Interactive Mathematics on the Internet
In recent years, an
increasing number of web sites have offered significant opportunities for
student to learn online, often through the use of Java applets. You can here
explore some good examples of this genre of material that are publicly
available, to consider their potential for mathematics learning. Note that your browser will need to be Javaenabled for many
activities, while many others will use Flash; free downloads for these are
available. Websites will generally offer advice on such technical matters.
A thorough examination of the examples below will take many hours, and
so will not be possible in a workshop of an hour or two. You will need to be very selective here and return later for
a closer look.
If you are in a workshop in a computer laboratory, do not download anything onto the
computers unless you are sure that this is acceptable to the owners.
Workshop Instructions
1. Work with a partner,
either sharing a computer or working side by side on separate computers.
Discuss what you see.
2. Using the brief
description as a guide, choose a site below that is (relatively) unfamiliar to
you and matches one of your interests. The year levels suggested are
approximate.
3. Read the suggestions
before accessing the site, so that you know what to look for and what to do.
4. Once on the site,
explore a couple of examples, to get a sense of what might be offered to
students that is not usually accessible to them:
How
could these applets be used by your students: at school? At home?
How
could the applets be used by a teacher: for demonstration? on an interactive
whiteboard? for class discussion? for ideas?
5. Repeats steps 2 to 4 to
choose a range of sites, in order to see a range of possibilities and
educational styles.
Level 
Website 
Suggestions 
312 
National Library
of Virtual Manipulatives A large and very
impressive collection of applets, with substantial support provided for
teachers and parents. Linked to the NCTM
Standards. Can be purchased separately as a CDROM, avoiding the need for
Internet access. 
Click on Virtual Library to access a grid of content
by level. Choose an element of the grid and then choose an applet to explore.
(You can choose an age level or a content band instead if you prefer. Be
careful: the grade levels provided are not very reliable.) Icons at the top
of each applet offer suitable activities and instructions. Use the Back icon
to return. For example, try Platonic
Solids in geometry 912 to see a 3D version. 
58 
A large collection of
welldesigned applets, also keyed to the NCTM
Standards, mostly aimed at middle school level, but many of which are
also potentially valuable for older students too. Excellent support materials
for teachers are included. 
Click on the Activities
hexagon to access a complete list. Each Activity has a What? How? And Why?
Link to help, if needed. [A great deal of help is provided, but this workshop
doesn’t suit a close scrutiny of this today.] Choose a range to suit your
interests. Some of my favourites are Histogram,
Transmographer,
Area
Explorer and Fire!!

512 
This large UK site based
at Cambridge University contains many different kinds of resources for both
students and teachers, including interactive resources. You will need to have
a Flash player or Java in order to use most of these. Instructions for this
are given on the site itself via the Help link on the left. A good overview
of the site overall is available at the tour, which
deals with much more than just interactive elements. 
Interactive resources
appear in many places on this site, and can also be found by using a search
command. The site uses the term ‘interactivity’ to refer to interactive
resources. A nice example to try is the virtual geoboard,
which can be configured in various ways and elastic bands easily used. There
are several probability
examples, including some using the wonderful DIME recording sheet. A
substantial set of resources is available here
as a result of a search using interactivity. Choose some of these in your
areas of interest. 
510 
This small collection of
games from BBC TV’s Maths File has a variety focussing on various aspects of
Number, Algebra, Data and Measurement, focussed on the UK curriculum. Your
browser needs to have Shockwave installed. (There is a download link if
necessary). The games are nicely produced, with an undertone of British
humour (eg when a ‘prize’ is won.) Advice is offered for teachers, and some
activity sheets for students are provided too. 
Many of these games
would work quite well on an interactive whiteboard, although they could also
be used by individual students. Click Hypatia or Pythagoras to spin the
wheel. To get a feel for the style, try Builder
Ted, which is concerned with ordering decimals, or Bathroom
Tiles, which involves transformations in the plane, or Train
Race, which is concerned with using and interpreting means and medians. 
812 
Walter Fendt has
compiled a number of Java applets related to secondary school mathematics, in
several languages as well as his native German. The English language applets
offer interesting ways of interacting with mathematical ideas across a range
of topics, especially geometry topics, and are designed for direct use by
students. 
To explore this site,
choose some applets in an area of your interest. While the applets are a
little old now, they still offer useful opportunities for students and
demonstration opportunities for teachers. Note that the suite of applets can
be downloaded for use in your own classroom, so that they can be used when
Internet access is problematic. (Don’t do this if you are in a computer
laboratory, unless it is your own.) 
912 
David Joyce has created
this interactive online version of the most successful mathematics text in
history, including all 13 books. Worth including here, even if Euclid has
faded from many of our curricula! 
There’s too much here to
deal with in detail, of course. I suggest going to the Table of
Contents and looking at some of the Quick Trip, such as Book I
Proposition 47. Dragging coloured points allows you to see Euclid’s work come
alive. 
713 
This British site
contains a large number of welldesigned Java applets across a range of areas
of mathematics. These are arranged in two groups: for ages 1116 and ages
1619. Applets are designed for students to be able to use by themselves, so
that onscreen icons are useful. 
There are many small,
singlepurpose applets, which would be useful to use on an interactive
whiteboard for whole class teaching, while some are best suited to individual
use. Try Reflections,
Algebraic long division
or Positive and negative
numbers to see a range of interesting examples. Instructions are helpful
and easy to find through the icons, usually starting with the i
icon at the top. 
712 
An entry into the
encyclopaedic MathWorld site, powered by Mathematica.
Although much of this site assumes very significant mathematical background,
it is still worth a look. The Platonic solids are nicely represented, and can
be rotated various ways using the mouse. The Wolfram Demonstrations Project
(below) also uses Mathematica. 
After using the mouse to
have a good look at the Platonic solids, including some of the links to
individual solids down the page a bit, select Live 3D
Graphics from the left column to access the extraordinary collection of
mathematical objects, or use either the search engine or links at the left to
explore this entire site. Don’t spend too long here for now … the idea is
just to see what is now possible and available on the web. Much of what is on
this site is not interactive. 
612 
This is a website of the
Freudenthal Institute for secondary education (students of 12 to 18 years
old) in the Netherlands. The main focus of the site is a collection of Java
applets, developed and studied by FI staff. Some of these applets are in
English; many are very interesting and offer environments for pupils to
explore ideas. 
Select the applets link
on the opening page. Then show the applets after selecting Abstract to get
some instructions Most of these applets are small and come with minimal
instructions, yet some are likely to be quite engaging by students or by a
class, especially the 3D applets. For example, try Building houses, Impossible
object or CutOuts, Nets. Each applet has an About link at the top, which
gives (minimal) information. 
712 
This website contains a
very large collection of interactive activities created by the idiosyncratic
Alexander Bogomolny to highlight elegant and powerful features of
mathematics. Many aspects of secondary
school mathematics are involved as well as many mathematical elements that
are not part of the school curriculum. 
As CTK is a very
extensive collection of several hundred applets, no choices will be
‘typical’. Equations
of the straight line, Riemann
Sums and Polygonal
Numbers illustrate three different possibilities, but you will find many
others with other selections. To see the philosophy underpinning the
collection, read the author’s manifesto, which
is also accessible on each applet page. 
912 
This excellent Austrian
website offers many Java applets that have been thoughtfully constructed to
offer sound educational experiences for students learning mathematics, mostly
at the senior end of secondary school. The site contains a good collection,
together with other useful advice on using them and information about their
development. 
Start with the Gallery,
which houses the collection. Try Analytical
geometry 1, for example, which contains three different applets. For each
applet, make sure that you read the Exercises and the Didactical Background
to see the thinking employed in constructing them. You may then like to
explore some other examples or to try an interactive test, some of which
(like this one on derivatives)
are in the form of puzzles. 
912 
David Eck’s pioneering
work was freely available for many years, and has now been rewritten as a
Java applet, available for free download. Various aspects of graphing
functions are involved, including differential and integral calculus. (See
also the JCM link below to see how these ideas here were extended.) 
Go to the Launch xFunctions button a little way
down the page to get a sizable version. xFunctions
is a suite of programs, not just a graphing package. Instead of the Main Screen,
try the Animation Utility, Derivatives Utility and Riemann Sums Utility to
see the opportunities provided to students to explore the mathematical ideas
involved. Lots of detailed help with the applet is provided on the page, but
there won’t be time to explore this in depth in the workshop. 
912 
Loci was
previously The Journal of Online
Mathematics and its Applications, a major publication of the MAA over the
past few years and a fertile ground for thinking about and discussing the
role of interactive multimedia in learning mathematics. While a lot of the
material is focussed on undergraduate mathematics, there are still
interesting ideas here relevant to senior secondary school. Reviews or work
using multimedia and interesting conversations about work in progress are
located on the site. 
The MAA Mathematical
Sciences Digital Library (MathDL) offers a new environment for teachers and
authors to consider together the educational prospects for multimedia on the
web. The articles refer to educational use of the multimedia applets (mathlets)
and frequently contain good practical advice about using them, from teachers
and developers. A quick browse of the Resources will give you a sense
of the material. Free materials are available (as for JCM below), often
requiring very little (but some) HTML expertise to install. An example of
this process is shown here, using a Dice applet
referred to in a recent article.

912 
Java
Components for Mathematics This website reports the
work of the JCM project, an outgrowth of the work of David Eck, author of xFunctions. The project has produced
Java components that can be freely adapted for teaching purposes, and this
website offers a good deal of help in showing users how to do the adaptation.
With only a minimal knowledge of HTML, mathematics teachers can use these
components for their own purposes. 
This site provides a
good deal of practical help for beginners, although a little knowledge of
HTML will be needed in order to take advantage of it. In order to use the
components provided, you will also need to download the JAR file and store it
on your website or computer to be accessed by the applets created. (A big
advantage of this is that you can use the applets on a computer without web
access.) There is an example here of an elementary
use of the EpsilonDelta applet,
essentially comprising the text copied from the JCM website and pasted into a
text file constructed using a basic text editor. There is a small amount of
extra text (all of which can be seen by viewing the Source of the page.) 
912 
Dynamic Geometry Both Cabri Geometry and Geometer’s Sketchpad provide a
mechanism for users to easily create their own dynamic geometry Java applets.
The outstanding Geogebra software
is available online and can also be downloaded to your computer for free. 
There are some
rudimentary examples of applets from Cabri
and Geometer’s Sketchpad in use here, together
with links to the software manufacturers’ pages, explaining the process in
some more detail. Try perpendicular
bisectors, for example. The Geogebra site will need a lot of time to
explore adequately, but is well worth while. 
610 
The TLF is an agency of
various Australian governments, sponsoring the development of a large number
of multimedia and online earning objects for Australian schools. Many of the
objects are for younger pupils, but the brief extends from K10, so there are
examples for older pupils too. 
Explore the site to find
out what the TLF is doing in your state, as well as other information about
how it works. Use the Access
Information link at left to find out more local details: select Schools to do this. There are some
sample objects at the link provided here, but you can’t actually use the entire set of objects from
this site. The entire mathematics and numeracy catalogue is accessible as a
(large) download from the web. 
K12 
MathTools is a Digital
Library of maths resources compiled by the Math Forum to help teachers teach
and students learn in K12 mathematics. The site has two major features: a
catalog of software and a discussion forum. Above all else, MathTools is a
community of users that includes students, teachers, mathematicians,
researchers, and software developers. 
This site (for teachers,
not students) contains links to many other sites on this list, and also
contains discussions among users of the various kinds of software. There is a
good searchable index on the opening page, which allows you to find resources
that fit a particular teaching or learning need. It’s a good idea to register
in order to take advantage of the special features only available to
registered users. Registration is free. 
612+ 
Wolfram Demonstrations Project This site has been
constructed using Steven Wolfram’s extraordinary Mathematica software. It contains a very large (more than 2000)
and growing number of interactive demonstrations that use dynamic computation
to illuminate concepts in science, technology, mathematics, art, finance, and
a remarkable range of other fields. While most are very sophisticated, many
are useful at school levels. Free software Mathematica Player needs to be downloaded to use independently of
the web. 
The versions shown on
the web are just quick Flash
animations; to explore their use, first download the software (only onto your own computer and then download particular demonstrations.
The site has a good About description, good FAQ file, search facilities and a
useful Topics list. The related menus on the right of the screen offer many
links. Start exploring with Nets of Polyhedra
or Two Dice
with Histogram or Elementary
School Mathematics to see some of the range. Select a demonstration to
preview on the web and then use the related menus to follow your interests.
In most cases, the (small) downloaded versions provide many more
opportunities than the preview shows. 
913 
This website contains
a lot of descriptive text, which is mostly wellwritten in a large font, and a number of interactive
elements constructed by the author, Murray Bourne from Singapore. The interactive
elements use LiveMath, Flash or Scientific Notebook. The first two of these need a free plugin
to be downloaded and installed, while the third requires the actual software.
While the main purpose seems to be to provide good explanations and descriptions
for secondary students, which it achieves well, the website has many nice
uses of interactive applets. 
The Flash highlights link at the top of the page provide some nice
examples of applets and will give you a quick check to see if you already
have the Macromedia Flash plugin
installed. The Sitemap will give
you a good sense of the scope of the material, which may provide useful
revision for students, regardless of animations. Instructions for installing
plugins are useful (but only install
them on your own computer!). There are many LiveMath applets on the site, with a range of quality. Some of
these would be useful for demonstrations and class discussions. 
1012 
This nice Java applet
has been constructed by Lisa Murphy at UIUC and can be used in the study of
functions, line graphs, slope, the derivative, and basic motion concepts.
Using a mouse, the student drags a stick figure back and forth across the top
of the screen between its home and its school. Below, graphs of the motion
appear as the motion progresses. 
This is a nice applet
for students learning about distancetime graphs and the early steps in
understanding derivatives, and a good alternative to using a motion detector.
The instructions on the home page are useful, but you can go directly to the
applet here.
While individual students might use this, it seems also that it would be a
nice class discussion tool. Some teaching materials are also available on the
site, which contains some physics applets too, concerned with water flow and
electricity. 
1113 
Mathematical Visualization Toolkit This is a large
Javabased package for plotting, both 2D and 3D as well as solving
equations, developed at the Department of Applied Mathematics at the
University of Colorado. The entire package can be downloaded as well as run
on the Internet. The interface is easy to use and the suite of capabilities
very powerful. 
Explore the tools for
graphing, drawing vector slope fields, numerical integration and linear
algebra, as well as the various standalone applets under the applications
menu. While much of this suite is more suited to early undergraduate work,
especially in calculus and engineering, much of it is also very useful for
the senior secondary years as well. 
1113 
This is a large
collection of Java applets for the Advanced Placement calculus AB and BC
courses in the US, produced by Tom Downey using the excellent applets at Webcompmath.
The applets cover the entire range of the courses and hence of introductory
calculus in general. Each applet comes in a version shown on a web page,
suitable for viewing in a browser window, and also as a separate resizable
window with larger fonts and line widths, suitable for projection on a
screen. The entire set of individual applets can be downloaded for classroom
use, with appropriate attribution. A very rich resource for students or
teachers of calculus. 
The various applets are
welldesigned and reminiscent of David Eck’s superb applets of many years ago
(see the xFunctions link above).
The applets come with readymade examples for students or teachers to
explore, as well as helpful suggestions for exploring them. In addition, the
various applets can be used independently (eg using your own functions rather
than preset functions). Many of the applets rely on a graphing window. Try Derivative Function
to see the general idea and then explore topics of particular interest. 
1113 
This online textbook, Online Statistics: An Interactive
Multimedia Course of Study, is most suited to an introductory level tertiary
statistics unit, although there is also much of interest here for senior
secondary school. The text has a number of interactive elements, especially
simulations and an analysis laboratory. There are various case studies
provided for the user. 
Check with the instructions
regarding your computer settings and requirements. Essentially, you need a
browser, Java and Quicktime (for
the multimedia, which is optional). Notice that different Mode settings are
possible. After checking the Instructions, go to the Contents page and choose
a topic of your interest, or go direct to the List of Simulations and
Demonstrations. There are some good simulations (which generally get you to
answer some questions in advance). Make sure you try the Analysis Lab too. 
1012 
This website is based
around the remarkable Trendalyser
software, which allows for graphs Gapminder is a nonprofit venture promoting
sustainable global development and achievement of the United Nations
Millennium Development Goals by increased use and understanding of statistics
and other information about social, economic and environmental development at
local, national and global levels. 
This wonderful website
allows a user to compare countries on two different social dimensions
simultaneously, as in a scatterplot, but also shows how the relationship
changes over time as well as showing population growth over time. Easier to
observe than to describe. Choose GapMinder
World 2006 to interact with these ideas, linking statistics and global
public health. (There is a Help key if necessary.) The website also has some nice
presentations of Hans Rosling using the software, which provide a powerful
new way to see statistics of these kinds. 
912 
The Centre for
Technology and Teacher Education has long been a pioneer of nice uses of
technology, and this collection of Flash applets is no exception. You will
need a Flash [player to use these, most of which are for older students
(although some are for primary students as well). 
These applets can be
downloaded for offline use, which is very handy (and generous of the developers).
Some of them also come complete with some suggested activities, although it
is fairly clear in most cases how they might be used. The Quincunx
and Sampling
Distribution applets provide powerful visual demonstrations, while
the Integer
Cars may help students make sense of operations with negative
numbers. Most of these would be useful with an Interactive Whiteboard for
whole class use. 
1013 
These Mathematics Web
Resources from Lawrenceville cover a wide range of areas, and would be very
useful for an Interactive Whiteboard for whole class use. The resources have
been made using GeoGebra and EquationPlotter and take the form of
Java applets. 
The applets are mostly
concerned with mathematics of the senior years or the first undergraduate
year, especially in algebra, calculus and trigonometry. Each offers a
flexible interface to explore key features. Try Unit
Circle or A
Function and its Derivative to get
a taste of the materials, which would provide good classroom
demonstration tools. 
18 
This suite of 32 Flash
applets was produced as part of the National Numeracy Strategy of the DFES in
the UK, and are designed to be used on an interactive whiteboard. They deal
with a range of mathematical areas, and will help turn an IWB into a teaching
tool with mathematical capabilities. 
While there is a range
among the set, teachers in primary school will certainly find some of these
to be of value. Importantly, the applets can be downloaded to a computer, so
do not require you to be online to use them after they have been downloaded.
Different versions are available online Try Decimal
Number Line and Fractions
to get a feel for the applets. Each has some (minimal) onscreen
instructions, and slightly more extensive teacher notes can be downloaded. 
711 
Described as a
prototype, this free webbased secondary school geometry textbook developed
by John Page incorporates student learning with interactive tools and Java
and Flash animations. Many of the tools and animations refer to specific
concepts or singlepurpose mathematics tasks, such as constructions, and
could be used for a range of purposes, including demonstrations in class or
on an interactive whiteboard. 
Most
of the materials refer to aspects of geometry, although this includes some
measurement materials as well. To see some examples, try Circumcircle, Coordinate Plane, Volume of Cube or Graphical Quadratic
Explorer. The material is wellsuited as a reference for students (hence
the title and the ‘textbook’ notion). Recently, the materials have been
expanded to include coordinate geometry and some graphing applets. Use the index to find
specific topics. 
812 
The CorePlus
Mathematics Program is a curriculum development program based at the
University of Western Michigan. CPMPTools is a suite of Javabased
mathematical software, specifically designed to support student learning and
problem solving in each strand of CorePlus Mathematics. This tools component
includes a number of Java tools organised into four curriculum areas:
algebra, geometry, statistics and discrete mathematics. 
You will need to
download a java webstart link in order to access the tools, which are very
powerful. The software includes four families of programs, depending on which
level of course (of 4) is chosen. Each contentarea tool includes specific
custom tools and builtin copies of many data sets used in investigations and
homework. For example, there is a CAS engine, a spreadsheet, data analysis,
simulation and dynamic geometry software, among others. The Help files will
provide detailed information. Although the software supports a particular
curriculum (CPMP), it is widely applicable. 
812 
Mathias Wandel’s
website is mostly concerned with woodworking and other practical pursuits.
This link however is intended to test your skill at gauging straightness and
squareness, important woodworking attributes, but may also provide
opportunities to practice and develop estimation skills. 
The connection to
mathematics here is not very strong, as the tasks are mostly concerned with
visual estimation. Nonetheless, mathematics is involved in finding centres of
circles, line segments and triangles (especially!); in finding parallelograms
and convergence points of lines and in bisecting angles. Move the cursor and
release to enter a point; results are tallied (so that you can see your
improvement – if any) and your success is located on a large histogram. 
K12 
This is a large and
wellorganised website with many kinds of materials for students, many of
which have an interactive element. The website provides separate links for different
mathematics topics. (Of necessity, the site also contains advertising, so be
careful with tapping links unless you are interested in what is being
advertised.) 
There are many
explanatory and descriptive materials on the site, mostly written competently
and easy for students to follow. Amongst the material, there are also many
interactive activities. A good example of an interactive is the zoomable
number line in the Number section. This allows the decimal number line to
be explored in interesting ways. 
Feedback or comments
on any of this are of course welcome.
Last modified on 26 April 2015