Bueller さん,
Thanks for your explanations and for correcting my mistake. I should have known かける is transitive…
On the other hand, I do not fully agree with your recommendation to use kanji here. This was a grammar point, furthermore at a beginner's level, so, from my own experience as a beginner myself, it confuses me a bit when I need to struggle with unknown kanji in someone else's post.
If you are fluent, it may seem awkward to you to read simple words written in kana, but the idea was to make it easy for everybody to read my question, and, more interesting, other people's answers.
I guess someone posting on a topic at a higher level might choose to use kanji.
Just IMHO !