Now, what sparks my interest is when you told Dad not to "touch" you.
My guess is that Dad has NOT been inappropriately touching you and that you just included the word "touch" in your letter to him to enhance your theme of "Leave me alone in all ways", right?? He's not a bad touching person right?
OR...because there are some households where dads do do bad touching I ask you this--just to be safe...because you DO meniton "he has a super HOT temper and has taken it out on you and your sister"...so to be safe, I ask you..this.==>
Has your Father been touching you in a way that is not right?
( a hug or pat on shoulder or to be just nice is fine...but perverted, threatening or cruel is unacceptable--and so if he has been touching you in a NOT NICE way, you need to tell your Mom and if she does not listen then tell a teacher, school councilor or a policeman...IF he has been touching you in a bad way that makes you feel unsafe/threatened or creepy--but REMEMBER:
It is NEVER OK to make up lies about your Dad so, never EVER tell something that is not true about Dad or anybody ok? (You could get in a lot of trouble and ruin a person's life--so only tell if it is true.)
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To answer your question now:
OK-assuming he's basically a nice guy but with some unresolved anger issues,
You need to realize that people who tend to be like pressure cookers, getting all mad and ready to blow up often have some self esteem issues.
When Dad said,
"Okay son, whatever makes you happy"...you can bet your bottom dollar that Dad said that with a lot of hurt and feeling like a failure and rather unloved...so please try to go easy on him...he doesn't mean that. He wants to be there for you and he wants YOU to want and need him too. He's only Human and he has feelings too...(It's hard being a parent--we're all still learning.)
In otherwords, maybe dad feels a bit frustrated when he tries to be a good dad and then feels like a failure or like the world is insulting him.
Anyhow-he's the dad and you're the child so he should really try to keep his cool.
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OK-How to apologize:
Do it face - to - face so that the problem is cleared up immediately.
Also, he's going to have a lot of respect for you being a big boy and facing him--and then he can see the sincerity in your eyes...and you can see also his sincerity.
Make sure to give Dad a big hug when you're done having your say and remind him that you know you don't show it often but you want him to know that you love him and look up to him as the greatest man and example setter in your life and that you relly want your relationship to work out with him.
Try not to discuss the details about what the fight was about--this isn't about just homework...those homework details are actually periferal--although they are important--you have to do your homework in a mature fashion...BUT, for now, the issue is more about how to communicate without being frustrated and misdirecting your anger--both you and your dad...you need to realize that--even if dad is still learning the very DIFFICULT and frustrating job of how to be a Dad, you too are learning and going through all sorts of chemical changes in your body and dealing with the pressures of school and of your parent's expectations...and you too need NOT to feel like a failure...you're not a failure. You're just learning...and so is Dad...and you're 2 sensative guys who get frustrated sometimes in expressing your emotions. It's common and I don't think your family is a mess or anything. Surely not one that can't be healed...you're all gonna be ok.
But remember--don't opt to email him..do it face to face...like the brave and sincere young man who is you...compassionate and open to suggestions.
Also, in emails, there's too much chance of misinterpreting communication...you need immediate results and open discussion...so please, no letters, no emails...but face to face like a strong young man.
If you ever feel like you are going to blow up in anger, take a deep breath, close your eyes and count to TEN...this is how to relax and gain perspective.
If dad asks why you are closing your eyes. tell him you are just composing yourself so you don't say or react in a way you will regret--. He can learn from your demonstrated technique.
As a child, you should just tell dad tht you are sorry you blew up and that you did that because you were frustrated and took it out on him.,
Acknowlege that you realise he was just trying to help you and that --in the tenison and frustration of it all--and you understand that words were exchanged which do not accurately reflect how you both feel about each other.
Tell him you appreciate his time and energy to help you out with the homework and that you respect him as a human being...and that it hurt you when he called you that bad name...and my advice to you is to sit and listen to others who are trying to help you...and not be impatient or so quick to act like a know it all...telling him to hurry up and give you the answer--because the answer is no good to you if you do not understand how one came to arrive at that answer...so give your dad credit....that he's been on the planet a little longer than you have and just maybe--he's therefore learned a few things and can be a useful person to teach you a thing or two.
Remember when he does help you--to thank him and praise him as being one who you feel admiration for...as parents, we can feel frustrated too...so just let him know he's on the right track as a parent...and all of this angst will blow over.
Just say,
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"Dad, I apologize for the harsh words we exchanged. I don't really want you out of my life. I value you., It's not so easy to be a teenager/young lad...and sometimes my emotions blow up...but i will try in the future to be more considerate of your feelings as a human being and I promise to show more respecta nd patience when you are trying your best to help me.
I love you and am proud to be your son."
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We've all been there kid...don't sweat it too much.
Always best to clear the air in any dispute with a sincere apology--and remember to keep your promise to try to be more patient.
(You can actually teach him about patience if you demonstrate it yourself.)