My kid is learning "Discovering Algebra", I downloaded the accompanied TI-84 plus calculator programs and installed them.

Now if he choose "PRGM / EXEC / LINES", the downloaded program will run -- x & y axes are shown then a moving dot draws out a line.

The problem is, even after he pressed "QUIT" to exit from the program, it seems still running in background -- that is, if he sets 2 lists and plot it, showing via "GRAPH", the line drawn by program "LINES" will still show.

How could we stop the "LINES" program from running in the background?
you run the token ClrDraw
It's highly likely that the LINES program is leaving a Y= graph equation behind. What happens when you graph a plot is it will regraph everything else too.

If you press Y= and scroll down to Y0, does it have a line equation in it? mx+b perhaps?

If it does then this is normal behaviour and you can be confident that the program is not running in the background. You can clear the equation to stop it from drawing, you also may need to reset graph windows and such to standard (Zoom, ZStandard).
You can also use the Reset menu under the Memory menu to reset the calculator back to default settings without deleting anything.

The TI-84 Plus has no concept of background programs. (It does have a concept of hooks, which are sort of like DOS-style TSRs, but that's not what's happening here.) To add some more information, the graphing screen is considered a separate "app" with its own configuration. BASIC programs can command the graphing screen to do things that change its configuration, and the graphing screen retains that configuration until the user or another program changes it. This is intentional design, since it lets users share data between built-in functions and user-created programs.

This model of sharing data between apps and letting BASIC programs reconfigure your calculator is a holdover from the late 70s to early 80s 8-bit BASIC microcomputer vibe the TI-84 Plus has. It's actually reasonably powerful and convenient once you're proficient (as long as you like 80s microcomputers, anyway). Back then, computers didn't have the CPU power and memory for more modern concepts like running more than one thing at a time and having a shared clipboard. Early popular home computers didn't even have persistent storage; if you turned the computer off, you had to type in the program you wanted to run again!

The TI-84 Plus shares quite a lot in common with TI's first graphing calculator, the TI-81, released in 1990. It should be unsurprising that the TI-81 gets a lot from the era of BASIC computers; one suspects many of the engineers that worked on it first learned to program on an 8-bit computer. Up until the 80s, it was widely believed by engineers that all computer users would have some degree of familiarity with computer programming, and the idea of users buying commercially-developed software was somewhat a novel concept when Microsoft was founded. A lot of the software design of the TI-84 Plus still reflects an ideal that all students should learn some computer programming, and thus nearly anything you can do manually on a TI-84 Plus can also be done by a BASIC program.

Though I'm not even 30 years old, I'm sure the idea of math classes incorporating computer programming made perfect sense in the 90s, and I still think it's a lost opportunity. Sadly, it doesn't seem to have caught on. Ironically, it's probably at least partly TI's fault for keeping the TI-84 Plus's price way too high, as they could have profitably sold it for $50 in 2004, an era when even a single shared family computer (much less Internet) wasn't something public school teachers could count on students having.
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