The dramatic structure of this act is beautifully arched.
It is obvious that the entire household is in on Petruchio's plan on how to "tame" Kate. In his soliloquy at the end of the first scene, he tells us his plan. He compares taming her to taming a falcon. Nothing will be good enough for her. He will find fault with everything from food to the bed. He concludes by telling us that, "This a way to kill a wife with kindness,/And thus I'll curb her mad and headstrong humour..."
The succeeding scenes show us his plan in action. Nothing is too good for her. He begins to wear her down.
In the final scene of the act, aka the sun/moon scene, Kathrine finally catches on and brings her own twist. When she approaches old Vincentio and greets him with, "Young budding virgin,..." she and Petruchio begin to operate on the same page. Between her "shrewishness" and his "madness" they have a lot of fun with the citizens of Padua.
By the end of the act and into Act V, Kate and Petruchio are working together like a well oiled machine. They are truly meant for each other.
By the end Act III of The Taming of the Shrew, it's becoming apparent that Petruchio's plan to tame Kate just might work--or at least he's going to die trying. By the end of Act IV, however, it's clear that Kate is onto his plan and has enacted a plan of her own.
Old habits are hard to break, of course, and she has plenty of shrewish reactions to how Petruchio is treating her. Once she has endured the food deprivation, the sight of her beautiful new clothes being torn assunder right in front of her, and being denied the chance to visit her father, Katherine has figured it out--if she agrees with him she will get what she wants. This lesson culminates in the famous "sun/moon" speech on the road to Padua and sets the stage for the final dramatic bet at Baptista's home.
The narrative becomes clear--the shrew has been tamed...by her choice.