Worked examples for standard algorithm exercise

We're now going to do a few example questions from the Khan Academy exercise on the standard algorithm. So we're asked which of the following correctly multiplies 74 times 8 using the standard algorithm. So pause this video and see if you can work on that before we do it together.

All right, now let's just remind ourselves what the standard algorithm is. In fact, let's just remind ourselves what an algorithm is. An algorithm is a series of steps that you can do to do something. So you'll often hear about a computer algorithm, but you can also have algorithms in math—just a method for doing something.

And the standard algorithm, the standard algorithm, that's the typical or the standard way that a lot of people will tackle a multiplication question or computation like this. But just as a reminder, in the standard algorithm, if we're multiplying 74 times 8, we would write the 8 in the ones place right below the 4 in the ones place. Then you multiply each of these places times the 8.

So you would start with the four times the eight. You would get 32. 32 you can express as two ones and three tens, so you'll put that three up there. Then you would multiply the seven times the eight. Seven times eight is 56, and that's going to be 56 tens because it's seven tens times 8 is 56 tens.

Plus the three tens you had before gets you to 59 tens, and so you would write over here, that's 59 tens. This would be 592. Now, when you look at the choices, that's exactly what happened here in choice C. Just for kicks, we can see what went wrong in these other ones.

Let's see. In this first one, when we multiplied the 4 ones times eight ones, according to this, this person somehow got three ones and two tens, and twenty-three. Four times eight is not 23. You could rule that one out. Here, when they multiplied the four ones times the eight ones, that would be 32, that's so it's two ones and then another three tens.

So sure, there should have been a three up here. That way when you multiply the seven tens times eight, you get 56. But then you had this other 3 tens, so you really need to get to 59 tens. So that's one reason why that one didn't work.

Let's do another example here, and this is going to be with a different type of question. Here we are told that Don starts to use the standard algorithm to solve 418 times five. His work is shown below. What number should Don replace Y with? Pause this video and see if you can figure it out.

Okay, so the way to think about this might at first confuse you a little bit because why, and why is there a Y there in the first place? But what they're really trying to get at is making sure that you or we understand what Don is trying to do when he's trying to do the standard algorithm.

As we just highlighted in the last example, the way that we would tackle this with the standard algorithm—actually let me write down 418 times five—we would say eight ones times five ones is going to be 40 ones. 40 ones we could write as zero ones and four tens.

So that looks like the place where Don stopped computing. He's on his way to solving the whole thing; he's just partially computed it so far. But just by doing that, we know what the Y should be. The Y should be four, so what number should Don replace the Y with? He should replace it with a four, and it's representing Y tens, or four tens.

Of course, you could keep going with this computation. If Don were then to say, okay, I have one ten times five, that would be five tens plus another four tens, that's nine tens. Then last but not least, if Don wanted to figure out, well, he's got four hundreds here times five, that's twenty hundreds, which you can express as zero hundreds and two thousands.

Or you could just view this as twenty hundreds. So they're not asking us to do the entire computation; we're just trying to figure out what Don did essentially in this first step. What number should he have written here instead of a Y? So Y could be replaced with the four; it's representing the four, which is really in the tens place, so four tens.