To overcome the difficulties in determining an inverse compressor function for a Gaussian source, which appear in designing the nonlinear optimal companding quantizers and also in the nonlinear optimal companding quantization procedure, in this paper a piecewise linear compressor function based on the first derivate approximation of the optimal compressor function is proposed. We show that the approximations used in determining the piecewise linear compressor function contribute to the simple solution for designing the novel piecewise linear scalar companding quantizer (PLSCQ) for a Gaussian source of unit variance. For the given number of segments, we perform optimization procedure in order to obtain optimal value of the support region threshold which maximizes the signal to quantization noise ratio (SQNR) of the proposed PLSCQ. We study how the SQNR of the considered PLSCQ depends on the number of segments and we show that for the given number of quantization levels, SQNR of the PLSCQ approaches the one of the nonlinear optimal companding quantizer with the increase of the number of segments. The presented features of the proposed PLSCQ indicate that the obtained model should be of high practical significance for quantization of signals having Gaussian probability density function.