Welcome!
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I have questions like how much of this do I share with my son and when should I share this information?
How much to share may vary greatly by family and even by child. The time to share is often when the child asks directly or indirectly (for example: expresses curiosity about metacognition, observes and comments on being different, wonders about intelligence, etc.) Some books which may be helpful for a child processing this information include The Survival Guide for Gifted Kids and 101 Success Secrets for Gifted Kids. Parents may wish to become familiar with any book prior to passing it along to their DC. A previous thread on this topic here.

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Is it rude or boastful to share this information with other parents?
Unfortunately, it may be taken that way. If others begin to seem overly curious for a number, an effective way to deflect can be nonchalantly mentioning that
- IQ scores occur in a range
- IQ score equivalencies may vary from test to test.

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How do I handle his constant need to debate with me, super sensitivity and extreme competitiveness?
Commiseration on forums can be helpful, as can reading books such as A Parent's Guide to Gifted Children, and Living with Intensity, and Emotional Intensity in Gifted Students. Kids may like the book Understanding Myself or Emotions! or How Do You Doodle? or The Boy Who Didn't Want to be Sad from Magination Press.

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He's thriving in public school
While a gifted kiddo is thriving, enjoy this time to read all you can. Although each child is unique and each family has a unique journey, you may wish to become familiar with what you may reasonably expect to encounter on the path ahead.

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should I be looking to enroll him in some type of giftedness program?
Many public schools may offer some type of gifted programming. You may wish to become familiar with your State policies and also your local school district policies.

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What if I put it in a gifted school and he doesn't perform well?
Depending upon the myriad issues which may be involved, you may choose to advocate or you may decide it was not the right "fit" and refine your criteria for your next search for a learning environment in which your child will thrive... best fit or least-worst fit.