You're connecting to the projector that shows the image on the board, not to the board itself. Most projectors just need a VGA adaptor.
You will not have the true interactive functionality of the Smartboard, however - all you'll be able to do is show the image on the board, but you won't be able to control the iPad from the Board. That would require a second connection between the Board and the iPad (the way the Board connects to the computer through a USB cable) as well as dedicated Smart software for the iPad (which doesn't exist yet).
I would like to share my setup at school to completely mirror iPad2 (and presumably iPad3) to your non-HDMI projector screen:
1. Apple TV - turn on Airplay $99.00
2. iPad2 - turn on Airplay (you already have)
3. Cables: a 15-pin VGA cable, an HDMI cable, a 3.5mm Audio Cable, Speakers
4. Last but not least, a box that has one HDMI input port, a VGA output port AND a 3.5 mm Audio Out port (search on amazon.com) I am using this one for about $36.00.
Here is how you do it:
Put both Apple TV and iPad2 on the same wireless network, and connect via Airplay (mirroring has to be on.)
Connect Apple TV to this HDMI TO VGA/AUDIO converter box
Connect this converter box to your VGA projector (with the VGA cable) and speakers (with the audio cable)
Pros: No laptop needed as an intermediate device, and any APPS can be used on your iPad because it is a complete 100% Airplay Mirroring. Don't need to buy an expensive HDMI projector. Many classrooms already have a vga or composite-based LCD projector. This will save lots of money.
If you are from a school/teacher and need more detailed help, please email me firstname.lastname@example.org.
Also we find that we can take a picture of a page of a textbook or a homework to be reviewed and play it with Educreation or Goodreader over the above setup and all students can learn and collaborate together easily. Thus saves paper. You might also want to get a stylus for easier writing on the iPad surface.
Hope this helps.