Possible Answers

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Chapter One
If Percy didn't have the pen to vaporize Mrs. Dodds, I don't think that he would have found another way to vaporize her because she is a monster, and has special features like talons to kill him, while Percy doesn't have any special features yet.

Chapter Two
I think that Grover didn't tell Percy what he remembered about Mrs.Dodds was true because Percy didn't know that he was a Half-Blood, and would think that Grover is crazy.   

Chapter Three
Sally Jackson's parents died in a plane crash when she was five years old, and her uncle raised her after that. 

Chapter Four
I think Gabe really does care for his son deep down inside.  He might be demanding and rude, but every parent loves their kid, and when Percy goes on his quest, Gabe tells the news to locate his son.

Chapter Five
Characteristics that make Chiron has to make himself a good camp leader is that he is very wise, smart, and encouraging.  He is wise because he disguised himself very well at Yancy Academy ( Percy's old school), smart because he knows what to help each student with at Camp Half Blood, and encouraging because he encouraged Percy with his quest to receive Zeus's master lightning bolt. 

Chapter Six
I do think that Clarisse inherited some characteristics from her father, the god of war, because she is very aggresive, competitive, and very strong just like her father.

Chapter Seven
I think it is undetermined that the camp does not know who Percy's father is for two reasons:  
1. He is new to the camp so the people at the camp do not know him enough.     2. The god who is his father, is not known to have any kids.

Chapter Eight
I do think Percy's excelling in sports has something to do with the fact that Percy is Poseidon't son. Example:  Just like if your mother has great gardening skills, you probably will too.

Chapter Nine
I think Zeus accused Poseidon of stealing his Master Lightning Bolt because at the time, the two brothers were in a fight.  Naturally when two people are in a fight, one person thinks that the other person is guilty of everything and vice-versa. 

Chapter Ten
I think Luke perishs Percy so he doesn't get suspicious about him being evil.  I think he doesn't assassinate Percy at the beginning so he could build a line of trust, and Percy would listen to Luke if they had that trust.

Chapter Eleven
The name of the restaurant the kids ate in was Aunty Em's Garden Gnome Emporium, the name of the monster they met was Medusa, and her hands were gnarled and warty, with sharp bronze talons.

Chapter Twelve
I think that Grover wants to be  a searcher still because he wants to try to be the first searcher who came back alive, and earn a reputation, even if it risks his life.

Chapter Thirteen
If I were in the place of the Enchida, I would do exactly what Percy did.  Take out Riptide, and start to fight her.  I would jump in the water to prove my bloodline because I have the gods to save me, and I would be the God of the Sea!

Chapter Fourteen
I would like to have the power of going in the water and not be soaked when I come out of the water.  It would help with my daily life because at the wonderful amusement parks I go to, there is always a water park, and I could never go because I would be soaked when I came out. 

Chapter Fifteen
I would be VERY frightened if the god of wars was sitting with me because he could kill you right there!  I would always be conscious of what I say to him. 

Chapter Sixteen
The ride Ares provided for Percy and his friends was a truck that said Kindness: International Humane Zoo Transport. Warning: Live Wild Animals.  The gift Ares gave in return for the favor was a blue nylon backpack.  The backpack had fresh clothes for the kids, twenty dollars in cash, golden drachmas, and Double Stuff Oreos.

Chapter Seventeen
I think the infinity symbol comes up when Annabeth hands the cab driver the green LotusCash card because they are the god's kids, and the gods support them all of the way.

Chapter Eighteen
I think that if the gods really wanted the Underworld protected, placed an animal which could be lured with a rubber ball because they never thought of who could figure out that the animal likes rubber balls because the animal would probably kill them first.

Chapter Nineteen
Percy didn't realize the Master Lightning  Bolt was in his backpack because when Ares gave him the backpack, the god put a spell on it that it would not place the Lightning Bolt in the backpack until Percy reaches the Underworld. 

Chapter Twenty
Athena is the god of wisdom.

Chapter Twenty-One
Yes, I do think Hades gave Percy's mother back because he retrieved his Helm of Darkness. Percy gave something back to him, so he would give something in return.

Chapter Twenty-Two
If I were in the place of Percy, and a scorpion was about to attack me, I would run away as far as I could, and run to the camp leader Chiron.  I would also use my powers that I had because I would be Poseidon's son.