When Lyddie's mother decides to go to her sister's farm, what do Lyddie and Charlie do? What does this show about Lyddie's relationship with her brother?

The answer to this question can be found in Chapter 1. Early in the chapter, a bear enters the house that Lyddie and her family lives in. Lyddie bravely fends off the bear to keep her family safe; however, Lyddie's mother takes the event with the bear as a bad sign. She no longer feels safe in the house, and she wants her entire family to go to her sister's place. Lyddie is not on board with that idea because she and Charles are worried that their father will come home to find his family gone. 

"But how will Papa find us if we've left home?" Charles asked. 

Lyddie tells her mother that she is unwilling to go to the other farm. She and Charlie agree to stay behind, take care of the farm, and wait for the possible return of their father. 

"I can't stop you to go," Lyddie said, "but I can't go with you.  I can't leave the farm." 

Charlie leaves with the family for two weeks to make sure that they make it safely to Uncle Judah's farm, and then he returns. Lyddie and Charlie successfully take care of the farm, feed themselves, and birth a calf.  All of this shows that Lyddie and her brother are close with each other. They get along with each other, and they are capable of working together as a team. It also shows that they have a similar work ethic and emotional attachment to their home. Finally, it shows that Charlie and Lyddie have a lot of trust in each other. They are each essentially placing their welfare, health, and safety in the hands of the other sibling.