Cactus: the centres of a triangle.
Hyde, Hartley
This is the first of two articles which describe how to use
JavaSketchPad to explore the centres of a triangle. This introductory
exercise is suggested in the GSP Workshop Guide.
Your students can use JavaSketchPad Interactive Geometry (JSP) at
home at no cost. They are likely to impress their parents with their
enthusiasm for geometry and all they need on their computer is a Java
enabled web browser and a suitable editor. They do not need to be
on-line to use JavaSketchPad
The start.html code shown, here in red, is ordinary html code which
generates the web page shown on the next page. It describes a simple 1?3
table. The top cell displays a heading and the bottom cell tells the
user how to drive the contents of the second cell.
The code for the second cell is shown in black text and consists of
a Java applet that directs the constructing of the geometric figure that
appears in the middle cell. You can explore this and other models at
http://users.on.net/~hhyde/cactus/jskpd. If you are using Windows you
may need to enable Java by visiting www.java.com.
You can download the JSP Applet from
http://www.dynamicgeometry.com/ JavaSketchpad/Download_Center.html as a
zip file. The unpacked JSP folder can be distributed to students who do
not have the internet. Unless you are a keen Java programmer you may
decide to regard the contents of the JSP folder as a "black
box"--it interprets any instructions from the line <applet
code="GSP.class" codebase="jsp" down to
</applet> and builds the geometric model that the numbered
instructions describe.
Users must place the JSP folder inside the project folder from
which they will launch their html files. The code is typed into a
suitable editor and saved into the project folder as start.html. To
avoid typing errors, it may be easier to copy the start.html code from
my user.on.net website and distribute it to students. The only items of
mathematical significance are the numbered instructions.
start.html
<head>
<title>Start</title>
</head>
<body>
<table width=640 border="0" bgcolor="#D0D0D0">
<tr>
<td>
<p> </p>
<h1 align="center">Start</h1>
<p> </p>
</td>
</tr>
<tr>
<td align="center">
<applet code="GSP.class" codebase="jsp"
archive="JSP4.jar"
width="600" height="480"
align=center>
<param name=BackRed value=240>
<param name=BackGreen value=240>
<param name=BackBlue value=240>
<param name=LabelSize value=24>
<param name="Construction" value="
{1} Point(470,150) [black,label ('A (drag)')];
{2} Point(350,450) [black,label ('B')];
{3} Point(100,60) [black,label ('C')];
{4} Polygon(1,2,3) [white];
{5} Segment(1,2) [thick,black];
{6} Segment(2,3) [thick,black];
{7} Segment(3,1) [thick,black];
">
Sorry, this page requires a Java-compatible web
browser.
</applet>
</td></tr>
<tr>
<td><p> </p><blockquote>
<p>The three points A, B and C define a
triangle.</p>
<p>The size and shape can be changed
by dragging any of the points A, B or
C.</p>
<p> <</p>
</blockquote>
</tr>
</table>
</body>
</html>
Most modern wordprocessors are unsuitable to use as editors because
they attempt to open an html file as a web page instead of code.
Sometimes it is possible to copy the source file from a browser window
and paste it into a word-processor. After editing, the code can then be
saved as a text file .txt and then manually changed from .txt to .htm.
This is very messy and time consuming. A dedicated text editor is far
easier to use.
Ideally we need a screen large enough to place the browser and the
editor windows next to each other. A degree of overlap still works well
because we have to click in the window we need to activate anyway.
If your students use Internet Explorer from a Windows platform they
can view source code by choosing View->Source. This opens the html
file using Notepad and the code can be modified and saved as an html
file. Each time they edit the source file (in this case start.html) they
can save the file, click to activate the browser window and then click
the refresh icon at the top of the browser screen. The browser will then
read and interpret the updated file.
On old Macintosh computers SimpleText could be used in much the
same way as Notepad. After exploring a new Macintosh with the usual
selection of iWorks and iLife applications I couldn't find any
software that can be used as a simple html editor. Even the new version
of TextEdit opens html files as a webpage instead of source code. It is
possible to drop html fragments into iWeb but it crashes if the code
includes Java. Safari users can download BBEdit, which is quite
expensive. Alternatively Macintosh users can install the FireFox browser
and use Nandu as the editor. Both of these downloads are free.
When your students open the start.html file using a Java enabled
browser such as Internet Explorer, Safari or Firefox they should find
that the page looks like this:
It is possible to manipulate this figure in the same way that we
use GSP or Cabri. However, to add to the figure, instead of reaching for
a convenient tool icon, we need to write more numbered instruction
lines. That is why a simple editor is so important.
I gave my class a JSP folder, a start.html file and a printout of
the JavaSketchPad Construction Grammar which I have saved at http:
//user s .on .net/~hhyde/cactus/jskpd as a .pdf. After a few nights of
exploration students were able to build quite sophisticated dynamic
geometry models.
[ILLUSTRATION OMITTED]
Our class studied the instructions in the start.html code. Items
{1}, {2} and {3} define three points based on a coordinate system that
counts from the top left-hand corner of the JSP screen. We changed the
coordinates of these points and observed what happened.
Instruction {4} defines a white triangle that identifies the
original triangle more clearly after many construction lines have been
added. Instruction {5} Segment(1,2) [thick,black]; describes a thick,
black line segment (AB) that joins the points defined by instructions
{1} and {2}. Instructions {6} and {7} are similar.
I sent my students home to add instructions to find the centroid of
the triangle.
The students had no trouble defining the midpoints D,E and F and
joining these to the opposite vertices. Exploring the different types of
intercepts can be tricky when curves are involved, but it isn't
hard to find the intercept of two medians to find the centroid (G).
Instructions {15}, {16} and {17} identify the segments DG, EG and
FG so that the ratio of these lengths to the corresponding medians can
be calculated. This allows the instruction {18} Ratio/Segments
(11,15,50,350,'Ratio AD/GD = ') to calculate the ratio of the
length AD, defined by {11}, to the length GD, defined by {15} and give
the answer at coordinates (50, 350).
[ILLUSTRATION OMITTED]
[ILLUSTRATION OMITTED]
If students get confused, it is a good idea to give each construct
a different colour to debug the code. When the centroid.html code is
understood, it is easy to modify the instructions to construct a
circumcentre.
The code is almost identical except that perpendiculars are
constructed at the midpoints instead of medians. Constructing the
circumcircle {15} provides a good check on the accuracy of the model.
Students adapt to this system quickly and they soon discover
features of the instruction grammar that enhance their models. The html
files that they create can be added to their private web pages. Nothing
drives student achievement quite so effectively as an opportunity to
show their work to the world.
JSP provides a simple macro level introduction to programming that
encourages further exploration. Students find the task enjoyable and
happily take files home to keep working--possibly using a different
platform.
Some of you are worried about the cost and availability of software
for any computers which may be supplied following the election promises.
This free dynamic geometry software will run on any Java enabled
browser, running under any operating system.
Hartley Hyde
cactus.pages@internode.on.net
The centres of a triangle