Questions & Answers about Я хочу написать письмо своим новым карандашом.
Why is the pencil in the Instrumental case here? There is no preposition like с (with).
In Russian, the Instrumental case is used on its own (without a preposition) to show the tool or instrument used to do an action. So, карандашом means "by means of a pencil" or "using a pencil". If you added the preposition с, it would sound like you and your pencil are writing the letter together as a team!
How do the Instrumental endings for своим новым work?
Since карандаш (pencil) is a masculine noun, the words describing it also need to be in the masculine Instrumental case. Regular masculine adjectives take the ending -ым (making новым), while possessive pronouns take -им (making своим).
Why does the sentence use своим instead of моим for "my"?
Свой is a special pronoun that means "one's own". In Russian, whenever the owner of the object is also the main subject of the sentence (here, Я is the subject), it is much more natural to use a form of свой. Using моим wouldn't be completely wrong, but своим is the standard way native speakers express this.
Why use написать instead of the imperfective verb писать?