Is this one CORRECT: Sarah lost her camera while skiing to the base?
This is incorrect.It gives an impression that camera and not sarah was skiing to the base.
The correct version is - While skiing to the base, Sarah lost her camera.
Reason- while skiing to the base is an adjectival modifier that modified the person/thing that was skiing to the base. The rule of noun modifier (adjective can modify only nouns) is that the adjective should touch the noun.The only exception to this is a participle phrase (set off by commas) at the end of the sentence.In all other cases even participle phrase must touch the noun.
For eg: I saw a man driving a car ( driving a car is participle phrase that modifies the nearest noun, which is man).
Driving a car, I saw a man. ( In this case Driving a car modifies I not man)
I saw a man, driving a car. (here driving a car can modify anything- either man or I because it is at the end of the sentence and set off by commas.Therefore, here driving a car is ambiguous and hence incorrect.)
Since I have cleared out the usage of participle phrase, it will also make it clear why "Sarah lost her camera skiing to the base" is incorrect.
There are 2 corrections to this
Sarah lost her camera,skiing to the base.
Skiing to the base,Sarah lost her camera skiing to the base.
I sent a card thanking my eighth-grade English teacher for teaching me about grammar and inspiring me to write.
This again is wrong.
"thanking my eight-grade English.... to write" teacher is a participle phrase not set off by commas at the end of the sentence.The usage as such implies that the card (nearest noun) was thanking eight-grade English teacher and not you!
Correction and a more precise one-
To thank my eight-grade English teacher for teaching me about grammar and inspiring me to write, I sent a card.
( see the entire noun modifier touches I and not card).
Thanking my eight-grade English for teaching me about grammar and inspiring me to write, I sent a card
Writing "thanking my eight-grade English.... to write" cannot be used even with commas because it is not idiomatic and logical.
We use participles such as thanking, causing ... to show the effect of something ( cause-effect relationship)
For eg. The stock market crashed, causing panic among investors.
This sentence means that the crash of the stock market made investors panicky ( effect of the crash)
We use infinitives such as to thank, to cause etc to show intent of someone.
The stock market crashed, to cause panic among investors.
This sentence as such is illogical because the doer of action " the people/news etc who caused the crash is not present".The more logical sentence is
The big banks caused crash in the stock markets, to cause panic among investors.
or
To cause panic among investors,the big banks caused crash in the stock markets.
This sentence shows the intent of someone (the banks) for doing something (causing the crash in the stock markets).
Back to the example- the person who sent a card intended to thank the other person.Therefore, use the infinitive.Also one last point " To thank....to write" is a prepositional phrase and can be adverbial as well as adjectival modifier. Here,however, it is an adjectival modifier, which modifies I.
