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To show that they did not acknowledge the Confederacy as a separate nation.
Clearly, it wasn't. The fort wasn't even fully built when it was attacked, it was basically unmanned and unarmed. On the other hand, South Carolina needed to protect Charleston and to keep the harbor open. When the rebellion started, South Carolina and Georgia were the only states on the east coast in revolt and they needed open trade routes across the Atlantic. Charleston and Savannah were the only decent ports available. As Grant and Sherman would eventually prove, Sumter was a trinket not worth fighting for. In the actual battle, there was one death on each side. The confederate was killed by his own cannon when it misfired and the federal was killed during a 100 shot salute by union troops after their surrender. The Confederates had allowed the ceremony although why the Union troops wanted to conduct it is beyond me. In any case, it was shortened to 50 shots, probably out of fear of more casualties. One Union soldier actually died as a result of the battle itself. Sumter was in Confederate hands until Sherman took it in 1865, two months before Lee's surrender. Now, if your teacher is really asking why the union thought it was important to hold Sumter, you can just say that Lincoln wasn't about to order his men to cut and run just because some terrorist traitors told them to get out.