It seems like someone copied the language wrong. As written, this should say something less than the monthly rate unless the move-in date is the first of the month. A standard language would say something like "the prorated portion of the monthly rent of $1200."
I'd change it to the correct number before signing the agreement. If the landlord wants to cross out the "monthly", let them. On the check, I'd put "Prorated rent for _______" and replace the blank with the move-in month and year. E.g. "Prorated rent for June 2016".
If you are the landlord, consider changing to something like
For the period from Tenant's move-in date, _________, through the end of the month, Tenant will pay to Landlord $_________, the prorated portion of the monthly rent of $1200.
I am not a lawyer. If you want to be sure, hire a lawyer to review the agreement. If you cannot afford a lawyer, there may be services available to help you. Often the local bar association can give you a referral (either to a regular lawyer or to a subsidized service).