1. I went to school by bike.
2. I went to school on a bike.
3. I rode a bike to school.
3-2. I rode a bike to go to school.
4. I took a bike to school.
5. I went to school by my brother's bike.
[Which ones are grammatical, and which ones are unacceptable?]
All except 3-2 are fine.
3-2 is wordy; you don't need "to go."
5 is awkward, but basically all right. Using "on" instead of "by" would be better.
All of the sentences you provided are grammatically correct. However, some of them may sound more natural or commonly used than others. Here is an analysis of each sentence:
1. "I went to school by bike." - This sentence is grammatically correct and commonly used. It indicates that you used a bike as your means of transportation to go to school.
2. "I went to school on a bike." - This sentence is also grammatically correct. It conveys the same meaning as the first sentence, but it may be less commonly used.
3. "I rode a bike to school." - This sentence is grammatically correct and commonly used. It clearly states that you used a bike for transportation when going to school.
3-2. "I rode a bike to go to school." - This sentence is grammatically correct, but it may sound slightly less natural or commonly used compared to the previous options. However, it still conveys the intended meaning of using a bike for transportation to reach school.
4. "I took a bike to school." - This sentence is grammatically correct and commonly used. It suggests that you chose to use a bike as your mode of transportation to reach school.
5. "I went to school by my brother's bike." - This sentence is grammatically correct, but it may sound a bit less natural than the previous options. It indicates that you used your brother's bike as your means of transportation to go to school.