How to keep text messages from going to my old iphone?

I recently upgraded to iphone 6 from iphone 4, and for some reason some of my friends (android and iphone users) couldn't seem to text or call me. Well I charged my old iphone and turned it on for my music and saw that all my missing texts and calls were on that phone and I tried everything to switch it off or deactivate it but I couldn't figure it out. Then I dropped the old phone on the concrete with no case and it broke into three pieces now i'm really stuck. Is there any way to fix this?

Y- you need to take the SIM card out of the old phone, and put it into the new phone

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