How do you know which version of the word 'the' to use?

The word for 'the' which you use depends on the gender of the noun that follows it, and also what the noun is doing in a sentence. In the sentence 'The dog bites the cat', the dog is doing the action, which makes it the subject of the sentence, whilst the cat is having the thing done to it, making it the object of the sentence. The dog, which is the do-er of the action, will be in the nominative case in German. So it is 'der Hund', whilst the cat, the object of the sentence, will take the accusative case, making it 'die Katze'. These are the two most important cases in German, so being able to identify who is doing the action and who is having it done to them are the most important parts when you're working out what articles to use in a sentence.

The other two forms an article (the word 'the') can take are the dative and the genetive. The dative is used to show that something is being done on behalf of another. So in the sentence 'I buy the book for the child', 'the child' takes the dative case, because that is who the action is being done for. This makes the neuter noun 'das Kind' into 'dem Kind'. Finally the genetive case, which is actually poorly understood by many native German speakers, indicates belonging. So if 'I buy the book for the child of the teacher', the 'teacher' is in the genetive case, because that is who the child belongs to. In the genetive you need to add the letter 's' to the end of the noun as well as change the article, so in this example, the noun 'der Lehrer' becomes 'des Lehrers'.

(For this sample answer I would supply the student with a table of definite articles divided by case and gender)