New cases are either "New" from the standpoint of A.) the date they were added as a case (ie. entry date); or B.) from the standpoint of the date of their first assistance (ie. assistance date) (which can also be relative to the assistance category being used to filter the report.)

A.) In a "Cases" report, newly added cases (regardless of any assistance given) for a given date range can be determined by using the "Filter by Entry Date". See screenshot here.

B.) In an "Assistance" report, the Report Summary: New Case Count = the number of cases that have never been assisted before the start date of the report. See screenshot here.
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The same applies to the Report Summary: New Household count = the number of households that have never been assisted before the start date of the report. See screenshot here.

If a household is split, and a new household with a new street address is formed, it will only be considered as a "New Household" if it meets the criterion of "a household never assisted before the start date of the assistance report". Creating a "new" street address does not automatically qualify it as a "new" household.