Dynamically tunable band stop filter based on metal-graphene metamaterials is proposed and numerically investigated at mid-infrared frequencies. The proposed filter is constructed by unit cells with simple gold strips on the stack of monolayer graphene and the substrate of BaF2. A stable modulation depth up to -23.26 dB can be achieved. Due to the cooperative effect of the "bright-bright" elements, the amount of the gold strips in each unit cell determines the number of the stop-bands, providing a simple and flexible approach to develop multispectral devices. Further investigations illustrate that the location of the stop bands not only can be adjusted by varying the length of gold strips, but also can be dynamically controlled by tuning the Fermi energy level of graphene, and deep modulation is acquired through designing the carrier mobility. With the sensitivity as high as 2393 nm/RIU of the resonances to the varieties of surrounding medium, the structure is also enabled to be an index based sensor. The results will benefit the on plane or integrated micro-structure research with simple structure and flexible tunability, and can be applied in multi-band stop filters, sensors and other graphene-based multispectral devices.