Stroustrup's Programming: Principles and Practice is probably the best match, it introduces programming as well as C++.
In my opinion the book of Stroustrup is the worst book for beginners. I never advice to start learning C++ from thsi book. It is not a book on C++. It is a book around C++. You spent much time to read it but your knowledge of C++ will be close to zero compared with the book volume.
Moreover it is written bad from the methodical point of view and I often encounter complaints of beginners that they do not understand what is written in the book.
From various reviews it does sound like the Stroustrup book would be good for someone who is new to programming, as well as C++. After all, the first step should be too learn how to program without getting too bogged down in the details of the language that's being used.
I bought a copy of "Programming: Principles and Practice" myself recently -- to see if it might help fill in the gaps of some of the junior programmers I've worked with -- and I was pleased to see that it focused on the programming. But then I've never tutored beginners...
After the programing book ("Programming: Principles and Practice" or a better, equivalent book) has been completed, then a more complete C++ book can be moved onto, which goes into the nitty gritty of the language.
From the comments made in Stackoverflow's "Definitive C++ Book Guide and List", does sound like the other books -- C++ Primer, Accelerated C++, and Thinking in C++ -- would need to be read in tandem with a how to program book.
They are absolutely free, but if you like them you can download the extended pdf set for a very reasonable cost, or even get the just published paperback, as I did, for the sheer convenience of being able to read it anywhere and highlight it with markers.
Everyone learns differently, but I really like the way this guy explains things!