Checking the validity of a private key is a really compute-intensive task to do by yourself, but you can do this without actually using the private key to send coins (so no need to do it using pen and paper).
On the bitaddress.org website the
Wallet Details tab lets you enter the private key and after clicking
View details your bitcoin address should be showed in either
Bitcoin Address Compressed or
Bitcoin Address field.
But of course, you should not give away your private key, especially not online.
The site is actually an all-in-one HTML page, that is it can be downloaded as one file and it will work offline as it does every calculation in your browser window (no data you entered will be sent to the server). You can save to your computer (Ctrl-S in your browser) and open it in your browser without Internet connection.
I'd suggest to do the check at least in a private browsing window, or a live system (keep reading).
If you would like to do an independent check, use a Bitcoin Core client and create a new empty wallet (if you have it running, shut it down, move the wallet.dat somewhere else and start it again), go offline (disconnect your machine from the Internet, you will not need synchronization anyways) and try to import the key as it is described in this question. The client will check the private key, if that's invalid then an error will be reported (I changed a character, it reported
Invalid private key encoding (code -5)). After this you can close the debug window and check your Bitcoin addresses in
Receiving addresses. After this, shut down your client and delete the wallet.dat. If you have moved the wallet.dat move it back now and keep using that.
Of course using this method writes your private key to the HDD/SSD of your machine, therefore even after deleting the wallet it is likely to be recovered using forensic/data recovery tools. If you'd like to avoid this you will need to use a disposable system, like a live OS that retains no data.
You can for example download Ubuntu and put it on a pendrive then boot from it, by default it will not save anything you do. You can also download the client from the Bitcoin Core download page (choose "Linux (tgz) 64 bit"), save it to the pendrive using your regular OS and then when you boot the Ubuntu you can simply extract it to the Desktop and run from there, all offline.
If you'd want to be even more careful then before booting the Ubuntu you should disconnect all other drives from your machine, leaving only the pendrive connected. Also, after using the Ubuntu and returning to your regular OS search for a method to securely wipe your pendrive and do that.
And another step would be a computer dedicated solely for this purpose, for example a Raspberry PI. Download a base image, put it on an SD card, copy the saved bitaddress webpage beside then connect a display, a mouse and a keyboard and power it on. The Bitcoin Core client has no official binary release for this platform but you can build it from source if you're adventurous enough.