Put the verb in the correct tense

[micaelo]

In the following fill-in-the-gap exercise: "While I ... (cook) the supper, the children played chess." I know the logical answer would be "was cooking", but would it be possible to use "cooked" as the second part of the sentence uses the past simple and those two actions happen at the same time?

[David L.]

Yes.
The trouble is, for people learning a new language and in exercises, being asked to fill in words, sentences like these don't just occur in isolation - there is a context, and which one the speaker would use depends on the meaning he wants to convey. Takes these contexts:

"We had been out fishing all day and came home with two big trout. While I was busy cooking them for supper, the children played chess."

Here, the 'sense' that is conveyed is that the emphasis is on cooking of the trout caught that day, and while I was busy doing that, the children were content to play chess.

"We decided to have a quiet night at home. While I cooked supper, the children played chess."
Both actions are in the past, and have equal importance in the conversation.

"We had had a great birthday party for my daughter, and the children were finally running out of energy. It was getting on in the evening, and while I cooked supper, the children were content just playing chess."
Here the emphasis is on the children and their activities, not on the 'cooking'.