Teaching a dog to smile takes a bit more work than some other tricks, but it can be achieved. Reinforce the behavior when the dog is happy and relaxed. Smiling is not the same behavior in dogs that it is in humans, and although it appears to us to be a smile, to the dog it is mild panting with the mouth slightly open -- a behavior that occurs when the dog is having fun, being scratched in just the right place, or after a game. The key is to teach your dog at just the right time that this is the behavior you want.
Begin when your dog is happy and relaxed, i.e., after a play session or when the dog is getting a good scratch. Gently lift the dog's lips on either side of its snout with your fingers from above. Hold a treat in your other hand.
Give the command, "smile," in a happy, high-pitched voice. As you do this, give the dog the treat. Pat and praise, using the same happy voice. Repeat this several times.
Use the dog's favorite toy for a game. Play fetch, ball, or tug. If you have more than one dog, do this one-on-one with the dog you are training so it has your undivided attention. At the end of the game, ask the dog to smile with a treat. If it offers the behavior, give a treat and lots of praise using the command word, such as "good smile!"
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Work with the dog when you're petting it. Find your dog's favorite pleasure point. Some dogs love to be scratched on the back at the base of their tail; others, just under the ears. Scratch the dog in its favorite place, then give the "smile" command. Treat the dog when it responds correctly.