改为：1.HAving thought he could sell the jar and getting some money for it,he immediately opened it.要点：当主句和从句的主语指同一人或同一事物的时候,从句中的谓语动词就可以用分词.分词有现在分词和过去分词,表示主动的用现在分词,表示被动的用过去分词,分词在时间上有一般式和完成式,与谓语动词同时发生用一般式（doing）,发生在谓语动词之前用完成式（having done).
2.The boy rushed out ,almost knocking me down.要点同上
3.Being interested in geography,I I find it quite easy to learn it well.要点同上
4.I heard someone calling my name.此句本身就是现在分在分词作宾语补足语,hear sb.doing.不需要再改.