Hi, Joel, you did a good job!!
Just please remember we use measuring word to modify nouns quite often which is pretty like the articles, THE, A, AN, in English:)
So in that case you put there, it would be better if you say: 我是妈妈的第三个孩子。
Furthermore, 三 is a cardinal number, 第三is an ordinal number, while 第三个 is what you can put before a noun as an adjective.
Hope it helps!
Have a nice day!
Candy