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For each integer s ≥ 1, we present a family of curves that are ${}_q$-Frobenius nonclassical with respect to the linear system of plane curves of degree s. In the case s=2, we give necessary and sufficient conditions for such curves to be ${}_q$-Frobenius nonclassical with respect to the linear system of conics. In the ${}_q$-Frobenius nonclassical cases, we determine the exact number of ${}_q$-rational points. In the remaining cases, an upper bound for the number of ${}_q$-rational points will follow from Stöhr-Voloch...