punkten
to score
To score points in a game, competition, or exam.
Die Mannschaft konnte in der zweiten Halbzeit noch zweimal punkten.
The team was able to score twice more in the second half.

to score / to impress
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to score
To score points in a game, competition, or exam.
Die Mannschaft konnte in der zweiten Halbzeit noch zweimal punkten.
The team was able to score twice more in the second half.

The verb 'punkten' (to score) comes directly from the noun 'der Punkt' (the point). Literally, it means to make or collect points.
In diesem Quiz kann man mit jeder richtigen Antwort punkten.
In this quiz, you can score points with every correct answer.
When you make a good impression, you 'punkten' (score points) 'bei' (with) a person 'mit' (with) a certain quality or action. So the structure is: punkten bei + dative + mit + dative.
Er hat bei seiner Chefin mit seiner Pünktlichkeit gepunktet.
He scored points with his boss through his punctuality.
The meaning 'einen guten Eindruck machen' (to make a good impression) is colloquial. In formal situations, you would usually choose other expressions.
Mit diesem Witz hast du bei der Gruppe echt gepunktet.
You really scored points with the group with that joke.
In the meaning 'Punkte erzielen' (to score points), the verb is used almost exclusively in the context of sports, games, or competitions.
Unsere Mannschaft muss in diesem Spiel unbedingt punkten, um die Meisterschaft zu gewinnen.
Our team absolutely has to score in this game in order to win the championship.
The meaning 'mit Punkten markieren' (to mark with dots) is very specific and is mainly used in crafts or technical contexts. It often appears in the passive.
Der Karton wird entlang der gepunkteten Linie gefaltet.
The cardboard is folded along the dotted line.
to win / gain
'gewinnen' and 'punkten' can both have to do with success in a game or competition. But 'gewinnen' means the overall success at the end: you are the winner. 'punkten' first means that you score points; so you can often score without winning the game. Colloquially, 'punkten' can also mean that you make a good impression or gain an advantage. Learners may confuse the words because both sound positive. As a practical rule: use 'gewinnen' for victory, and 'punkten' more for individual points or a good impression.

Mit ihrer ruhigen Art punktet ihr im Büro sofort bei den neuen Kollegen.
With her calm manner, she immediately scores points with the new colleagues at the office.
Mein Bruder möchte das Turnier gewinnen, weil er lange dafür trainiert hat.
My brother wants to win the tournament because he has trained for it for a long time.