My iPhone 6 sometimes acts like someone is remotely controlling it

I've had my iPhone 6 since about December. For a few weeks now, when I'm in Safari, it will randomly open a new window and start doing stuff on its own. Tonight it got so weird that it kinda scared me. I walked over to pick my phone up off the table, and it was on the calculator screen and looked like someone was doing their algebra homework with it. Numbers were coming up and it was adding and subtracting, but no one was touching the phone. I closed everything and turned off the phone. About 5 minutes later I turned my phone back on, and went to my settings to turn off Bluetooth in case that had anything to do with it, and as I was scrolling down, it started scrolling back up. I took my finger off the screen and it kept going through the settings and got to a screen that asked for my passcode so I turned it off again. What could be going on? Is this something I can fix? I'm worried about my passwords and credit card info. We don't have an Apple Store where I live, or I'd just take it in. Thanks in advance for any help/suggestions.