Machine unlearning has become a pivotal task to erase the influence of data from a trained model. It adheres to recent data regulation standards and enhances the privacy and security of machine learning applications. Most existing machine unlearning methods perform well, however, they typically necessitate access to the entirety of the remaining data, which might not be feasible in certain scenarios. In this work, we present a new machine unlearning approach Scissorhands, which operates effectively with only a subset of the training data. Initially, Scissorhands identifies the most pertinent parameters in the given model relative to the forgetting data via connection sensitivity. This process involves reinitializing the most influential top-$k$ percent of these parameters, resulting in a trimmed model for erasing the influence of the forgetting data. Subsequently, Scissorhands retrains the trimmed model through a min-max optimization process, seeking parameters that preserve information on the remaining data while discarding information related to the forgetting data. Our experimental results, conducted across five distinct datasets and utilizing both CNN and ViT, demonstrate that Scissorhands, despite utilizing only a limited portion of the training data, showcases competitive performance when compared to existing methods.