By the end of the book Chains by Laurie Halse Anderson, how has Isabel’s sense of her identity changed?

By the end of the book, Isabel has become a proactive and confident girl who takes control of her life and her freedom.

At the end of the book.  Isabel is no longer somebody's slave (literally and figuratively).  Technically, that is how the book begins too.  When Miss Mary Finch died, she gave Isabel and Ruth their freedom.  Unfortunately, Mr. Robert Finch does not honor that agreement, and he quickly sells Isabel and Ruth to the Locktons.  

The Locktons are incredibly harsh slave owners.  They mistreat both Isabel and Ruth.  Isabel is practically worked into the ground, and Madam Lockton is unhappy with anything that Isabel does.  At one point, Madam Lockton even has the letter "I" branded on Isabel's cheek.  

The entire time that Isabel is working for the Locktons, she dreams of freedom; however, Isabel has no clear idea of how to achieve it.  She desires freedom, but she isn't willing to take any drastic steps to achieve it.  Even when Curzon dangles the possibility in front of Isabel, she turns him down because it is too dangerous.  

Isabel does eventually work up the courage to begin spying on the Locktons, but she isn't only hoping that the Patriots grant her freedom.  She wants her freedom, and she doesn't care where it comes from.  At one point in the book, Isabel seeks to help the British because they offer her a chance to be free as well.  

The branded letter "I" on Isabel's cheek is symbolically the biggest indicator of how Isabel changes from the beginning of the novel to the end. Madam Lockton tells Isabel that the mark stands for "Insolence," and for most of the novel, that is how Isabel considers herself.  Near the novel's close, though, Isabel remembers that her father had a similar mark; however, her father treated his mark as a badge of courage, strength, and distinction.  Isabel decides that this is what she must do, and she decides that the "I" is her mark of distinction.  It now stands for "Isabel."  

This is my country mark.  I did not ask for it, but I would carry it as Poppa carried his.  It made me his daughter.  It made me strong.

I took a step back, seeing near my whole self in the mirror. I pushed back my shoulders and raised my chin, my back straight as an arrow.  

This mark stands for Isabel.

With her newfound attitude about herself, Isabel digs deep and finds a new inner courage and strength.  She will no longer sit back and take the Lockton's abuse.  She will no longer wait for freedom to come to her.  Isabel will run away from the Locktons and take her own freedom.  In order to do this, Isabel steals a pass, rescues Curzon, and rows herself and Curzon to freedom.   Isabel will be in charge of her life from now on.  Nobody else.