Lori - a similar question recently.

My thoughts copied below. Substitute ‘ une brossé à dents ‘ for ‘ une boîte à bijoux ‘

“ Sara - when it is ‘something for something’ it is just  ‘ qqc à qqc ‘ without a definite article, so it remains ‘ … une boîte à bijoux ‘. Whether a plural or singular, feminine or masculine noun follows the preposition ‘ à ‘ is not relevant in this construct.  The noun here is compound ‘ une boîte à bijoux ‘ is the full noun form.

There is also no definite article used with ‘ qqc de/en qqc ’, indicating made of something.

 (There are some exceptions, of course. Some already noted in QandA under the lesson)

However, the definite article is included in the case of foods ‘made with or flavoured with’ - hence à la/à l’ or the contractions au/aux are used as appropriate to the following noun.

The lesson could be clearer describing this - in particular the example of ‘tarte aux pommes’ could be moved to the final section, rather than being in the introductory examples, and the use/non-use of definite articles could be specifically noted for clarity.

(It would be better to have all the examples in the lesson proper with their relevant section in my view, rather than clumped at the start) “

There's been a long discussion about une boîte à lettres and boîte aux lettres. It turns out that the latter form, boîte aux lettres, is the most commonly used one although both are deemed correct. So there is some inconsistency and no rule is perfect. It depends, I guess, whether you think of the mailbox as a box for a (generic) letter or a box for (all the) letters.