Tips on adding filter media from an established aquarium to a new aquarium?

I’ve had a 30 gallon freshwater up and running for three months now. I’d like to start a 12 gallon and was thinking of using some filter media from my 30 to get ‘er jump started. I have an Aquaclear 70 for my 30, and the 12 gallon comes with a smaller Aquaclear filter. Could I take the sponge out my 30 (with beneficial bacteria in tow), cut it, and put it in my 12?