Often you will see tennis players warm-up using the service boxes (short court) as the full-court. Why? It's a great way to not pull muscles by running too soon. It can teach short back swings, watching the ball, controlling the ball. Adjusting to different speeds and spins, patience, placement and footwork.
 
As a parent you can use this "short court" to teach your kids the basics. (You might want to go to a professional to learn the grips and basics). During your time on the "short court" you can have a very valuable conversation with your child. Not just about tennis.
 
As they get better and better you can play points using only the "short court." As they get better, and they will very fast, give them only one of the service boxes on your side, where as you get both sides of the service box on their side. It can make it very competitive and fun and challenging. This will gradually teach them the fun in competing.
 
As they get better, they will start taking a longer backswing and follow through when they move back to the baseline. But, before that, start them with a bumping or blocking motion until they gain control!
 
The game is built on "motor memory." This method is excellent for developing motor memory. Also, use the new foam balls as they start. They will not hurt them if they get hit, the rallies will be longer and you will spend more time hitting rather than chasing the ball everywhere. As their game matures go to the regulation ball.
 
Don't forget how valuable the "short court" warm-up can be. It's fun and you can still instill the competitive nature of the game. Also, it works just as well when you warm-up for your adult matches.
 
Give it a try!
 
Good luck on the courts!