The development of an efficient and powerful machine learning (ML) model for materials property prediction (MPP) remains an important challenge in materials science. While various techniques have been proposed to extract physicochemical features in MPP, graph neural networks (GNN) have also shown very strong capability in capturing effective features for high-performance MPP. Nevertheless, current GNN models do not effectively differentiate the contributions from different atoms. In this paper we develop a novel graph neural network model called GATGNN for predicting properties of inorganic materials. GATGNN is characterized by its composition of augmented graph-attention layers (AGAT) and a global attention layer. The application of AGAT layers and global attention layers respectively learn the local relationship among neighboring atoms and overall contribution of the atoms to the material's property; together making our framework achieve considerably better prediction performance on various tested properties. Through extensive experiments, we show that our method is able to outperform existing state-of-the-art GNN models while it can also provide a measurable insight into the correlation between the atoms and their material property. Our code can found on - https://github.com/superlouis/GATGNN.