The correct option is board / bored.

To determine the correct words for the blanks, we need to analyze the context of the sentence and the meanings of the homophones "board" and "bored".

1. Analyzing the First Blank:
The sentence describes an action performed "while teaching": "Writing too many things on the ________...".
In a classroom setting, a teacher writes on a physical surface, which is commonly referred to as a board (such as a blackboard or whiteboard). Therefore, "board" is the correct spelling for the first blank.

2. Analyzing the Second Blank:
The sentence describes the effect of writing too much on the students: "...could make the students get _________."
If a teacher writes excessively on the board, students are likely to lose interest, become disengaged, or feel weary. The adjective that describes this state of feeling is bored. Therefore, "bored" is the correct spelling for the second blank.

Conclusion:
Combining the two words in their respective positions gives us "board / bored", which makes the complete sentence: "Writing too many things on the board while teaching could make the students get bored."