Learn Hardcore Icelandic: Hann vinnur að áætluninni.

vinnur is the present tense of the verb vinna, which means “to work.” In Icelandic the simple present can translate both “works” and “is working.” So Hann vinnur að áætluninni means either He works on the plan or He is working on the plan.

Why is the preposition að used here, and what case does it take?

Here að is a preposition meaning “on” or “toward” in the sense of working on something. It always governs the dative case in Icelandic. That’s why the noun that follows—áætluninni—appears in the dative singular form.

How is the dative definite form áætluninni constructed from áætlun?

Icelandic marks definiteness with a suffix. For a feminine noun like áætlun (an -un stem), the definite dative singular ending is -inni. You take the stem áætlun-, add the article marker -in-, and then the dative ending -ni. Together they fuse as áætluninni.

Could I say Hann vinnur við áætlunina instead? What’s the difference between vinna að and vinna við?

Both can be used with “to work on,” but with a slight nuance. vinna að often focuses on working toward a goal or contributing to something (the plan in this case). emphasizes the activity or task itself (“to labour at the task”). In the context of “working on a plan,” is more idiomatic.