We propose the design of event-triggered (ET) discrete-time sliding mode (DTSM) control for a high-order discrete-time system via a reduced-order model-based approach. This design includes a triggering mechanism using a reduced-order state vector and a controller based on the modified Bartoszewicz' reaching law for a reduced-order model of the system, to stabilize the uncertain high-order system. The main advantages of using a reduced-order vector in the event condition are the low-order synthesis of the controller and the sampling pattern, which may be sparser than the full vector-based design. This motivation arises from the fact that relaxing a few components of the state vector in the triggering mechanism may decrease its rate of violation. An added advantage of the proposal is that the transmission of the reduced-order vector, particularly in a network-based implementation, can outperform the full-order based design due to the severe challenges that exist in the data network. The robust performance for the closed-loop system is achieved using the DTSM control. We show that our proposal guarantees the stability of the full-order plant with the reduced-order triggering mechanism. The control execution is Zeno-free because of the inherent discrete nature of the control. The efficiency of the proposed method is shown using the simulation results of a numerical example.