The primary Montessori fraction manipulative is a set of fraction circles, showing 1, and the fractions 1/2-1/10. 

The first thing children learn, is the names of the fractions, and the work they do to learn those names is to trace the fractions from the set, and write the names next to them.

The next thing they do is to add fractions where the denominators are the same.  These sums are generally more than 1, so children are learning how to think of fractions as repeated units (so it's OK to be bigger than 1) and they are learning about converting to mixed numbers.  Another thing they are doing is using manipulatives to simplify the fractions in the answers.  As a teacher for this sort of assignment, you need to have enough manipulatives for children to do all of the problems you give (they shouldn't need more fourths at a time than there are in the manipulative set they have available), and there should be a reasonable variety of problems that do or don't need to be converted to mixed numbers, and do or don't need to be simplified.

Next students address equivalent fractions.  This should be done both ways: taking a fraction and writing many equivalent fractions for it, and taking a complex fraction and simplifying it.  There is only a limited amount of work that can be done with the fraction circles and pre-made fraction manipulatives.  After an introductory set with carefully chosen problems that is done with the fraction circles, children do the same work with some sort of a rectangle or square model.  My favorite version of this uses folding paper squares (not a Montessori lesson, but very good).  The most flexible student-made  ways of doing and showing this is using rectangle models that are drawn on grid paper. 

Next, students learn to add fractions with unlike denominators.  There are two manipulative stages to this work.  The first is done with fraction circles, so the teacher needs to be careful to give only problems where the common-denominator fraction is available.  The second is done with either folding paper squares or with rectangles drawn on grid paper.

After adding, students study subtraction, multiplication and division.  Each of these is learned first with fraction circles, and later with grids of some sort.

Sharing problems that result in fractional parts. (If 5 children share 3 brownies, how much does each child get?)

Measuring problems that result in fractional amounts (This paper tape shows the length of my arm span.  How many feet long is it?)

Adding problems in well understood contexts (If daddy at 1/2 of a pie on Monday, and 2/3 of a pie on Tuesday, how much pie did Daddy eat?)

Subtraction problems (more on this next week)

Multiplication problems (that can be reduced to a statement like 1/2 of 3/4)

Division problems: ones you find in US resources are usually measurement (The vet gave me 9 large pills.  My puppy should take 2/3 of a pill per day.  For how many days should he take the pills?)

Division problems in sources from countries like Japan or China are often partition division, sometimes with rates (If I walk 2/3 of a mile in 1/2 an hour, how fast am I going?)