We can slip this asignment up into different parts and each

1.slip this assignment up into:把assignment分成（几个部分）
slip sth.(e.g.assignment) into:to put sth somewhere quickly,quietly or secretly (牛津高阶词典第6版)
也就是说前者相当于devide...into,后者是将某物快速、安静或秘密地（不易被察觉地）置于某地,例如：Anna slipped her hand into his.
I managed to slip a few jokes into my speech.
2.each 后面这个句子的主语,即“we”中的每一个人,on our part是指依照各自的分工（角色）来做（work),part有角色的意思,相当于role,这里就应该是指根据每个人不同的特长来确定的分工、工作.所以这句话是说每个人按照各自的分工来做（根据前面slip...up into后的结果）