Explaining to students why to use double instead of float (unless there's a very good reason)
Luckily this is quite simple to make clear, even if you are steering clear of the IEEE standard.  Honestly, even binary is not needed here.

I would start by borrowing a page from Scheme, and say that we can't make blanket guarantees about the exactness of decimal numbers.  For instance, when we try to express $2/3$ in decimal, and it doesn't work!

All the more so when we are restricted to 10 digits.  We write $0.666666667$.  Why?  It's not that 2/3 is $0.66666667$, it's just that we can't precisely represent $2/3$ in decimal, and at the 10th digit, we just round to get as close as we can.

This is really a problem with any number that has a decimal place.  There are infinite numbers that we can't represent!

The computer has the same problem with decimal numbers as we do.  This isn't a fault of the computer, it's just a feature of floating point numbers, and the computer has to work around it, just like we do. 

Now, we do have a way of increasing that precision: we can increase the number of digits allowed.  It doesn't solve the problem, exactly, but it does get us closer to the true numbers.

That's the difference between a float and a double.  A double allows more digits.  And that's why it's usually a better idea to use a double.