You can use your method, or do without the cartridge entirely and custom cut a sponge to fill the same compartment. Then you just rinse the sponge really well instead. However, depending on the cartridge type, there is no need to replace that often. If you are using a cartridge with carbon, the carbon will cease functioning in 3-5 days and just become another place for bacteria to thrive. If you rinse the whole thing off to remove any large debris, you can keep using the same filter for many moons (think I've used the same one for 6 months in a lightly stocked tank before trashing it and going with just a sponge).
If you are using a cartridge that contains some sort of ammonia treatment, once the tank is cycled there is no need for this stuff, as it can impact the bio-bed.