What should i said? Ich Danke Der Frau or Ich Danke Die Frau
Some verbs (and also prepositions, for that matter) imply the dative case. 'danken' is one of them. See here for a short overview: http://german.about.com/library/verbs/blverb_dativ.htm @Dieter: The method with the questions 'wer/was?, wessen?, wem?, wen/was?' is useful for German school children who already know the correct cases but have difficulties to give them the corresponding names. In order to decide which of the four questions to ask, one already has to know the correct case ;-)
Oh, @Dieter_Drawe is right. It is Ich danke der Frau, because 'danken' is a dative verb, which means it puts its object in the dative (indirect object) case instead of the accusative (direct object) case. Which, if you're just doing the Phrases module, you don't know anything about yet. And even then, if you don't know that 'danken' is a dative verb, you end up giving bad advice like I just did.
In summary: it is 'Ich danke der Frau', and future lessons will explain why it is 'der' instead of 'die'.