"I thank our cook."
Translation:Ich danke unserer Köchin.
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"danken" is a verb which takes the dative, so whomever you thank will need the appropriate dative endings on all articles and adjectives. In this case, a male cook would be "ich danke unserem Koch" (der Koch, dat. dem Koch) and a female would be "ich danke unserer Köchin" (die Köchin, dat. der Köchin). I hope that helps.
It is dative -you are correct. There are a few verbs in German which always demand the dative. 'danken (to thank)' and 'helfen (to help)' are two very common ones. Some additional verbs and more explanation can be found here: http://german.about.com/library/verbs/blverb_dativ.htm
If you are presented wordblocks, they are guaranteed to enable building one of the possible answers, not all of them. "cook" can be a male or a female cook. In German those are two different words. So both "underem Koch" and "unserer Köchin" are correct answers. Most probably "unserer" and "Köchin" were among the word blocks, so you could form the correct answer "Ich danke unserer Köchin".
If, however, you use "Koch", it must be "unserem". In a freetext answer you can type "unserem". Using wordblocks, you should not use "Koch" if there is no "unserem".
The German term "der Chef/die Chefin" is "the boss (i.e.; employer); chief" in English. A cook is "der Koch/die Köchin". The term "chef" meaning "head cook" in English comes from the French 'chef', short for 'chef de cuisine', literally "head of the kitchen" (see etymonline.com). If you want to specify the head cook in German, you can use "der Chefkoch/die Chefköchin".
As the two of you have documented, this appears to be a case where Duo has focused on one particular gender, omitting the other. (This was masked when I worked the exercise because I had "word bank" - "Koch" was featured, but "unserem" was not, forcing the answer to be "unserer Kochin")
When putting a noun into another case, in most cases only the article changes. So the dative of "der Koch" is "dem Koch". There is no "e" in the end of "unserem Koch".
Because Duolingo ignores capitalization (and you incorrectly didn't even capitalize "Koch"), it takes "koche" for the existing verb form "koche" like in "ich koche" ("I cook"), which is definitely wrong.
In this sentence the verb is "danken", not "bedanken". The "für" is used to express what you are thanking for (with both verbs). But this is not present in the English sentence, therefeore it is not used in the German translation as well.
If the sentence were "I thank our cook for the excellent meal", it would be "Ich danke unserem Koch für das ausgezeichnete Essen". If you want to use "bedanken", the construction is different, because it is used reflexicely: "Ich bedanke mich bei unserem Koch für das ausgezeichnete Essen".