Neural network training is computationally and memory intensive. Sparse training can reduce the burden on emerging hardware platforms designed to accelerate sparse computations, but it can also affect network convergence. In this work, we propose a novel CNN training algorithm called Sparse Weight Activation Training (SWAT). SWAT is more computation and memory-efficient than conventional training. SWAT modifies back-propagation based on the empirical insight that convergence during training tends to be robust to the elimination of (i) small magnitude weights during the forward pass and (ii) both small magnitude weights and activations during the backward pass. We evaluate SWAT on recent CNN architectures such as ResNet, VGG, DenseNet and WideResNet using CIFAR-10, CIFAR-100 and ImageNet datasets. For ResNet-50 on ImageNet SWAT reduces total floating-point operations (FLOPs) during training by 80% resulting in a 3.3x training speedup when run on a simulated sparse learning accelerator representative of emerging platforms while incurring only 1.63% reduction in validation accuracy. Moreover, SWAT reduces memory footprint during the backward pass by 23% to 50% for activations and 50% to 90% for weights. Code is available at https://github.com/AamirRaihan/SWAT.