Why Lattice Multiplication Works

Understanding why lattice multiplication works

In the last video we did a couple of lattice multiplication problems

and we saw it was pretty straightforward.

You got to do all your multiplication first

and then do all of your addition.

Well, let's try to understand why exactly it worked.

It almost seemed like magic.

And to see why it worked I'm going to redo this problem up here

and then I'll also try to explain what we did in the longer problems.

So when we multiplied twenty-seven--

so you write your two and your seven just like that-- times forty-eight.

I'm just doing exactly what we did in the previous video.

We drew a lattice, gave the two a column and the seven a column.

Just like that.

We gave the four a row and we gave the eight a row.

And then we drew our diagonal.

And the key here is the diagonals, as you can imagine,

otherwise we wouldn't be drawing them.

So you have your diagonals.

Now the way to think about it

is each of these diagonals are a number place.

So for example, this diagonal right here, that is the ones place.

The next diagonal, I'll do it in this light green color.

The next diagonal right here in the light green color,

that is the tens place.

Now the next diagonal to the left or above that, depending on how you want to view it,

I'll do in this little pink color right here.

You could guess, that's going to be the hundreds place.

And then, finally, we have this little diagonal there,

and I'll do it in this light blue color.

That is the thousands place.

So whenever we multiply one digit times another digit,

we just make sure we put it in the right bucket

or in the right place.

And you'll see what I mean in a second.

So we did seven times four.

Well, we know that seven times four is twenty-eight.

We just simply wrote a two and an eight just like that.

But what did we really do?

And I guess the best way to think about it, this seven--

this is the seven in twenty-seven.

So it's just a regular seven. Right?

But this four, it's the four in forty-eight.

So it's not just a regular four, it's really a forty.

Forty-eight can be rewritten as forty plus eight.

This four right here actually represents a forty.

So right here we're not really multiplying seven times four,

we're actually multiplying seven times forty.

And seven times forty isn't just twenty-eight, it's two hundred eighty.

And two hundred eighty, how can we think about that?

We could say that's two hundreds plus eight tens.

And that's exactly what we wrote right here.

Notice: this column-- I'm sorry, this diagonal right here,

I already told you, it was the tens diagonal.

And we multiplied seven times forty.

We put the eight right here in the tens diagonal.

So that means eight tens.

Seven times forty is two hundreds.

We wrote a two in the hundreds diagonal.

And eight tens.

That's what this two eight here is.

We actually wrote two hundred and eighty.

Let's keep going.

When I multiply two times four.

You might say, oh, two times four, that's eight.

But what am I really doing?

This is the two in twenty-seven.

This is really a twenty and this is really a forty.

So twenty times forty is equal to just eight with two zeros.

Is equal to eight hundred.

And what did we do?

We multiplied two times four and we said, oh, two times four is eight.

We wrote a zero and an eight just like that.

But notice where we wrote the eight.

We wrote the eight in the hundreds diagonal.

Let me make this a different color.

We wrote it in the one hundreds diagonal.

So we literally wrote--

even though it looked like we multiplied two times four and saying it's eight,

the way we accounted for it,

we really did twenty times forty is equal to eight hundreds.

Remember, this is the hundreds diagonal,

this whole thing right there.

And we can keep going.

When you multiply seven times eight.

Remember, this is really seven-- well, this is the seven in twenty-seven,

so it's just a regular seven.

This is the eight in forty-eight, so it's just a regular eight.

Seven times eight is fifty-six.

You write a six in the ones place.

Fifty-six is just five tens and one six.

So it's five tens in the tens diagonal and one six. Fifty-six.

Then when you multiply two times eight,

notice, that's not really just two times eight.

I mean we did write it's just sixteen when we did the problem over here,

but we're actually multiplying twenty.

This is a twenty times eight.

Twenty times eight is equal to one hundred sixty.

Or you could say it's one hundred--

notice the one in the one hundreds diagonal-- and six tens.

That's what one hundred sixty is.

So what we did by doing this lattice multiplication,

is we accounted all the digits. The right digits in the right places.

We put the six in the ones place.

We put the six, the five, and the eight in the tens place.

We put the one, the eight, and the two in the hundreds place.

And we put nothing right now in the thousands place.

Then, now that we're done with all the multiplication,

we can actually do our adding up.

And then you just keep adding,

and if there's something that goes over to the next place,

you just carry that number.

So six in the ones place, well, that's just a six.

Then you go the tens place.

Eight plus five plus six is what?

Eight plus five is thirteen.

Plus six is nineteen.

But notice, we're in the tens place.

It's nineteen tens or we could say it's nine tens and one hundred.

We carry the one up here, if you can see it, into the hundreds place.

Now we add up all the hundreds.

One hundred plus two hundred plus eight hundred plus one hundred.

Or, what is this?

One thousand two hundred.

So you write two in the hundreds place.

One thousand two hundred is the same thing as two hundreds plus one thousand.

And now you only have one thousand in your thousands diagonal.

And so you write that one right there.

That's exactly how we did it.

The same reasoning applies to the more complex problem.

We can label our places.

This was the ones place right there.

And it made sense.

When we multiplied the nine times the seven,

those are just literally nines and sevens. It's sixty-three.

Six tens and three ones.

This right here is the tens diagonal.

Then we got six tens and three ones.

When we multiplied nine times eighty-- remember, seven hundred eighty-seven,

that's the same thing as seven hundreds plus eight tens plus seven, just regular seven ones.

So this nine times eight is really nine times eighty.

Nine times eighty is seven hundred twenty.

Seven hundreds-- this is the hundreds place.

Seven hundreds and twenty-- two tens just right there.

And you can keep going.

This up here, this is the thousands place.

This is the ten thousands.

I'll write it like that.

This is the hundred thousands place.

And then this was the millions place.

So we did all of our multiplication at once,

and accounted for things in their proper place based on what those numbers really are.

This entry right here,

it looks like we just multiplied four times eight and got thirty-two,

but we actually were multiplying four hundred-- this is a four hundred-- times eighty.

And four hundred times eighty is equal to three two and three zeros.

It's equal to thirty-two thousand.

And the way we counted for it-- notice, we put a two right there,

and what diagonal is that?

That is the thousands diagonal.

So we say it's two thousand and three ten thousands.

So three ten thousands and two thousands.

That's thirty-two thousand.

So hopefully that gives you an understanding.

I mean it's fun to maybe do some lattice multiplication and get practice.

But you know sometimes it looks like this bizarre magical thing.

But hopefully from this video you understand that all it is

is just a different way of keeping track of where the ones, tens, and hundreds place are.

With the advantage that it's kind of nice

and compartmentalized, it doesn't take up a lot of space.

And, it allows you to do all your multiplication at once,

and then, switch your brain into addition and carrying mode.