verteilen
to distribute
To hand something out to several people or bring it to different places.
Der Lehrer verteilt die korrigierten Klassenarbeiten an die Schüler.
The teacher distributes the corrected class tests to the students.

to distribute / spread
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to distribute
To hand something out to several people or bring it to different places.
Der Lehrer verteilt die korrigierten Klassenarbeiten an die Schüler.
The teacher distributes the corrected class tests to the students.

'Verteilen' (to distribute/spread) is an inseparable verb with the prefix 'ver-'. It is derived from the verb 'teilen' (to divide/share). The prefix 'ver-' here often describes the process of bringing something to different places or people.
Die Hilfsgüter wurden an die Bevölkerung verteilt.
The relief supplies were distributed to the population.
When you state who receives something, use the preposition 'an' with the accusative. The structure is: 'etwas an jemanden verteilen' (to distribute something to someone).
Die Flugblätter wurden an die Passanten verteilt.
The flyers were distributed to the passersby.
To describe the surface on which something is spread out, you use the prepositions 'auf' (on, with dative) or 'über' (over, with accusative).
Verteile die Sauce gleichmäßig auf der Pizza.
Spread the sauce evenly on the pizza.
The reflexive form 'sich verteilen' (to disperse/spread out) is often used for groups of people or things that spread out in a room or area. It describes movement from one point to many.
Nach der Veranstaltung verteilten sich die Besucher in den umliegenden Cafés.
After the event, the visitors dispersed into the nearby cafés.
to share / divide
'teilen' and 'verteilen' can both mean that something goes to several people or places. 'teilen' often emphasizes sharing something or making it into parts, for example 'einen Kuchen mit jemandem teilen' (to share a cake with someone). 'verteilen' emphasizes the action of giving something to several people or bringing it to different places. For example, you can 'Zettel in der Klasse verteilen' (hand out papers in class) or 'Creme auf der Haut verteilen' (spread cream on the skin). The words can be confused because when you distribute something, it is often divided first. If shared use is important, 'teilen' fits; if handing out or spreading out is important, 'verteilen' is a better fit.

Sie verteilt morgen die Einladungskarten an alle Nachbarn.
Tomorrow she will hand out the invitation cards to all the neighbors.
Heute teilst du dir das Zimmer mit deinem Bruder.
Today you are sharing the room with your brother.