Achild with down syndrome is born with the genotype aaa. the father is genotype aa and meiosis occurs normally. non-discjunction occurs in the mother, at which stage in meiosis did it occur?

Nondisjunction results when chromosomes in gametes (egg and sperm cells) do not separate properly during meiosis (cell division of gametes in 2 stages), producing gametes which have more or less than the normal number (23) of chromosomes. 

Nondisjunction occurs during anaphase of either Meiosis I or Meiosis II.  Anaphase is the splitting and pulling of the chromosomes, lined up at the center of the cell, to the edges or poles of the cell. Shown in the figures below are examples of when nondisjunction occurs in Meiosis I (Anaphse I) and in Meiosis II (Anaphase II).

(Figures source: http://study.com/academy/lesson/nondisjunction-in-meiosis-definition-examples-quiz.html) 

Cell division is important to single celled organisms because it allows the cells to reproduce. through cell division, cells make exact copies of themselves. when single celled organisms go through cell division, they make exact copies of themselves which is how they reproduce.