There are a lot of methods of double digit multiplication. Personally, I don't feel that they are all created equal. Some are cumbersome, some are confusing, and many don't carry forward. Consider the following three methods: the Familiar, the Frustrating, and the Fabulous.
The Familiar. Yes, there is the traditional algorithm that most of us learned in school. This method is still taught, but it's one of several methods.
And yes. When done correctly, it works. But the honest truth is that it isn't always done correctly. Things go wrong (often it's that pesky zero in the second partial product that represents our multiplying by a factor of 10). So, having alternatives is a good thing.
But then there are alternatives like this:
Yes, it looks vaguely familiar, but upon closer inspection, it's just confusing. Why am I starting with my 10's? How am I supposed to proceed? What is that cross hatch thing in the middle supposed to be? Can I expand this to more digits?
Sure, I can create an explanation for this method, but why? Does it follow into any other methods? Will my student see this again? And the answer to both of those last questions is "No." As far as I can tell, this is a stand alone method. It doesn't help with understanding the traditional algorithm. It isn't useful in algebra or other higher math. And, trying to expand it to three digit multiplication is migraine inducing. In my opinion, it is a dead end which confuses both parents and students.
I vastly prefer this method:
This is another evolution of our array method.
The box on the right has the two numbers we are multiplying on the exterior of the box. They have been separated into 10's and ones for ease of figuring. Each interior box has a set of partial products, which added together (on the left) give us our total. (As the student advances, the factors being multiplied in each little box don't need to be written. This significantly speeds up this method.)
If you are having trouble understanding this method, think of it as an area problem. You have a rectangle that is divided into smaller rectangles. By calculating the areas of the interior shapes, you can find the total area.
Why do I like this method? Well, it builds on earlier visual methods. The individual computations are easy. You can easily expand it into more digits (you just add more cells to your box). And, most importantly for me, this method is built upon in algebra. It replaces (or acts as a partner to) foiling in algebra.
Ah yes, our frenemy, foiling. The old "First, outer, inner, last" method that plagued my high school brain.
If someone had shown me the box method back in the day, I would have been a much happier camper.
The box method? So much sweeter. Notice how the x factors that can be combined line up on a diagonal? So nice. My high school students do not know the panic foiling used to give me, because they use this method instead. I love a multi-use tool. It slices, it dices, it reduces mistakes in multiplying and combining.
So hang in there. Some of the methods being taught will work for your student, and some won't, but there is light in the spectrum!