In Jeanne DuPrau's The City of Ember, Lina finding what's left of the piece of paper and deciphering some of its meaning advances the plot by leading towards the climax of the story. Likewise, Doon agreeing with her that the instructions look important and joining with her in the search for the clues in the Pipeworks that the paper points to further leads to the story's climax, the climax being Linan and Doon finding the way out of Ember the Builders had intended, in order to save the rest of Ember's people. All of their actions in finding the paper and working towards deciphering it count as rising action.
In chapter 7, Lina glues together what piece of the paper she has and begins studying it in her spare time. Once she recognizes the message speaks of a "river, a door, and the Pipeworks," she decides to show the paper to Doon since Doon is taking the problems of the city so seriously, and, so far, no one has taken what Lina has found seriously. When Doon examines the fascinating box and the paper, he very quickly agrees that the paper looks important. He is puzzled about the door, though, since he recently found a mysterious locked door, but it wasn't near the river. Together, they begin exploring, which counts as more rising action.
Together, they go to look at the locked door Doon found in the Pipeworks. Later, another moment of rising action occurs when Doon returns to find the key in the door and opens it to discover the mayor is hording supplies for himself in the room. At that point, Doon starts feeling discouraged and begins thinking the instructions aren't useful after all. However, when Lina learns from her friend Clary that the full title of the paper is "Instructions for Egress," Doon's excitement returns, and in chapter 13, he returns with Lina to her home to work out some more of the puzzle, leading the story closer to its climax. Together, they decipher that somewhere in the Pipeworks, they'll find a "stone marked with E" and further "down the river bank," they'll find a ledge where the door is near, the door they presume leads out of the city (Ch. 13).