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...
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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, theybegin 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).