Learning Mandarin through Song
One way to make language learning fun for your child is to include familiar songs in the lessons. It is easier to memorize words when they are set to music. Also, your child may already be familiar with the song in a different language, and the familiarity will work to your advantage. My daughter loves the Disney movie Mulan. Learning the lyrics to familiar songs from that movie could be a fun way to acquire Chinese vocabulary.
However, the lyrics to Disney movies are a bit sophisticated, so it might be better to start with a folk song whose words are easy to learn and whose melody is a catchy tune. When I was in Taiwan, I kept hearing a song about a butterfly. It was in books of children's nursery rhymes, but it also appeared on commercials whose theme was keeping Taiwan beautiful through conservation. Here is one version:
The Butterfly Song
One of the things to watch out for when your child learns the lyrics to a Chinese song is that singing the song doesn't confuse him about the tones of each word. Chinese is a tone language. Each word is one syllable, and each syllable is assigned a tone. If you get the tone wrong, then you get the wrong word. When a song is sung, the individual tones of each word tend to disappear into the melody. The way to get around this is to have your child first memorize the lyrics, as if the song were a poem to be recited. Once the tones are in place correctly, you can add the melody to the performance.