle of launch. The birds are launched one unit above the ground. So, their initial point is (0,1). Adapting Allian's research, the parabolas can be derived for different decisions. Consider three different launch angles of 30, 45, and 60 degrees; we get:

y = 1 + 0.577x - 0.070x² for 30°

y = 1 + x - 0.106x² for 45°, and

y = 1 + 1.732x - 0.211x² for 60°.

Below, we see the curves for a bird slung at these angles. We see that for this screen only a 30-degree shot will result in an air strike.

What height will a bird be when it crosses the vertical line seen below if shot at 45 degrees? To answer this, we must determine where the dotted line crosses our x-axis.

Remember, 1 unit = the height of a slingshot. If we lay slingshots down one after another, we see that the dotted line is 5 units from the launching spot. So, we need only to set x = 5 in the formula y = 1 + x - 0.106x² = 1 + 5 - 0106(25) = 3.35. In the picture below, we see the projected height of the bird when it crosses the dotted line. Good enough for an arial view of the pig's fortress.

When you play Angry Birds, you visually estimate the coordinates of your target and use this to determine your initial angle. In North Carolina Kristianna Luce motivates topics of Algebra 1 using Angry Birds, which was a connection introduced in my Charlotte Teachers Institute seminar Math Through Pop Culture.

To begin, Luce had her students play the game, much as we did to begin this article. Soon, she directed the conversation toward asking what would be the highest point for a bird. Such a point is called vertex of the parabola. Given our work here, let's find the highest point for a bird launched at 60 degrees. This occurs at x = -b/2a. For 60 degrees, a trajectory follows y = 1 + 1.732x - 0.211x². So, a = -0.211 and b = 1.732. The vertex occurs at x = 1.732/(0.411) = 4.214 at a height equalling 1 + 1.732(4.214) - 0.211(4.214)² = 4.552. Note from the picture above that the top of the initial screen is about 4 slingshots high.

Luce's students used their knowledge of Angry Birds to help with algebra problems on their end-of-course exams. We'll consider two sample problems released by the North Carolina Department of Public Instruction. One question asks for the roots of the equation 0 = 9x² - 49. What would this be asking in the context of birds and pigs? Luce's class had discussed the point where a bird hits the ground; this is a root of the parabola and can be found using the quadratic formula. Our problem corresponds to an angry bird following the path y = 9x² - 49 and requires finding the points where this curve hits the ground, with a point existing on either side of the slingshot. Another sample end-of-course problem asks for approximate solutions to the equation x² + 4x = -2. Here a bird follows the trajectory y = x² + 4x, and we are interested in the x coordinate when the bird is 2 units below the ground.

We use math every day. In fact, given the mathematical thinking required in Angry Birds, math is used a lot every day! Further, Angry Birds can help in math class. Frustrated with math? Play Angry Birds and remember math and its application can be fun.

Follow Tim Chartier on Twitter: www.twitter.com/timchartier

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mathequality
1st Yr., 2nd Career HS Math Teacher
10:19 PM on 07/03/2012
[...continued] Fortunately, there are many true, real-world applications of math that exist all around us, which includes how the software programmer determined how to show the flight of the angry bird, or draw other objects on the display screen, etc. Whether one actively 'uses' math every day is debatable though, unless you are redefining the word "use," or its usage. As a mathematics teacher with an electrical engineering degree, I explain how mathematics is used by mathematicians, engineers, computer programmers, etcetera in cell phones, smart phones, iPods, Wiis, etcetera, as well as how mathematics can model actions, or phenomena, within those devices. I dare not claim they are 'using' mathematics, or 'doing' mathematics, in those situations since I do not believe that is factually correct, albeit they might be using or doing something that may itself be modeled, described, created, programmed, or conceived using mathematics.

If you can help me see your viewpoint, I will be most appreciative. I do not want to be overly constrained by reality, or at least my perception of it. So, please set me straight, or curved, as the situation may warrant! :)
mathequality
1st Yr., 2nd Career HS Math Teacher
10:19 PM on 07/03/2012
I appreciate your sincere desire to use Angry Birds, or similar apps / games, to connect students with mathematics, Tim. However, I struggle with comments like yours, or other claims like "math is fun!", which require use of a reality distortion device, aka "wishful thinking." I believe most students see through statements that they are employing mathematical thinking when they play AB, even after seeing a teacher explain the similarity between the mathematics of quadratics and the trajectory of an angry bird. The same is true with playing baseball, football, soccer, or horseshoes as well as doing just about any other action in life.

As a teacher, maintaining the trust of my students is critical; I believe making assertions that one does math when playing anything aside from Sudoku, Blackjack, Boardwalk, or similar is misleading at best, and incorrect at worst; even in the cases I mentioned, one may play them without necessarily doing math. [continued...]
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dsws
No owning ideas. Limit only commercial use.
10:42 AM on 06/13/2012
Just because something can be described mathematically, that doesn't mean it's done mathematically in actual practice. I don't play Angry Birds, but in other stuff I play, I've noticed that I'm just looking at actions, triggering stimuli, and results. I wouldn't know whether Angry Birds has parabolas, more realistic ballistic flight with air resistance, gliding flight, lookup from a table that doesn't follow any physics at all, or what. I would just know that clicking in one place gets success and clicking in another place gets failure.
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12:46 PM on 06/12/2012
Angry Birds is just simple Newtonian physics so I get frustrated when people complain that they don't know physics yet the get high scores on AB. AB-Space is a lot more interesting (to me, anyway) because gravity is no longer constant.
12:04 PM on 06/12/2012
Whenever people talk about the math of AB it's always about the simple ballistic motion, never the more interesting physics of how the structure responds when hit.
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Tim Chartier
05:33 PM on 06/13/2012
Do you have content that describes this? I've always been interested in how the structures fall and when it decides that things are static. Hitting a structure often involves anticipating where you see its weakest point being. Is this the way you see it or in a much different way?
06:23 PM on 06/14/2012
If you mean how AB does it I don't have any specific information (other than that they use "Box2D"), but out in the wild the subject is called rigid body dynamics, or rigid body physics. It's like you say. You have to think of where an impact will do the most to destabilize the structure.