Hi Dave,
Are you aware of categories in the gradebook? You could divide yours up into the 4 marking periods (1 period = 1 category). If each category builds on each other to form a total course grade, then set the course aggregation to be Mean of Grades, or Simple Weighted Mean. However, if you want the course total to only reflect the total for the current grading period, set the aggregation to Weighted Mean of Grades, then set the current period category to be worth 100% of the course, setting the other (non-current) periods to be worth 0%. You would have to fix the weightings on the roll-over of each grading period, always making the current period worth 100% of the course grade.
Hope this helps.
Michael