What are three things Kit does for Prudence in The Witch of Blackbird Pond?

The first time Kit helps Prudence is on the boat trip from Saybrook to Weathersfield.  Prudence drops her doll overboard, Captain Eaton will not turn back for it, and Prudence's mother only gives her a scolding.  Taking things into her own hands, Kit leaps overboard and retrieves the child's toy, not realizing that, in Puritan New England, for a girl to swim is scandalous.

Another way Kit helps Prudence is in teaching her to read.  Considered worthless and stupid by her mother in particular, Prudence is not allowed to attend the Dame School with the other children.  Kit invites Prudence to meet at Hannah Tupper's house, and, discovering her to be an apt pupil, teaches her to read and write.

The most significant way in which Kit helps Prudence is through the kindness and friendship she extends towards the little girl.  Prudence is starved for attention, left mainly to fend for herself by her overbearing and self-righteous mother and her silent father.  When Kit takes an interest in her, Prudence blossoms into a lovely and capable little girl and develops a sense of self-worth which has always been denied her.  She is even able, near the end of the book, to stand confidently in a courtroom before a judge to speak in defense of her mentor.

What did Prudence do to get Kit's attention at dame school in The Witch of Blackbird Pond?

Prudence follows Kit to the dame school and leaves bunches of flowers, "buttercups and wild geraniums", on the doorstep.  The third time she does this, Kit catches "a sudden motion" at the open doorway out of the corner of her eye.  She goes outside, and as she is picking up the bouquet, spies "a shadowy figure" among the trees whom she recognizes as little Prudence Cruff.

Kit had first met Prudence on the boat coming up the Connecticut River.  Prudence had dropped a precious doll into the water, and Kit had leaped into the river to rescue it for her, to the shock of all present.  Since then, Kit has not seen much of Prudence.  Although the little girl basically runs wild, because "nobody cares where (she) go(es), just so's (she) get(s) the work (at home) done", she is not allowed to attend the dame school with the other children because her mother thinks she is "too stupid" to learn.

Despite her mother's disapproval, Prudence longs to go to school.  She often stands outside the door listening when classes are going on, and thinks she "could learn as good as (the others)".  Although Kit invites her in to give the lessons a try, Prudence refuses, because she thinks someone will tell on her and then she will be in trouble with her mother.  Kit then arranges to teach the child herself at Hannah's cottage in the Meadows, where they can meet in secret, away from others who might report Prudence's activities to her mother (Chapter 11).