Java Applets: Tools for Learning/Teaching Mathemtics: As mathematics teachers we are always looking for ways to better illustrate complex mathe matical concepts. Visual presentations encourage experimentation and discovery learning, allowing students to construct their own knowledge. But these are laborious and time consuming when done by hand on a chalkboard or even when done on a computer that is not equipped with the right software. It is a given now, that the proper use of technology should be an integral part of every mathematics course, but mathematics teachers are always worried about the time "wasted" teaching the technology itsel f. You often hear things like: "We are supposed to be teaching mathematics not Maple, Matlab or Mathematica." Are such software packages appropriate for use in undergraduate mathematics or do they represent an over kill? A powerful, machine independent and easy to learn/use software is clearly needed.
Recently, such a software, Java, was introduced and has been spreading like a wild fire. Java and the Web are providing teachers with the tools to create interactive hands-on demonstrations that lead to deeper understanding of difficult mathematical concepts. They are increasing the chances of success in mathematics for students with learning disabilities. Students are able to access these interactive demonstrations anytime, anywhere using any type of platform.
Properly designed java appletts, allow the student/teacher to expolre many complicated mathematical concepts without spending a lot of time learning a particular software package. All one needs to do is push buttons, drag points or slide a bar, while concentrating on what is important, mathematics, not the technical code required to do the mathematics. In terms of their educational function applets fall into the following categories: Simulations (virtual experiments), information retrieval, and self-testing (interactive quizzes).
Traditionally, instruction using the Web conjured up the image of a book in the vertical plane that has no pedagogical advantage. On the other hand, Java applets are a very powerful and dynamic medium of instruction, they allow for interactive exploration of topics in mathematics. The fact that these applets are interactive, promote discovery learning and can be accessed anytime, anywhere provided a connection the internet give them great pedagogical advantages over other modes of electronic instruction.
To demonstate the power of java, the participants will visit a number of excellent websites dedicated to mathematical java applets and investi
gate applets dealing
with a variety of mathemtical concepts.
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What is Java: Java is an object oriented programming language that is architecturally neutral.
What is an Applet?
A Java program is executed as either an application which runs stand-alone from the command line or a
s an applet that runs under a Web browser.
Applications and applets start program execution in different ways; an application has a main() method and an applet has a init() method inititiated by the browser.
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What do you need to run Java?
From http://java.sun.com/ you need to download and install JavaTM 2 Platform, Standard Edition (J2SE 1.4) released in March 2002.
The essential Java 2 SDK, tools, runtimes, and APIs for developers
writing, deploying, and running applets and applications in the Java programming
language. Here you will find a compiler javac, a debugger and an applet viewer.
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How do you ex ecute a Java Applet?
To insert an applet in a web page you use the applet tag. A
t the very least you need to tell the browser where the code for the applet is, and how
much space it will occupy on the web page. As an example,
< applet CODE=" buff.class " WIDTH="160 " HEIGHT="100" ></applet>
will insert the applet whose initial file is called buff.class in a rectangle of width
100 pixels and height 100 pixels into the web page.
To insert remote applets you use the URL. The following for instance
will insert an applet called buff.class that is at the site http://www.angelfire.com/wa/hurben/images/.
< applet CODE=" buff.class&q
uot codebase=" http://www.angelfire.com/wa/hurben/images" WIDTH="160 " HEIGHT="100" ></applet>
Excellent Math Java Applets Sites:
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Some of My Java Applets:
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Interactive Web Pages Using Java Sketchpad:
Good news! You don't have to learn java to make interactive Web pages, you can use Java Sketchpad.
All you need are a copy of Geometer's Sketchpad and a free
copy of the Java Sketchpad applet and Converter from Key
Curriculum Press.
How Does it Work?
The following examples were done using Java Sketchpad.
import java.awt.*;
import java.applet.*;
import java.awt.event.*;
public class My6JavaApplet extends Applet implements ActionListener
{
Button button1 = new Button();
Button button2 = new Button();
Label label1 = new Label();
Button button3 = new Button();
Button button4 = new Button();
Label label2 = new Label();
int
vxmin,vymin,vxmax,vymax; //screen coordinates
double xmin,ymin,xmax,ymax; //imaginary coordinates
double dx; //interval of x
double a,b; // parameter
double da,db; //interval of a
/* initializing */
public void init(){
String str,str1,str2;
xmin=-5;xmax=5;ymin=-5;ymax=5;
vxmin=0;vxmax=300;vymin=0;vymax=300;
dx=0.1;
a=1;da=0.2;b=1;db=0.5;
setLayout(null); //free layout
setSize(400,300); //
size of this applet
add(button1);
button1.setLabel("+");
button1.setFont(new Font("Dialog", Font.PLAIN, 16));
button1.setBounds(320,40,20,20);
button1.addActionListener(this);
add(button2);
button2.setLabel("-");
button2.setFont(new Font("Dialog", Font.PLAIN, 16));
button2.setBounds(350,40,20,20);
button2.addActionListener(this);
add(label1);
str1="a = "+Double.toString(a);
label1.setText(str1);
label1.setFont(new Font("Dialog", Font.PLAIN, 16))
;
label1.setBounds(320,60,60,40);
add(button3);
button3.setLabel("+");
button3.setFont(new Font("Dialog", Font.PLAIN, 16));
button3.setBounds(320,120,20,20);
button3.addActionListener(this);
add(button4);
button4.setLabel("-");
button4.setFont(new Font("Dialog", Font.PLAIN, 16));
button4.setBounds(350,120,20,20);
button4.addActionListener(this);
add(label2);
str2="b = "+Double.toString(b);
label2.setText(str2);
l
abel2.setFont(new Font("Dialog", Font.PLAIN, 16));
label2.setBounds(320,140,60,40);
}
public void paint(Graphics g)
{
double x1,y1,x2,y2;
String str;
//axes and scale
scale(g);
//graph
g.setColor(Color.blue);
x2=xmin;y2=func(x2);
for(x1=xmin;x1<=xmax;x1=x1+dx){
y1=func(x1);
Line(g,x1,y1,x2,y2);
x2=x1;y2=y1;
}
g.drawString("y=a(x-b)(x-b)",320,200);
Font f =
new Font ("SansSerif",Font.PLAIN,10);
g.setFont(f);
str="y = "+Double.toString(a)+"(x-" +Double.toString(b)+ ")" +"(x-"+ Double.toString(b) +")";
g.drawString(str,305,220);
}
/* Convert imaginary coordinates to screen cordinates. */
public int mapX(double x){
int sx;
sx=vxmin+(int)((x-xmin)/(xmax-xmin)*(double)(vxmax-vxmin)) ;
return sx;
}
public int mapY(double y){
int sy;
sy=vymin+(int)((ymax-y)/(ymax-ymin)*(double)(vymax-vymin));
retu
rn sy;
}
/* Drawing Line */
public void Line(Graphics g,double x1,double y1,double x2,double y2){
g.drawLine(mapX(x1),mapY(y1),mapX(x2),mapY(y2));
}
/* Defining function */
public double func(double x){
double y;
y=a*(x-b)*(x-b);
return y;
}
/* axes and lattice */
public void scale(Graphics g){
double x,y;
//back color
g.setColor(new Color(255, 255, 192));
g.fillRect(vxmin,vymin,vxmax-vxmin,vymax-vymin);
// scale
g.setColor(Color.lightGray);
for (x=xmin;x<=xmax;x=x+1){
Line(g,x,ymin,x,ymax);
}
for (y=ymin;y<=ymax;y=y+1){
Line(g,xmin,y,xmax,y);
}
// xy-axes
g.setColor(Color.black);
Line(g,0,ymin,0,ymax);
Line(g,xmin,0,xmax,0);
}
public void actionPerformed(ActionEvent event)
{
String str1,str2;
Object object = event.getSource();
if (object == button1){
a=a+da;
a=(double)(Math.
round(10*a))/10;
repaint();
str1="a = "+Double.toString(a);
label1.setText(str1);
}
if (object == button2){
a=a-da;
a=(double)(Math.round(10*a))/10;
repaint();
str1="a = "+Double.toString(a);
label1.setText(str1);
}
if (object == button3){
b=b+db;
b=(double)(Math.round(10*b))/10;
repaint();
str2="b = "+Double.toString(b);
label2.setText(str2);
}
if (object == button4){
b=b-db;
b=(double)(Math.round(10*b))/10;
repaint();
str2="b = "+Double.toString(b);
label2.setText(str2);
}
}
}
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import java.awt.*;
import java.applet.*;
import java.awt.event.*;
public class My8JavaApplet extends Applet implements ActionListener
{
Button button1 = new Button(); // Plot Button
Button button2 = new Button(); // Init Button
Button button3 = new Button();
Button button4 = new Button();
Button button5 = new Button();
Button button6 = new Button();
Label label1 = new Label();
Label label2 = new Label();
Image buffer; // image buffer
Graphics myGC; // graphics
for the buffer
int vxmin,vymin,vxmax,vymax; //screen coordinates
double xmin,ymin,xmax,ymax; //imaginary coordinates
double dx; //interval of x
double angle; //angle
double d_angle; //interval of angle
double cx,cy,cr; //center and radius of circle
double a,b; // parameter
double da,db; //interval of a
/* initializing */
public void init(){
Stri
ng str,str1,str2;
xmin=-3;xmax=7;ymin=-1.5;ymax=1.5;
vxmin=0;vxmax=500;vymin=0;vymax=150;
dx=Math.PI/60;
angle=0;d_angle=Math.PI/60;
cx=-1.5;cy=0;cr=1;
a=1;da=0.2;b=1;db=0.2;
setLayout(null); //free layout
setSize(500,150); //size of this applet
add(button1);
button1.setLabel("Plot");
button1.setFont(new Font("Dialog", Font.PLAIN, 12));
button1.setBounds(420,5,30,20);
button1.addActionListener(this);
add(button
2);
button2.setLabel("Init");
button2.setFont(new Font("Dialog", Font.PLAIN, 12));
button2.setBounds(460,5,30,20);
button2.addActionListener(this);
add(button3);
button3.setLabel("+");
button3.setFont(new Font("Dialog", Font.PLAIN, 12));
button3.setBounds(430,40,20,10);
button3.addActionListener(this);
add(button4);
button4.setLabel("-");
button4.setFont(new Font("Dialog", Font.PLAIN, 12));
button4.setBounds(460,40,20,10);
button4.addActionListener(this);
add(l
abel1);
str1="a = "+Double.toString(a);
label1.setText(str1);
label1.setFont(new Font("Dialog", Font.PLAIN, 12));
label1.setBounds(430,60,40,10);
add(button5);
button5.setLabel("+");
button5.setFont(new Font("Dialog", Font.PLAIN, 12));
button5.setBounds(430,100,20,10);
button5.addActionListener(this);
add(button6);
button6.setLabel("-");
button6.setFont(new Font("Dialog", Font.PLAIN, 12));
button6.setBounds(460,100,20,10);
button6.addActionListener(this);
add(label2);
str2="b = "+Double.toString(b);
label2.setText(str2);
label2.setFont(new Font("Dialog", Font.PLAIN, 12));
label2.setBounds(430,120,40,10);
// create off-screen image
buffer = createImage(vxmax-vxmin,vymax-vymin);
// get graphics
myGC = buffer.getGraphics();
setLayout(null);
setSize(500,150);
}
public void paint(Graphics g){
String str;
drawGraph(); //draw graph to the buffer.
g.drawImage(buffe
r,0,0,this); //copy buffer to the screen
g.setColor(Color.blue);
g.drawString("y=sin(x)",300,20);
g.setColor(Color.red);
str="y = "+Double.toString(b)+"sin(" +Double.toString(a)+ " x)";
g.drawString(str,300,140);
}
public void drawGraph() {
double x1,y1,x2,y2,y3,y4;
//axes and scale
scale(myGC);
//unit circle
myGC.setColor(Color.black);
Line(myGC,cx,ymin,cx,ymax);
Circle(myGC,cx,cy,1);
x1=cx;y
1=cy;
x2=cx+cr*Math.cos(angle);y2=cy+cr*Math.sin(angle);
Line(myGC,x1,y1,x2,y2);
myGC.setColor(Color.gray);
Line(myGC,x2,y2,angle,Math.sin(angle));
Line(myGC,x2,y2,angle,Math.sin(2*angle));
//graph1
myGC.setColor(Color.blue);
x2=0;y2=Math.sin(x2);
for(x1=0;x1<=angle;x1=x1+dx){
y1=Math.sin(x1);
Line(myGC,x1,y1,x2,y2);
x2=x1;y2=y1;
}
//graph2
myGC.setColor(Color.red);
x2=0; y3=b*Math.sin(a*x2);
int i;
for(x1=0;x1<=angle;x1=x1+
dx){
y4=b*Math.sin(a*x1);
Line(myGC,x1,y4,x2,y3);
x2=x1;y2=y1;y3=y4;
}
}
/* Convert imaginary coordinates to screen cordinates. */
public int mapX(double x){
int sx;
sx= vxmin+(int)((x-xmin)/(xmax-xmin)*(double)(vxmax-vxmin)) ;
return sx;
}
public int mapY(double y){
int sy;
sy= vymin+(int)((ymax-y)/(ymax-ymin)*(double)(vymax-vymin));
return sy;
}
/* Drawing Line */
public void Line(Graphics g,double x1,double y1,double x2,double y2){
g.drawLine(mapX(x1),mapY(y1),mapX(x2),mapY(y2));
}
/* Drawing Circle */
public void Circle(Graphics g,double x,double y,double r){
g.drawOval(mapX(x-r),mapY(y+r),mapX(2*r)-mapX(0),mapY(-2*r)-mapY(0));
}
/* axes and lattice */
public void scale(Graphics g){
double x,y;
String str;
//back color
g.setColor(new Color(255, 255, 192));
g.fillRect(vxmin,vymin,vxmax-vxmin,vymax-vymin);
g.setColor(Color.lightGray);
for (x=Math.PI/2;x<=Math.PI*2;x=x
+Math.PI/2){
Line(g,x,-0.05,x,0.05);
}
for (y=-1;y<=1;y=y+1){
Line(g,-0.05,y,0.05,y);
}
// xy-axes
g.setColor(Color.black);
Line(g,0,ymin,0,ymax);
Line(g,xmin,0,xmax,0);
}
public void actionPerformed(ActionEvent event)
{
String str,str1,str2;
Object object = event.getSource();
Graphics g=getGraphics();
if (object == button1){
angle=0;
//plotting animation
while(angle<2*Math.PI){
angle=angle+d_angle;
paint(g);
}
}
if (object == button2){
angle=0;
repaint();
}
if (object == button3){
a=a+da;
a=(double)(Math.round(10*a))/10;
repaint();
str1="a = "+Double.toString(a);
label1.setText(str1);
}
if (object == button4){
a=a-da;
a=(double)(Math.round(10*a))/10;
repaint();
str1="a = "+Double.toString(a);
label1.setText(str1);
}
if (object == button5){
b=b+db;
b=(double)(Math.
round(10*b))/10;
repaint();
str2="b = "+Double.toString(b);
label2.setText(str2);
}
if (object == button6){
b=b-db;
b=(double)(Math.round(10*b))/10;
repaint();
str2="b = "+Double.toString(b);
label2.setText(str2);
}
}
}