What is more powerful than a graphics calculator?
As a mathematics teacher, one of the most useful educational tools that modern technology has made widely available is the ability to instantly draw beautiful graphs of mathematical functions, allowing you to visually illustrate the solutions of a problem.
Here are some awesome packages which are all available for free online - without the need to create an account or sign up for anything. You can even download some of them to your computer or mobile device for offline use. Not only can these programs plot graphs, they can also do interactive 3D geometry, statistics, algebra and calculus!
My top three online maths programs...
No doubt there are many other programs out there, but these three should more than cover your needs for graphing and equation solving. They are also very helpful as an extension resource for inquisitive maths students who like to explore and discover things for themselves - the programs are very intuitive and most young people can quickly figure out how to navigate the basic features. For more advanced features there are also comprehensive help centres and Youtube tutorials.
1. Geogebra https://www.geogebra.org/
I met the creator of this site, Markus Hohenwarter, at a presentation at Cambridge University's Education Faculty while on my PGCE teacher training in 2005. At the time, the main graphing packages available were commercial programs such as Autograph and Geometer's Sketchpad. Markus felt strongly about putting Geogebra in the public domain so that students and teachers would be able to use it for free, and to this day it remains open-source software. Have a play, it's lots of fun...
There is also a huge searchable database of free resources which teachers can use for lessons: https://www.geogebra.org/materials/
Recent additions include a graphing calculator for iPhones - the app is available for free download from itunes (click on the image):
2. Desmos https://www.desmos.com/
This is a relative newcomer (the company started in 2012) - actually one of my students recently made me aware of this program and I was instantly impressed by its attractive presentation and ease of use. Described by its creators as beautiful, free math, this graphing software also has a teacher section which has plenty of engaging and visually impressive free resources at https://teacher.desmos.com/
See this short clip on youtube for a preview of the free desmos online resources and lesson ideas
3. Wolfram Alpha http://www.wolframalpha.com/
Wolfram Alpha is much broader than a graphing calculator - its creators describe it as "the world's definitive source for instant expert knowlege and computation," and liken it to the all-knowing computer from Star Trek (if you've no idea what I'm talking about, click here for some vintage Star Trek footage of Captain Kirk using the 1960's precursor to iphone's Siri!).
The Wolfram interface take a little bit of getting used to, but essentially you just type in any function (e.g. "x^3 + 6x^2 - 5") and Wolfram Alpha instantly churns out the graph, along with a host of related information such as minimum/maximum point, the factorised form, the roots, the derivative and the integral! (click the picture below for a live example)
The user-interface is perhaps not as intuitive as other sites and in order to unlock its full capabilities you have to upgrade to Wolfram Alpha Pro (paid subscription), but even so the free version has some pretty serious computational capability and is useful for advanced problem solving.
There are also downloadable app versions which you can purchase for iphone and other smartphones: http://products.wolframalpha.com/mobile/
For university-level mathematics, Wolfram has a program called Mathematica. There is no free version unfortunately, but if you are looking for serious, heavy-duty computational power then Mathematica has it by the truckload!
So that's my top three - if you have any favourites that I haven't mentioned feel free to drop me a line and let me know...