angry birds mathematics - parabolas vectors

8/10/2019 Angry Birds Mathematics - Parabolas Vectors

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334 MATHEMATICS TEACHER |Vol. 107, No. 5 December 2013/January 2014

Sometimes the simplest concept can fascinate completely. Whocould have guessed that launching a red bird toward a towerof blocks inhabited by egg-stealing green pigs would become aspopular a video game as it has? Toddlers to 90-year-olds have

become enamored of this simple game rooted in the mathemati-cal concepts of projectile motion. From the moment I first launched a

tiny, angry red bird toward those thieving pigs, I became intrigued by thethought of using this game to motivate mathematics students, their teach-ers, and their future teachers. As a professor of mathematics educationand a teacher of high school precalculus, I have developed a way to usethe elements of the game Angry Birds as a platform to engage my stu-dents with the concepts of parabolas and vectors.

Vector properties and the birds frictionless

environment help students understand the

mathematics behind the game.

Parabolasa

ndMathem

John H. Lamb

Copyright 2013 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.This material may not be copied or distributed electronically or in any other format without written permission from NCTM.


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Vectors

SCREEN:COURTESYO

FROVIO;TABLET:THINKSTOCK

tics


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opposite force to stop the flight of the bird. Eachelement of the game has a certain mass and is actedon by the constant downward, or negative, verticalforce of gravity. Gravity pulls the bird downwardand causes the pigs and towers to crumble, depend-ing on their masses. The games programmers did

not add horizontal air resistance, which wouldaffect the flight of the bird; therefore, the horizontalvelocity of flight remains constant, vertical velocityis governed by gravity, and the resulting trajectoryof the bird becomes a parabola.

Parabolas and quadratic functions appear in cur-ricula from upper middle school through calculus.The remainder of this article describes how I haveused Angry Birds to engage all my students andfocuses on how high school precalculus students

were challenged to explore Angry Parabolas, AngryVectors, and the partnership of these mathematicalconcepts within Angry Projectile Motion.

ANGRY PARABOLASWhen exploring the Angry Parabola, I begin by

launching a bird and asking a simple question suchas, What mathematics do you see? A birds flight

path is revealed by the white dots that appear fol-lowing a launch (see fig. 1). As students share theirobservations, I urge them to discuss the force, angles,and, more specifically, the parabolic trajectory.

The class discussion then moves toward deter-mining the equation of the trajectory of the bird.Because the game is played in two-dimensionalspace, a vertical and horizontal coordinate system

must be defined before doing so. If we use the vertexform for a quadratic function,f(x) = a(x h)2+ k,determining the vertex and another point on the par-abolic trajectory will allow us to find a value for a.

At this time, figure 1remains posted whereit is visible to the entire class, and students markoff their coordinates before providing the angrycoordinate system shown in figure 2. I want stu-dents to understand the importance of establishinga standard unit of measurement in both dimensions,imperative for any graphical representation of two-

dimensional functions. Students begin by suggestingAmerican standard units of measure such as inches.

However, given the size of the screen (mobiledevice, tablet computer, or projection on the white-

board), they soon realize that other units of measureare preferable. Students then move to ideas such asusing the slingshot height as the unit of measure. Ithen suggest using the diameter of the Angry Bird asthe standard unit of measure.

Once a coordinate grid has been established,

students approximate the location of the vertexthrough visual inspection, using the horizontaland vertical distances from the top of the sling-shot. Two additional points can be determined by

The desire to use computers and computergaming in mathematics education dates to work

like that of Feurzeig and Papert (1968, 2011).They developed the computer programing lan-guage LOGO specifically for education, resultingin students learning, through a game, to programan on-screen turtle to complete a maze (Papert1980). These early computer gaming ideas haveevolved, incorporating better graphics and plac-

ing a greater emphasis on specific mathematicalconcepts. Kebritchi, Hirumi, and Bai (2010) foundthat high school students who played an algebra-infused computer game significantly outperformedstudents who did not. The game-playing studentswere found to score significantly better on bothschool benchmark tests and assessments specific tothe content of the game. Although research is notconsistently positive, much of it supports the con-

jecture that students learn mathematics from com-

puterized games (Egenfeldt-Nielsen 2007).

Egenfeldt-Nielsen (2006) described one type ofcomputerized game as one that simulates a part ofthe world that is simplified and constructed to facili-tate working with concrete objects. When interact-ing with objects in microworlds, we are learningabout the objects properties, connections, and appli-

cations (p. 198). The game Angry Birds couldbe categorized as this type of microworld

game in which students interact with the

properties, connections, and applicationsof projectile motion. The vector proper-ties of the slingshot and the frictionless

environment in which the bird travelscoupled with the entertaining story line,exceptional graphics, and catchy sound

effects provide a motivational resource forteachers to guide students toward understand-

ing the mathematics behind the game.Projectile motion is governed by Newtons three

laws of motion:

1. Objects stay at rest or move with a constantvelocity unless acted on by some force.

2. The acceleration of a body is directly propor-tional to, and in the same direction as, the net

force acting on the body and is inversely propor-tional to its mass. Thus,F= ma, whereFis thenet force acting on the object, mis the mass ofthe object, and ais the acceleration of the object.

3. When one body exerts a force on a second body,the second body simultaneously exerts a forceequal in magnitude and opposite in direction tothat of the first body.

In Angry Birds, the slingshot exerts a force ontothe bird at a certain angle that sends the bird intomotion until the tower of pigs exerts an equal and


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Vol. 107, No. 5 December 2013/January 2014 |MATHEMATICS TEACHER 337

approximating the coordinates of a launched birdin midflight or by using the starting position at thetop of the slingshot, which would be at the origin.

Obviously, a certain amount of guesswork is asso-ciated with this activity, and human error playsa role. Because the main idea is for students toexplore the concepts of quadratic functions withinan engaging connection to this popular game, we

accept the human error that necessarily occurs.I require students to approximate the coordi-

nates of the vertex point to the nearest tenth of anangry unit. Using the coordinates (20.2, 4.8) forthe vertex and (0, 0) for the point on the curve,

we determine the quadratic function for the AngryParabola in figure 2. We first substitute the origininto the vertex form of the quadratic function toobtain 0 = a(0 20.2)2+ 4.8 and then find that

a 0.0118. Our equation then becomesf(x) =

0.0118(x 20.2)2+ 4.8, which can be converted tostandard form:f(x) = 0.0118x2+ 0.47672x.

Formative and summative assessments areimplemented with my students by providing imagesof Angry Bird trajectories superimposed on theangry coordinate plane (see the appendix) forstudents to calculate the parabola, its vertex point,zeros, and other points on the modeled pathway.With my precalculus students, I use this activityearly in the course to help students review theirknowledge of quadratic functions and how to deter-mine functions through modeling.

ANGRY VECTORSVectors are defined by magnitude and direction.

Magnitude can be expressed by distance, speed,or translation. The direction of a vector is oftendenoted as its angular bearing from a constantdirection, such as an angle of elevation. The AngryParabola is a result of two factors: the slingshotforce and its angle. The combination of these two

factors creates the Angry Vector.I provide students with figure 3and discuss all

the Angry Vectors that can be created in the game.The magnitude of the Angry Vector can be defined

by the length of the slingshot band as it is stretched,illustrated in figure 3by the circles ranging from

one-fourth to full power. The direction of the AngryVector is determined by the angle of rotation of the

bird about the top of the slingshot. Using figure 3,my class discusses the Angry Vectors created in all

four quadrants as well as the probable trajectoriesfollowing a bird launched with those vectors.

Each location of an Angry Bird, as it rotates aboutthe slingshot, is the initial point for an Angry Vector.The coordinates of this point can be determined ineither a rectangular or a polar coordinate system. Iguide my class, using figure 3, through an explora-tion of rectangular and polar coordinates of variousinitial points. The rectangular coordinates can bedefined by the Angry Bird units in the horizontaland vertical distances from the origin, whereas thepolar coordinates can be defined by the radius, or

force from the initial point to the origin, and the

angle of rotation. The polar coordinates then lendthemselves to defining the Angry Vector for eachtrajectory. Because the pig target in the game is tothe right of the Angry Birds, I have my class focus oninitial points in quadrant III, which will provide themost successful launches (see fig. 4, p. 339).

My class then completes a table of polar andrectangular coordinates for these initial quadrant IIIpoints according to their locations when the slingshot

is at full power, three-fourths power, one-half power,and one-fourth power (see table 1). Students usetheir knowledge of the unit circle or their knowledge

Fig. 1 The parabolic trajectory of the Angry Bird is high-

lighted during the launch.

Fig. 2 The diameter of an Angry Bird is the unit of measure

on both axes.

Fig. 3 Circular trigonometry can be used to determine the

prelaunch position of the Angry Bird.


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Vol. 107, No. 5 December 2013/January 2014 |MATHEMATICS TEACHER 339

of polar equations to complete the table. Once theclass has completed the table, discussion related tothe magnitude and direction of the Angry Vectors isused to urge students toward the final element of theAngry Bird mathematics: Angry Projectile Motion.

ANGRY PROJECTILE MOTION

The magnitude of the Angry Vector within thekinematic components of projectile motion is theinitial velocity of the Angry Bird, Va, resulting fromthe slingshots force. The direction of the AngryVector is determined by the standard positionangle, q(see fig. 5). As discussed earlier, the AngryBird travels in a parabolic trajectory because theonly force programmed to act on the bird, after it is

launched, is that of the downward pull of gravity.The Angry Birds position (the ordered pair (x,y))

after a given amount of time is governed by twokinematic equations:

x x V t

y y V t gt

x

y

1

2

0 0

0 0

2

= +

= +

In these equations, V0xand V0yare the initialvelocities in the horizontal and vertical directions,

respectively; gis the downward force of gravity(9.81 m/s2); tis time; andx0andy0are the initialrectangular coordinates of the object being pro-

jected. Class discussion then moves toward thediscovery of the Angry Vectors initial velocity, Va,and direction, q(see fig. 5).

Right-triangle trigonometry yields a relationship

between the initial velocity of the Angry Vectorand the measurements of V0x, V0yand q:

V

V

V

V

V

Va a

y x y

x

sin ; cos ; and tan .0 0 0

0

= = =

The class can then use the standard quadraticequation form ofy= ax2+bx+ cto solve for Vaand

q. The following verifies what a, b,and c equal in

terms of V0x, V0y, andg:

x x V t tx x

Vxx

0 0

0

0

= + =

We use this value for t and substitute as follows:

y y V t gt

y Vx x

Vg

x x

V

yV

Vx

V

Vx

g x x x x

V

yV

Vx

V

Vx

g

V

xgx

V

xgx

V

g

V

xV

V

gx

V

x

V

Vx

gx

V

y

y

y

x x

y

x

y

x

x

y

x

y

x

x x x

x

y

xx

y

x

x

1

2

1

2

2

2

2 2

2

2

0 0

2

0 0

0

0

0

0

2

0

0

0

0

0

0

2

0 0

2

0

2

0

0

0

0

0

0

0

2

2 0

0

2

0

2

0

2

0

2

2 0

0

0

0

2

0

0

0

0

2

0

2 0

( )

( ) ( ) ( )

( ) ( )

( )

( )

= +

= +

= + +

= + +

=

+ +

+ +

Following this verification, the class can take anAngry Parabola in standard form and determineVaand q. We use the parabolay= 0.0118x

2+0.47672x (whose equation we determined earlier)andg = 9.81 angry units/s2, and begin by equat-ing the two expressions for a:

Fig. 4 Students find polar and rectangular coordinates of

the Angry Bird prelaunch positions.

Fig. 5 Angry Vectors are defined by the initial velocity and

angle of elevation of the launched Angry Bird.


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V

V

x

x

0.0118 9.81

2

9.81

2(0.0118) 20.3882

0

2

0

( ) =

=

We then do the same for our representations ofb:

V

V

V

y

x

y

0.47672

0.47672 20.3882 9.7195

0

0

0

=

=

Then tanq= V0y/V0xtanq= 0.47672 q25.4881. We have sinq= V0y/VaVa = V0y/sinqVa 9.6885/sin 25.1881 9.6885/.4303 22.5864.

Thus, the initial velocity of the Angry Bird isnearly 23 angry units per second at an angle ofslightly more than 25 degrees.

DISCUSSIONAll my students have become more engaged andintrinsically motivated to learn the mathematicalconcepts emphasized in these lessons using Angry

Parabolas, Angry Vectors, and Angry ProjectileMotion. Middle school students seemed to clamor tomeasure angles and distances so that they could geta chance to launch an Angry Bird toward a tower ofpigs. Algebra students, despite moaning that I haveruined their favorite game, were able to understand

how scientific elements of vectors directly con-nected to algebraic modeling and prediction through

parabolas. Teachers embraced the possibilities ofusing these concepts in their classrooms, and many

contacted me with their adaptations.Teachers of mathematics have an important

role in developing student thinking. The conceptstaught throughout mathematical coursework help

build critical thinkers who will become moreproductive employees and leaders in their futurecareers. Too often, students fail to see a connection

between the mathematics that they are learning andits application in the real world. Using Angry Birdscan help students explore mathematical concepts in

ways that have direct appeal.

BIBLIOGRAPHYEgenfeldt-Nielsen, Simon. 2006. Overview of

Research on the Educational Use of Video Games.

Digital Kompentanse1 (3): 184213.

. 2007. Third-Generation Educational Use of

Computer Games.Journal of Educational Multi-

media and Hypermedia16 (3): 26381.

Feurzeig, Wallace, and Seymour A. Papert. 1968,

2011. Programming-Languages as a Conceptual

Framework for Teaching Mathematics. Preface

by Bob Lawler.Interactive Learning Environments

19 (5): 487501. doi:http://dx.doi.org/10.1080/

10494820903520040

Kebritchi, Mansureh, Atsusi Hirumi, and Haiyan Bai.

2010. The Effects of Modern Mathematics Com-

puter Games on Mathematics Achievement and Class

Motivation. Computers & Education 55 (2): 42743.

doi:http://dx.doi.org/10.1016/j.compedu.2010.02.007

Kim, Sunha, and Mido Chang. 2010. Computer

Games for the Math Achievement of Diverse Stu-

dents.Educational Technology and Society13 (3):

22432.

Papert, Seymour. 1980.Mindstorms: Children, Comput-

ers, and Powerful Ideas. New York: Basic Books.

JOHN H. LAMB,[email protected], is an

associate professor of mathematics edu-

cation at the University of Texas at Tyler.

He teaches elementary and secondary

school mathematics methods at the university as

well as precalculus at an area high school andresearches the testing culture, integrated math-

ematics education, and rural education.

APPENDIXBelow are two images suitable forstudent assessment. Foradditional similar images, go to

www.nctm.org/mt046.

angry birds mathematics - parabolas vectors

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