If the plastic thing broke off the sprocket which is at the rear wheel, this has little to do with the chain coming off the front chain ring. From the sounds of things the change is bouncing (caused by the spring tension being very low in the rear derailleur) or the chain having stretched or overly long. A guard on the inside of the chain ring would stop the result, but a new cheap rear derailleur might stop the cause. Get your LBS to check it out if you are unsure.
