A new method of calculating microlensing magnification patterns is proposed that is based on the properties of the backward gravitational lens mapping of a lattice of polygonal cells defined at the image plane. To a first-order approximation, the local linearity of the transformation allows us to compute the contribution of each image-plane cell to the magnification by apportioning the area of the inverse image of the cell (transformed cell) among the source-plane pixels covered by it. Numerical studies in the κ=0.1-0.8 range of mass surface densities demonstrate that this method (provided with an exact algorithm for distributing the area of the transformed cells among the source-plane pixels) is more efficient than the inverse ray-shooting technique (IRS). Magnification patterns with relative errors of ~5×10-4 are obtained with an image-plane lattice of only 1 ray per unlensed pixel. This accuracy is, in practice, beyond the reach of IRS performance (more than 10,000 rays should be collected per pixel to achieve this result with the IRS) and is obtained in a small fraction (less than 4%) of the computing time that is used by the IRS technique to achieve an error more than an order of magnitude larger. The computing time for the new method is reduced to below 1% of the IRS computing time when the same accuracy is required of both methods. We have also studied the second-order approximation to control departures from linearity that could induce variations in the magnification within the boundaries of a transformed cell. This approximation is used to identify and control the cells enclosing a critical curve.