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Try to wait until the snake isn't shedding and has gone to the bathroom before feeding it, as it's more likely to want to eat. Many snakes are more active at twilight, so you should feed them in the late afternoon.Some snakes are nocturnal and won't be comfortable eating during the day. Use tongs to hold the rodent by its tail. As you lower into the cage, move the head of the mouse towards the head of the snake in range of its tongue. With a garter snake, you will need to move it around a bit, as they're more attracted to movement than other snakes. Always use tongs, as snakes use heat and smell to guide them. They may strike at your hand if you don't! You can also feed your snake in a separate feeding enclosure, so that the snake doesn't associate you putting your hand in their cage with eating. The snake will use its tongue to check out the rodent. Once it does, it will typically strike the rodent and take it from your tongs. Be sure to drop the rodent when it does! Once your snake gets used to the bait, you may be able just to place the rodent in the cage for the snake to take.
Feed at a comfortable time for your snake. Lower the rodent into the cage by its tail. Wait for the snake to inspect and take the rodent.