So I wrote about that Chinese multiplication video and how it's not really that great a method if you have large digits. I gave the example of \(78\times 89\) to illustrate why. Here's a nice way to do it that is perhaps a little more visually pleasing than the standard algorithm we all learned in school. I'll explain it in a second, but here's an animation of it.
Here's what you do. Draw a box and divide it into rows and columns for each digit in your factors. In this case, it's a \(2\times 2\) grid. Then draw diagonals passing southwest to northeast in each box. For each pair of digits, multiply them and place the answer in the corresponding box, one on each side of the diagonal line. For example, the \(8\) in \(78\) times the \(9\) in \(89\) gives a \(72\) in the lower right corner. Then add along the diagonals. If a sum is more than \(9\), carry the appropriate number to the next diagonal and add it along with the numbers you find there.
In this example, we get the product \(78\times 89 = 6,942\). Neat, huh?
Really, this is the same algorithm we use, but it has the advantage of not requiring you to keep up with shifting things over in the successive rows when you stack up the various intermediate products. I try to use this method when I multiply, but to be honest I am so used to the old-timey way I learned in elementary school that I usually default to that. But I encourage you to give this a try the next time you find yourself calculatorless.