When you multiply, the first thing you do is take one number, and you think of it as a set of a certain size. It doesn't actually matter which of the two numbers you use, as long as you're consistent about it. So we'll say it's the first number, to keep things consistent.

So we have 51 objects. We'll think of this as a sit of little blocks, one inch on a side. They could instead be 1 centimeter, 1 mile, or whatever; what matters is each of these little blocks means 1. And we have 51 of them. Let's put them in a line. There are 51 blocks in the line, so the line is 51 long.

The next thing you do is you make a number of sets, the same size as that first one. The number of sets you make is equal to the second number (or really, whatever number you didn't use in the first step).

• If the second number is 0, then you don't actually have any sets. Throw away the first one, and your answer is zero. This is always true, no matter what the first number was: if you don't have any sets, you don't have anything.

• If the second number is 1, then you already have your one set. You made it in that last step. So that's your answer. 51 times 1 is 51. This is, once again, always true no matter what the other number is. (If it's zero times one you have a set that doesn't have anything in it, so you still have nothing.

• If the last number is anything else, you're going to make more sets, same as the first. Remember that that first set counts. So let's say the second number is 2, you have two of your lines of blocks, each of length 51.

Let's point all of these lines in the same direction, but then stack them on top of each other. So if the lines are horizontal, we stack them vertically, and vice versa. This turns our line into a rectangle. For our 51 * 2 problem, it's 51 long and 2 high. For 51 * 52, it's 51 long and 52 high. This is the key to understanding multiplication: what you're really doing is making rectangles.

Now, the answer to our multiplication goes back to the blocks: how many do you have, in total? Each block, remember, is 1. You could just count the blocks, but those numbers can get pretty big pretty quickly: even for 51 * 2 there are 102 blocks total, and for 51 * 52 there are actually thousands of blocks (2652 of them, to be precise). You really don't want to count all those blocks yourself, if you can avoid it: counting will work, but it takes a long time. And that's where all this stuff where we fiddle with these numbers comes into play. It's all just shortcuts to avoid having to count out those blocks. But in the end, it's all about making rectangles. And that goes in reverse: every rectangle you make is a multiplication problem.

Does this help?