7 ms·

Elegant visual way to multiply numbers by hand

Avshalom 17y ago
Cute but it does kind of predictably fall down when you're doing 9s and 8s instead of 2s and 3s.
- deleted 17y ago
  [deleted]
  - Avshalom 17y ago
    I meant that when you're doing say 987 * 659 you end with an entire page of lines and having to count up 9 groups of cross each > 40, for instance in the case of the middle column: accurately count and add 81, 40, 42. The method still works fine it just becomes a hassle with a lot of chance for mistake (miscounting mostly) and looks way less elegant. I like it too though, I just wouldn't teach it to a kid and then tell him to do 98*79.
    - RiderOfGiraffes 17y ago
      Sorry, I deleted my comment because it took me a moment to realise what you were saying. Yes, you're right. Here's how I avoid the problem. When I've use this method as an adjunct I've put two lines with a circle around them then labelled the "bundle" with how many lines there are. That way I can "have lots of lines" without actually drawing them all. The diagrams look similar, the method clearly works, and when I have used it, it's led the students to try to work out why it works, occasionally with success. I've found it to be a useful extra tool for teaching these things. BTW, I'm not the person in the video, it's just that I know and have used this technique.
      - Retric 17y ago
        I would like to mention to you that this is really just another way to visualize traditional multiplication. 999 x 999 results in adding stacks of 81's but with this method you add vertically then horizontally where the traditional method does horizontal then vertical. aka vertical then horizontal: (81 + 81 * 2 * 10 + 81 * 3 * 100 + 81* 2 *1000 + 81 * 1 *10000) Horizontal then vertical: 999 * 9 + 999 * 90 + 999 * 900 = (81 + 810 + 8100) + (81 + 810 + 8100) *10 + (81 + 810 + 8100) * 100 If you chose diffrent groupings you can merge the two.
yardie 17y ago
123 * 312 Using the first example I would have thought it would be 371376, but no using some math "voodoo" 7 is added to 1 making it 38376. How are we supposed to know where these numbers are merged? Gotta buy the book, I guess.
- RiderOfGiraffes 17y ago
  I'm wondering if you are serious or not - it's hard to tell. It's really pretty obvous why the "1" from one column goes into the next column over to the left, isn't it?
  - yardie 17y ago
    If you have to use the same rules as the standard multiplication method then how is this anymore elegant than doing it the old fashioned way. So instead if counting numbers now you count dots, how revolutionary
    - Avshalom 17y ago
      because to perform 312 * 123 you draw 12 lines, count a total of 36 intersections and perform a single addition. you did not have to actually know any multiplication. and the rules for are hung on a visual framework instead of just being "the rules." No one said it was revolutionary (well maybe some youtube commenters..)
      - bitwize 17y ago
        No, fuck you. The rules aren't just "the rules". They are directly derivable from the property of distributivity of multiplication over addition. E.g., 12 * 39 = 12 * (30 + 9) by the property of place value, and hence 12 * 39 = (12 * 30) + (12 * 9) by distributivity. Then you do the same thing again with the 12s to get the two subresults, and... well, you get the idea. This is the basis of the standard algorithm, and it is derived directly from the properties of multiplication itself. But this line thing... this is a trick. And, might I add, a trick that may actually cause more work and use up more scratch-paper space for the figurer. Using a "visual framework" without meaning is no way to teach math. The point is that the standard algorithm is the shortest path from point A to point B, and that is why it should be taught first and foremost.
klochner 17y ago
This method essentially converts the typical tabular calculations into a lattice, where the x-axis is the 10's coefficient. Cool way to think about it, but I doubt it would give any extra insight to someone learning how to multiply. It also loses a lot of the "elegance" when you have to carry over sums > 10.
- Avshalom 17y ago
  Maybe not insight but it gives you an approach to multi-digit multiplication that you can use even if you're still sketchy on single digit. I know as a kid I'd occasionally resort to just adding a number up the appropriate number of times. and while that works for 67 * 5 it's super error prone for a 6 year old if that 5 is a 9 or worse a 30. This might (might) give 6 year old me an approach to that, sure it doesn't change the need to be able to count and do a little addition but it's a way to break the problem down that I sure wouldn't have thought of.
  - bitwize 17y ago
    That's like saying it gives you an approach to running that you can use even if you're still sketchy on walking. There's only one proven way to teach arithmetic, and that's drill and repetition of the basic number facts and the standard algorithms. It's worked for centuries.
    - Avshalom 17y ago
      Just because somebody hasn't mastered step A yet doesn't mean they won't occasionally need/want step B.
Avshalom 17y ago
So playing around with this it also works for multiplication with decimals using say a dashed line between the appropriate digits. The location can be read off by finding the intersection of the dashed lines and putting it in front of that column's digit. It's significantly less elegant, particularly when that column yields a multi-digit count but it does work visually. I'm sure I'm not the first person to realize this, but it's cool that decimals can be integrated into the method while keeping vaguely in the spirit of the idea.
- eru 17y ago
  It also works with binary numbers.
nato1138 17y ago
how does one work this out if there is a zero in one or in both numbers... I played around with this on a napkin at a bar and couldn't figure it out...
- Avshalom 17y ago
  I guess you'd just have to figure out some way to represent it, say a squiggly line or something, then remember not to count it's intersection, or you could just leave a big blank spot, but too many big blanks might throw off the ease of vertical grouping
- fairlyodd 17y ago
  Imagine (or draw) a dotted line where the zeroes are..Count the dotted intersections as contributing zero to the diagonal addition step.

Cute but it does kind of predictably fall down when you're doing 9s and 8s instead of 2s and 3s.

deleted 17y ago
[deleted]
- Avshalom 17y ago
  I meant that when you're doing say 987 * 659 you end with an entire page of lines and having to count up 9 groups of cross each > 40, for instance in the case of the middle column: accurately count and add 81, 40, 42. The method still works fine it just becomes a hassle with a lot of chance for mistake (miscounting mostly) and looks way less elegant. I like it too though, I just wouldn't teach it to a kid and then tell him to do 98*79.
  - RiderOfGiraffes 17y ago
    Sorry, I deleted my comment because it took me a moment to realise what you were saying. Yes, you're right. Here's how I avoid the problem. When I've use this method as an adjunct I've put two lines with a circle around them then labelled the "bundle" with how many lines there are. That way I can "have lots of lines" without actually drawing them all. The diagrams look similar, the method clearly works, and when I have used it, it's led the students to try to work out why it works, occasionally with success. I've found it to be a useful extra tool for teaching these things. BTW, I'm not the person in the video, it's just that I know and have used this technique.
    - Retric 17y ago
      I would like to mention to you that this is really just another way to visualize traditional multiplication. 999 x 999 results in adding stacks of 81's but with this method you add vertically then horizontally where the traditional method does horizontal then vertical. aka vertical then horizontal: (81 + 81 * 2 * 10 + 81 * 3 * 100 + 81* 2 *1000 + 81 * 1 *10000) Horizontal then vertical: 999 * 9 + 999 * 90 + 999 * 900 = (81 + 810 + 8100) + (81 + 810 + 8100) *10 + (81 + 810 + 8100) * 100 If you chose diffrent groupings you can merge the two.

123 * 312 Using the first example I would have thought it would be 371376, but no using some math "voodoo" 7 is added to 1 making it 38376. How are we supposed to know where these numbers are merged? Gotta buy the book, I guess.

RiderOfGiraffes 17y ago
I'm wondering if you are serious or not - it's hard to tell. It's really pretty obvous why the "1" from one column goes into the next column over to the left, isn't it?
- yardie 17y ago
  If you have to use the same rules as the standard multiplication method then how is this anymore elegant than doing it the old fashioned way. So instead if counting numbers now you count dots, how revolutionary
  - Avshalom 17y ago
    because to perform 312 * 123 you draw 12 lines, count a total of 36 intersections and perform a single addition. you did not have to actually know any multiplication. and the rules for are hung on a visual framework instead of just being "the rules." No one said it was revolutionary (well maybe some youtube commenters..)
    - bitwize 17y ago
      No, fuck you. The rules aren't just "the rules". They are directly derivable from the property of distributivity of multiplication over addition. E.g., 12 * 39 = 12 * (30 + 9) by the property of place value, and hence 12 * 39 = (12 * 30) + (12 * 9) by distributivity. Then you do the same thing again with the 12s to get the two subresults, and... well, you get the idea. This is the basis of the standard algorithm, and it is derived directly from the properties of multiplication itself. But this line thing... this is a trick. And, might I add, a trick that may actually cause more work and use up more scratch-paper space for the figurer. Using a "visual framework" without meaning is no way to teach math. The point is that the standard algorithm is the shortest path from point A to point B, and that is why it should be taught first and foremost.

klochner 17y ago

This method essentially converts the typical tabular calculations into a lattice, where the x-axis is the 10's coefficient. Cool way to think about it, but I doubt it would give any extra insight to someone learning how to multiply. It also loses a lot of the "elegance" when you have to carry over sums > 10.

Avshalom 17y ago
Maybe not insight but it gives you an approach to multi-digit multiplication that you can use even if you're still sketchy on single digit. I know as a kid I'd occasionally resort to just adding a number up the appropriate number of times. and while that works for 67 * 5 it's super error prone for a 6 year old if that 5 is a 9 or worse a 30. This might (might) give 6 year old me an approach to that, sure it doesn't change the need to be able to count and do a little addition but it's a way to break the problem down that I sure wouldn't have thought of.
- bitwize 17y ago
  That's like saying it gives you an approach to running that you can use even if you're still sketchy on walking. There's only one proven way to teach arithmetic, and that's drill and repetition of the basic number facts and the standard algorithms. It's worked for centuries.
  - Avshalom 17y ago
    Just because somebody hasn't mastered step A yet doesn't mean they won't occasionally need/want step B.

Avshalom 17y ago

So playing around with this it also works for multiplication with decimals using say a dashed line between the appropriate digits. The location can be read off by finding the intersection of the dashed lines and putting it in front of that column's digit. It's significantly less elegant, particularly when that column yields a multi-digit count but it does work visually. I'm sure I'm not the first person to realize this, but it's cool that decimals can be integrated into the method while keeping vaguely in the spirit of the idea.

eru 17y ago
It also works with binary numbers.

nato1138 17y ago

how does one work this out if there is a zero in one or in both numbers... I played around with this on a napkin at a bar and couldn't figure it out...

Avshalom 17y ago
I guess you'd just have to figure out some way to represent it, say a squiggly line or something, then remember not to count it's intersection, or you could just leave a big blank spot, but too many big blanks might throw off the ease of vertical grouping
fairlyodd 17y ago
Imagine (or draw) a dotted line where the zeroes are..Count the dotted intersections as contributing zero to the diagonal addition step.