Best Integrated Geometry, Algebra and Graphing and STEM (free and online)
Over twenty years ago, I served as Project Officer for a national teacher professional development project for the Australian Association of Mathematics Teachers, working with representatives of each of the state and territory Mathematics Teacher Associations to explore and develop resources for the new phenomenon of the World Wide Web. Of the outcomes of the AAMT EdNA DAME project (Yes, Will Moroney, I am looking at you for that name!) [Education Network Australia Delivery Alternatives for Mathematics Education], two in particular retain a special fondness for me.
The AAMT Discussion List for Mathematics Teachers (discuss@aamt.edu.au) which, all these years later, remains a vibrant and important forum for teachers from across the country and beyond to share and learn, and
A project of my own, called Live Mathematics on the Web, in which I sought out and explored the possibilities for interactive browser-based mathematics learning opportunities which were beginning to appear, especially in the areas of algebra and graphing, but also for geometry and number. At the time, these tended to be based on browser plug-ins and Java applications, and a smattering of JavaScript. Only the last survives as viable in today's web experience, along with HTML5, which brings amazing opportunities for free, dynamic learning experiences for students on any platform, computer, phone or tablet.
Here I present some of the best free browser-based classroom-ready teaching and learning tools for mathematics and STEM. Feel free to use them and share them with your students.
Imagine dynamic geometry built on a foundation of computer algebra. So side lengths, angles, etc can be defined algebraically as well as numerically, and results can be derived in general form.
Then add constraint-based construction to make it even easier to use. Do you want a right-angled triangle? No problem - just define a general three sided figure and add the right angle later.
Derive Pythagoras' Theorem? Easy!
Have students explore algebraic relationships through convincing visual means using geometric constructions.
GXWeb is a free browser-based symbolic geometry tool that combines constraint-based dynamic geometry with computer algebra. From Saltire Software who offer the amazing Geometry Expressions and Mathematical Illustrations, this surprisingly powerful tool for classroom learning offers students a pathway to learning and understanding algebra through visual and dynamic geometric representations.
The algebraic objects that you create in GXWeb can be copied as text or code by tapping on them, and this can then be pasted into GeoGebra (using the Input bar), Check My Steps: An Algebra Toolkit on these pages or, if you are using the Desktop version, into Desmos, TI-Nspire, Mathematica...
The full-featured Geometry Expressions software, of course, offers many more amazing opportunities for learning mathematics, even beyond its unique blending of constraint-based dynamic geometry and computer algebra. The ability to create GX documents and then export these to a variety of formats is a powerful feature - the focus in these pages, of course, lies in exporting to the web, using HTML5 and JavaScript. Like GeoGebra, GX can be the foundation of powerful online assessment and exploration tasks for both mathematics and STEM.
The activities which follow demonstrate this interesting and powerful feature that supports interaction between JavaScript on the web page with GX web apps, potentially serving as a powerful tool for online assessment.
Two friends decide to meet during their lunch hour: if they agree to wait for x minutes, what is their chance of meeting?
A beach race begins at a point 4 km offshore and ends at the end of 10 km beach. If I swim at 4 km/h and run at 10 km/h, at what point on the beach should I aim to come ashore to minimise my race time?
A small square is constructed within a larger unit square as shown. Can you find the algebraic model for the area of the smaller square as the point X changes?
What is the likelihood of someone else in your class sharing the same birthday with you? How big would your group need to be for this to be better than 50% - actually smaller than you might think!
Suppose you have two samples from a population (for example, before and after samples from a medical treatment or exam results on a common test for two parallel class groups) and you wish to know if there is a significant difference between these - or whether any differences might just result from chance or error. We can use probability distributions to decide!
Every real number, rational and irrational, can be represented as a continued fraction. While normal fractions can only represent rational numbers, continued fractions are different. Rational numbers produce finite continued fractions, while irrationals become infinite continued fractions.
GXWeb Continued Fraction Collection 2: What might appear to be the simplest of continued fractions turns out to be anything but simple - and related to a family of functions with important applications to physics - welcome to the Bessel Function Family!
The factorial function is well known - \(3! = 3 \cdot 2 \cdot 1 = 6, 4! = 4 \cdot 3 \cdot 2 \cdot 1 = 12,\) \( n! = n \cdot (n-1) \cdot (n-2)\)... But what happens in between? The Gamma function includes all values, not just the integers, and has some interesting properties.
Rational and Irrational Numbers
Approximations
GXWeb Continued Fraction Collection 4:
Imagine a population, say of fish in a pond. If the pond is fixed in size and limited in the amount of food which it can provide, then the population of fish cannot grow unbounded. But what happens over the next few years becomes the basis for chaos theory.
Rational and Irrational Numbers
Approximations
GXWeb Continued Fraction Collection 5: The modern musical scale is based upon a geometric sequence - each successive note is produced my multiplying by the twelfth root of 2 (\( \sqrt [12] 2 \) or \( 2^{1 \over 12} \) - approximately 1.059463). Now what if you could turn any continued fraction into a melody?
Rational and Irrational Numbers
Approximations
Every once in a while in Mathematics we come across problems in which the numbers just do not seem to behave themselves. They appear simple enough at the outset, but we quickly discover that our mathematical intuition fails us, and the solution - if it comes at all - arrives as the result of a long and tortuous process.
Most of the curves you see on a computer screen or printed page - everything from text fonts to animations - are generated mathematically using Bezier Curves (sometimes called Bezier Splines).
GeoGebra is an exciting blend of Algebra and Geometry using an integrated graphing and geometry environment. It is freely available for all platforms and has a wealth of supporting sites. Explore the sample files below.
Did you know?
The GeoGebra documents that you create can be embedded in a web page.
AND your students can interact with those applets from the web page, allowing you to create dynamic worksheets, which might be used for feedback or assessment!
The activities which follow demonstrate this interesting and powerful feature that supports interaction between JavaScript on the web page with GeoGebra web apps, potentially serving as a powerful tool for online assessment.
Area of a triangle
Linear inequalities
Pythagoras' Theorem
Linear functions
Calculus extension
Pythagoras' Theorem
Similar triangles
Calculus extension
Area of a triangle
Pythagoras' Theorem
Calculus extension
Similar triangles
Pythagoras' Theorem
Volume of a cone
Chance and probability
Factorial functions
Algebraic modelling
Area of a circle
Volume and Surface Area of a cylinder
Area of a triangle
Pythagoras' Theorem
Similar triangles
Exponential and Logarithmic functions
Recursive functions
Parametric functions
Rational and Irrational Numbers
Approximations
Means and Sequences
Geometric Constructions
Would you like some step-by-step help with your algebra?
Before we head into the amazing potential of Bluetooth Low Energy coupled with JavaScript to make web pages powerful tools for the teaching and learning of STEM, it is worth noting that the internal sensors available within mobile devices are readily accessed from modern browsers.
Accelerometer, Gyroscope and Magnetometer are exciting and important tools for exploring motion and orientation in a variety of ways, and it is surprisingly easy to create web pages that support the observation, collection and manipulation of data from such sources. While desktop computers may not lend themselves to this domain (yet!) phones and tablets are the perfect tools for students to explore the science and mathematics of motion!
Of course, all devices, mobile or desktop, are location aware these days, and geolocation is another potentially powerful tool for STEM learning. Again, with relatively simple JavaScript coding, students and teachers may access, not only the GPS coordinates of their current position but can track movement and even show this on an embedded Google map right in their browsers.
QR Codes are an excellent way to share data and information with those around you. Most commonly used to share web links, they can also be a quick and convenient way to transfer data with colleagues or even assessment responses between students and teacher.
This JavaScript utility makes this process simple to add to your own web page!
Heart rate monitors were among the earliest adopters of Bluetooth Low energy technology, and so it is not surprising that there are (relatively) simple solutions for offering these great tools within a web page.
The BLE heart rate script transmits a value using the Bluetooth SIG Heart Rate Profile and should work with most common heart rate monitors.
Once again, we combine the BLE sensor with GeoGebra's graphing and spreadsheet capabilities to develop your own personal fitness trainer.
Explore Musical Tones, Add a Piano Keyboard (with GeoGebra and BLE LaunchPad support)
Explore the world of musical tones created in your browser through Web Audio...
You might also like to try a combination of Web Bluetooth, GeoGebra and music using Mike Bostock's amazing arc piano! Explore the mathematics and STEM basis for music with this free online tool!
If you would like to use GeoGebra as a tool for STEM, then what about pairing it with the amazing data collection available using the TI CC2650 SensorTag?
The TI SensorTag is a small, inexpensive device which packs a powerful punch - it is a BlueTooth Low Energy (BLE) tool with sensors for temperature, humidity, barometric pressure, accelerometer, magnetometer, gyroscope and more.
Used with the Chrome browser on Mac, Windows (Chrome Canary), Android or ChromeBook, students can easily collect, view and analyse a wealth of real world data for mathematics and science, from the middle years through to seniors. From building their own weather station, to exploring the mathematics and physiology of exercise and dance the opportunities for rich mathematical exploration are truly exciting. All in a tiny device that fits in your hand and costs around $AUD100!
Another great STEM application can grow from students learning to code for MicroController Unit (MCU) boards, like the TI LaunchPad series or Arduino boards.
TI LaunchPads are microcontroller development kits from Texas Instruments. They come in a variety of flavors to address various project needs. All LaunchPad kits include everything needed to begin developing applications in minutes, priced from $USD9.99 to $USD19.99. That's right - Blink LEDs and spin motors, all for less than $USD20. These units can be the basis for everything from simple coding to connecting a variety of sensors and other peripherals. They are easily programmed via USB from Mac or PC, and can be linked to BLE (Bluetooth Low Energy) modules, and even wifi to send and receive data both locally and remotely.
We can now use Web Bluetooth to enable free and effective communication with these programmed boards, using the Chrome browser on Mac, Windows (Chrome Canary), Android and ChromeBooks or WebBLE app for iOS! Coupled with the versatile mathematical capabilities and programmability of GeoGebra, students can control these boards remotely, using them to collect and analyse data, even to control BLE robots! Step-by-step tutorials lead from beginner to robotics developer!
What if you had a BLE-enabled LaunchPad and access to either mobile device internal sensors, or even a TI SensorTag - what could you do?
Actually - quite a lot, but one application that springs quickly to mind would be to use an accelerometer to drive your LaunchPad robotic vehicle!
Build your own TI Launchpad Robot (Tap the image to switch robots!)
Build your own TI Innovator Hub Robot (Tap the image to switch robots!)
BLE TI Rover (Tap the image to switch robots!)
In this example, we use the 3D accelerometer on either the internal device sensors or the SensorTag to send the forward, back, left and right commands to the sketch on board the robot - and it is So. Much. Fun!
Just connect the LaunchPad robot and either internal device sensors or SensorTag, tap the "Drive Using Accelerometer" button and practise your remote control skills!
OR... Just connect your LaunchPad, tap the "Drive Using Accelerometer" button and use the acc_x and acc_y GeoGebra accelerometer sliders to control your Robot!
Follow the tutorial step-by-step directions to build and program your LaunchPad robot - and you can even use a TI-Innovator Hub or go the next step on and use the TI Innovator Hub Rover
All these LaunchPad BLE documents should work well using the Google Chrome browser on a recent Mac (mid-2012+), Windows 10 with Chrome Canary, Android device or ChromeBook. LaunchPad BLE (but NOT TI SensorTag!) will work on iPad with the WebBLE app, but this is an experimental build and may not always work seamlessly. For example, I find that I need to CONNECT, then DISCONNECT, and then CONNECT again to establish a reliable connection using WebBLE. However, after this, it generally works well.
The latest versions of Chrome (69+) have these features enabled by default. Back when I started exploring, I got mine running on my Mac without too much trouble: I did go in and switch on Chrome's experimental flag to support Web Bluetooth but this was pretty straightforward. You can enable it turning on the enable-experimental-web-platform-features flag: In the Chrome browser, type in the web address chrome://flags/. This action results in a list of experimental browser features available to you. You want the Web Bluetooth one.
The simplest solution is probably to download Chrome Canary for your platform, which is their beta software, enabling pretty much everything by default.
If you have any questions or would like to share your experience with these utilities, please drop me an email!
My First Web Page: Create Your Own Live Mathematics and STEM Web Pages
Would you like to be able to quickly and easily create interactive mathematics and STEM resources for your students? Live documents that can be freely accessed on any device or any platform with Internet access?
The tutorials in this series offer a quick and easy way to get started creating your own web-based resources, even if you are a complete beginner! Each lesson is designed to take only a few minutes, with the option to explore further if desired. The first lesson offers a taster of what lies ahead - from the simplest of web pages to adding the power of GeoGebra - graphing, computer algebra, dynamic geometry and interactive statistics - to your own lessons and activities!
Later lessons add access to mobile device internal sensors and BlueTooth Low Energy (BLE) capabilities to these Live web pages, supporting active engagement with key STEM principles and the ability to develop engaging activities for students.
And some other JavaScript/HTML5 mathematics goodies..
Welcome to the Canvas element and the bountiful possibilities for adding live graphics to your web page! You will be amazed at what can be done with just a few lines of code. What are you waiting for? The time to explore is NOW!
Or what about taking your web pages to another level and using some of those Lua skills you have been working on to quickly and easily bring your web page to life?
Imagine being able to physically grab, drag and drop the elements of mathematics - arithmetic, algebra, geometry and statistics!
This is classroom technology assisting learning in the best possible way - intuitive and powerful interactions between students and the mathematics they are learning.
The two examples featured here demonstrate beautifully the potential of good classroom technology to enhance and transform the learning process.
DudaMath is an integrated environment for interactive exploration of mathematical concepts and problem solving.
Graspable Math allows students to explore algebra online.
Move terms fluently to solve equations and explore the power of algebra without the frustration for free!
Like DudaMath, students are empowered to explore the concepts behind the manipulations.
I was pleasantly surprised by the feature set of this free web resource from CASIO. Not quite up to the high standard set by GeoGebra, no integrated graphing and geometry, and a requirement to pay for the complete (CAS) functionality, but a very clear and easy to use mathematical workspace for students.
Graphing Functions
ClassPad.net offers mathematical functions required for education, and various graphing capability.
The simultaneous display of graphs and a slider function helps deeper understanding of mathematics.
Statistics and Analytics
A Spreadsheet can be used in combination with a graphing function to draw statistical graphs.
It allows you to analyze statistical data using various approaches within a single screen.
Geometry Functions
You can create freehand graphics using the Geometry functions including the ability to freely alter the shape of graphics and automatically calculate length, area, angle and so on.
If you have never heard of a Mathenaeum before, you are probably not alone, but visit this site and you will find the most wondrous, captivating and useful collection of mathematical explorations, games and learning tools that I have ever seen.
Hold on - you and your students are in for one amazing ride!
In particular, I would point to the excellent Algebra Tiles activities - I have been a fan of algebra tiles for decades and these are among the best I have come across for students to build firm concrete foundations for their algebraic thinking and facility.
iNZight lite lets you import your own data set or explore one of the many example data sets; visualize and conduct statistical analysis on the data, and even modify the data, to explore hidden secrets behind the data.
The desktop application iNZight is a simple data analysis system which was initially designed for high school students to help them explore data quickly and easily. It still serves this function but other demands have seen extensive additions to its capabilities. For example, it now provides very flexible multivariate visualisation capabilities and supports 3D graphics, maps, time series, and multivariable regression analysis. iNZight lite is an online version of the software, which requires only access to the internet and a modern web browser, thus making it accessible to a wider range of users.
Graphing Functions
Statistics and Analytics
Geometry Functions
