Writing too many things on the ________ while teaching could make the students get _________.

The correct option is board / bored.

Let us break down the sentence step-by-step to understand why this option is correct:
The sentence has two blanks that need to be filled with the correct homophones—words that sound the same but have different spellings and meanings.

1. Understanding the first blank:
The context of the sentence is "writing too many things on the ________ while teaching". In a classroom setting, a teacher writes on a physical writing surface. The noun for this writing surface is board (such as a blackboard or whiteboard).
Therefore, the first blank must be board.

2. Understanding the second blank:
The context of the sentence is "could make the students get _________." This describes the feeling or mental state of the students when too much information is written. The adjective that describes feeling weary or uninterested due to lack of dullness or tedious repetition is bored.
Therefore, the second blank must be bored.

Combining the two words in order gives board / bored, which matches the correct option.