Carried Numbers

Hi,
Having scant information on carried numbers I was asked by someone how to they would work for a multiplication problem such as:

574
x823

By applying the basics I decided to “build” the problem in 2 steps. Step 1 will take care of the multiplication with each new carried line above the previous. Then in step 2, add the 3 lines which also has carried numbers.
In such a situation, one has to apply what one knows and hope that it’s presentation is clear to the student. TNs may help in this process as well, such as: Problem solved in 2 steps. And again perhaps one to explain the carried numbers in layers for the multiplication (see example).
There are 2 different ways shown to carry the numbers. The top problem is explained above.
The 2nd (bottom) solution is to add the 2nd carry line [u]under[/u] the 1st, then the 3rd carry line under the 2nd. What you end up with is still 3 carry lines but each “newest” line is kept close to the columns of numbers in the problem making it easier to do the math w/o the interference of previous carried numbers. This way you don’t have to look way at the top then then look way down to align numbers, they're right next to each other.
Lgically, this is the best way I came up with and anxious to hear your comments. TN’s of course will be needed but they don’t need to be long ones, more of identifiers such as “1st carried line”, “2nd carried line”, etc. as in the example.

Thanks,
Fred