rechnen
to calculate
To solve mathematical problems or perform calculations.
In der Grundschule lernen die Kinder lesen, schreiben und rechnen.
In elementary school, children learn to read, write, and calculate.

to calculate / to expect
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to calculate
To solve mathematical problems or perform calculations.
In der Grundschule lernen die Kinder lesen, schreiben und rechnen.
In elementary school, children learn to read, write, and calculate.

The verb 'rechnen' changes its meaning fundamentally depending on whether it is used on its own, reflexively, or with a preposition. Always pay attention to the words that directly follow 'rechnen' in order to understand the correct meaning.
Kannst du das im Kopf rechnen?
Can you calculate that in your head?
The expression 'rechnen mit' means 'erwarten' (to expect) or 'voraussehen' (to foresee). The preposition 'mit' always takes the dative.
Die Polizei rechnet mit vielen Demonstranten in der Stadt.
The police expect many demonstrators in the city.
The reflexive form 'sich rechnen' means that something is profitable or worthwhile. The subject of the sentence is the thing that brings the benefit.
Es rechnet sich nicht, für diese kurze Strecke ein Taxi zu nehmen.
It isn't worth taking a taxi for this short distance.
The construction 'jemanden/etwas zu etwas rechnen' means to consider someone or something part of a group. The preposition 'zu' requires the dative here.
Man rechnet dieses Restaurant zu den besten der Stadt.
This restaurant is counted among the best in the city.
to hope
'hoffen' and 'rechnen mit' can both talk about the future, but they do not express the same feeling. 'hoffen' means that you wish for a positive result. 'mit etwas rechnen' means that you expect something to happen, often because it is likely. The result does not have to be positive. The words are easy to confuse because both have to do with uncertain situations. If the wish is important, 'hoffen' fits. If the assessment or probability is important, 'mit etwas rechnen' fits.

Wegen der Baustelle rechnet er am Montag mit Verspätung.
Because of the construction site, he expects delays on Monday.
Vor der Reise hoffen alle auf gutes Wetter.
Before the trip, everyone is hoping for good weather.