(This is where the inset play begins. We're still in Sly's bedroom at the Lord's house, which is apparently big enough for a live theater performance. Sly and his "wife" watch the play from a lofted space above the stage.)
The inset play opens on a street in Padua, where Lucentio and his trusty servant Tranio have just arrived. While shooting the breeze, Lucentio reveals that his dad has sent him to the famous college town so Lucentio can round off his education. Tranio reminds Lucentio that studying philosophy is all good and well, but they need to have a little fun with the ladies, too. Lucentio agrees.
Just then, Lucentio and Tranio spot Baptista, and his gorgeous daughters (Bianca and Kate) talking with Bianca's suitors. Lucentio and Tranio pause so that they can eavesdrop.
Baptista tells Gremio and Hortensio to stop begging him for permission to marry Bianca, who isn't allowed to get hitched until Kate marries. But, if they like, they're welcome to court Kate and take her off Baptista's hands.
Gremio scoffs at this and says he would rather "cart" Kate than marry her.
(We interrupt this program for a history snack. "Carting" refers to the way "shrewish" women were legally punished in 16th-century England for being disobedient and mouthy. Convicted "scolds" were sometimes tied up and strapped to the back of a cart so they could be paraded around town and publicly shamed into keeping their mouths shut.)
Kate yells at her dad and says he's out of line – he's making her a laughing stock and treating her like a prostitute in front of these two chumps. When Hortensio interjects and tells Kate she'll never be married unless she learns to pipe down, Kate gives him a tongue-lashing. This involves Kate talking about herself in the third person and denying she's interested in marriage.
The ever-observant Tranio says Kate is crazy and way too mouthy. Lucentio is hot for Bianca because she is nice and quiet, which is just how he likes his women.
Meanwhile, Kate picks on Bianca. Bianca plays the good girl and tells her dad she'll spend all her time studying until he says it's time for her to get engaged.
Gremio and Hortensio complain that Baptista is keeping Bianca penned up like an animal, but Baptista holds his ground. Before he leaves he asks the suitors if they can recommend any teachers for his precious daughter, a hint that they should rustle up some good tutors if they want to make him happy.
Baptista says he's got to run and needs to talk with Bianca, but Kate is free to hang out with the guys if she likes. Kate gets huffy about being told what she can do and storms off.
The suitors agree together that they need to find someone to marry Kate so they can have free access to Bianca. So they take off to hatch a plan, leaving nosey Lucentio and Tranio to discuss what's just happened.
Lucentio is love struck and starts to say all kinds of cheesy things about how he burns for Bianca. He asks Tranio for some advice, so the servant tells him to quit his yammering and think of a plan already.
Tranio and Lucentio decide that Lucentio will disguise himself as a teacher so he can give Bianca some "private tutoring."
Since Lucentio is expected to be seen in Padua studying and schmoozing all his dad's rich friends, they decide that Tranio will disguise himself as Lucentio.
They exchange clothes: now Lucentio is disguised as a teacher named Cambio and Tranio is disguised as Lucentio.
Along comes Biondello (another one of Lucentio's servants). Lucentio and Tranio tell Biondello that they're in disguise because Lucentio killed a man and needs to flee the city in case there are witnesses who can identify him. They explain that Tranio will pretend to be Lucentio in order to keep up appearances. Biondello buys their story and they exit the stage.
One of the Lord's servants (remember the Induction scenes at the beginning?) who has been watching the inset play with Sly and Bart, asks Sly why he has dozed off. Sly denies he fell asleep and says he likes the play.
(Note: Nobody knows why, but this is the last time we hear from Sly in The Taming of the Shrew. We can assume he remains on stage – perhaps he's fallen asleep again or there are lost lines and scenes that don't appear in Shakespeare's manuscripts. In a closely related play called The Taming of a Shrew, which was written and performed around the time of Shakespeare's play, Sly's character speaks throughout the performance, commenting on the play as it goes along. At the end of the performance, Sly runs off to tame his own wife. Some directors and editors like to include these scenes, especially the ending, even though there's no evidence Shakespeare wrote them. Other directors decide to cut Sly out of the Induction altogether.)