Why lattice multiplication works Understanding why lattice multiplication works
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.
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