Thinking diagonally with multiplication grids Jason 22 Sep 2006

63 comments Latest by a

Yesterday’s Chicago Tribune ran an article about teaching math in school [registration required] that included something I’d never seen before: The multiplication grid.

I don’t have any idea if this is better/worse/harder/easier, but I just found it interesting. It’s refreshing to see alternate solutions (grid) to more common approaches (stacked).

63 comments so far (Jump to latest)

John Resig 22 Sep 06

This confused me at first, but then I tried it with two different-sized numbers - 54 and 371. A couple points:
- It feels easier when the larger number is on the top, versus on the side.
- Making a straight grid really helps confusion.
- When you’re left with strange numbers like: ‘1, 9, 9, 13, 4’, just carry the extra digit over from number to number - resulting in an answer of ‘20034’.

As a whole, I’d rate it as being similar, but a little bit easier, to do than the traditional method.

Somehow, I wouldn’t be surprised if teachers were against making a change to this system - at least not until book publishers were on board with the change, too.

Jacob Patton 22 Sep 06

That’s a neat trick, thanks! Speaking of simplicity, you might find it interesting how the Trib simplified Ivy Hall Elementary’s original directions.—much simpler!

Shaun Inman 22 Sep 06

Not being a pigeon, this looked really interesting at first. After playing with it though, there doesn’t appear to be any benefit over the stacked approach if the sum of any of the diagonals is more than 10 (eg. 36 x 36). “Carrying the one” clockwise is far less intuitive than carrying along a single axis, right to left.

Tony Karrer 22 Sep 06

You must not have kids in elementary school. It was fun when mine came to me with homework in math that I couldn’t do. Luckily they had a math night to teach us parents all the new ways they teach multiplication, division, etc.

Benjy 22 Sep 06

While cool, this doesn’t seem to save any time and takes up more space on a page. But I bet the paper companies are happy!

Benjy 22 Sep 06

While cool, this doesn’t seem to save any time and takes up more space on a page. But I bet the paper companies are happy!

jonathan 22 Sep 06

Although I live abroad, my children just spent 3 months back in a US school where the multiplication grid was taught to them.
My daughter, in particular, latched onto it immediately and found it extremely easy to work with.

I haven’t seen this taught elsewhere and no one I’ve spoken to in this part of the world are familiar with it either.

Very cool stuff.

jonathan

Rik Lomas 22 Sep 06

I was first shown this technique in at high school, years ago, and I used it all through my maths degree, and still use it today. Although it is a slightly strange way of working, once you get up to speed on it, it’s a very useful and productive way of working out harder multiplication, especially when you start multiplying in hundreds and above.

jonathan 22 Sep 06

Although I live abroad, my children just spent 3 months back in a US school where the multiplication grid was taught to them.
My daughter, in particular, latched onto it immediately and found it extremely easy to work with.

I haven’t seen this taught elsewhere and no one I’ve spoken to in this part of the world are familiar with it either.

Very cool stuff.

jonathan

Adam Sanderson 22 Sep 06

Both methods work, however I think that the diagonal one is pretty damn cool. It encapsulates carrying digits better in my opinion.

The only downside to the grid method, is that it seems a little less obvious why it works.

Try them both with something a little larger ie: 237*54

michael 22 Sep 06

neat, but takes me much longer than the method i learned in 1st. nonetheless, cool to know.

Benjamin Schroeder 22 Sep 06

This is my very favorite way to multiply polynomials (e.g. “x squared + y squared” times “2x cubed + 3”). What’s really nice is that it scales to any number of terms.

Peter Cooper 22 Sep 06

It’s called Elizabethan Multiplication. At least, it was when I learnt it as a kid (not at school, but from a book full of ‘weird / cool’ stuff). I used to do really large ones (10-50 digit numbers) for fun and multiplication practice until Sudoku came along :)

Josh 22 Sep 06

I remember learning these in second grade (1991 or so), but I always found the traditional method easier/faster. It’s certainly nice to be able to choose from more than one method to effectively solve a problem.

Peter Fitzgibbons 22 Sep 06

So far, I think Adam Sanderson has it most closely.

This diagonal method, which I have never seen before 9:16 PM CST 9/22/2006, does indeed encapsulate (isolate?) the carrying and ‘shift’ of the (more traditional?) ‘stacked’ method.

I can see (I don’t remember if my 2nd grade schoolmates had trouble) that there might be a subset of the children in class that have trouble one or another way with understanding easily the ‘shift’ of each multiplicand, and/or how to correctly ‘carry’ the 10^1x value of each subtotal.

I will happily teach my children the diagonal method.

If you want to save paper, write smaller.

warren 22 Sep 06

i always do multiplication by using the distributive law. e.g., 36x27 = 36(20+7) = 720+252 = 972. you can break down the problem into easier parts and then just add up the results; of course, each one could be further broken down; 36x20=36x2x10=72x10=720. multiplying a number that’s a multiple of 10 is easy.

i think that would be easier for children since it’s just using what they already know in a new way.

both the way multiplication is commonly taught and this diagonal method strike me as profoundly unintuitive and cumbersome. the fact that working with large numbers is no different from smaller ones is lost.

Daniel 22 Sep 06

Less is more, right?

Of course that could be used to describe the benefits of each one of these methods over the other. The traditional one has less (visible) digits floating around the system, the other one force you to do less at a time (first all multiplication, then all addition).

To me it seems that the separation makes it harder to get a grip on what is actually going on. How do you organize the digits from the grid/diagonal-method when you count “in your head”, i.e. without paper?

Daniel 22 Sep 06

Warren - but I thought that was exactly what the traditional way was visualising?

z 22 Sep 06

one more trick.

in real life it is often to get a “good enough” result.

so

36 x 27 is roughly 30 x 30 which is 900-ish.

if thats not precise/good enough, a slightly more involved (but still simple) method would be to 36 x 30 less 3 times 36, which is 30 x 30 + 6 x 30 = 900 + 180 - 108 = 900 + 72 = 927.

it probably sounds involved, but does save some brain cycles. basically trying as much as possible to multiply single digit numbers.

Anonymous Coward 23 Sep 06

Eventually the top-down list method is probably more efficient but in terms of teaching new concepts, a visual metaphor like this is excellent.

Dean 23 Sep 06

I’ve got a fantastic new device called a calculator that does this for me.

John Koetsier 23 Sep 06

Seems far more complicated to me … at least if you have to draw your grid every time.

Also, I agree with the posters above who think this might be less obvious and less intuitive. The answer appears “by magic” through apparently unrelated actions and odd placement of preliminary solutions … which I think might reduce the level of mathematical learning that is happening as the student is solving problems.

Sam 23 Sep 06

I was taught this grid in primary school about 7 years ago and, although it is longer and less intuitive than the normal long multiplication method, I have used it all the time.

An extra tip is that you can also do this with decimals, simply take all the decimal places out (remembering how many places there was), and add the sum of the decimal places to the answer. For example:

3.5 x 2.7 has two decimal places (5 and 7). We multiply 35 and 27 which gives us 972, we then add those two decimal places back again, leaving us our final answer of 9.72.

Sam 23 Sep 06

I was taught this grid in primary school about 7 years ago and, although it is longer and less intuitive than the normal long multiplication method, I have used it all the time.

An extra tip is that you can also do this with decimals, simply take all the decimal places out (remembering how many places there was), and add the sum of the decimal places to the answer. For example:

3.5 x 2.7 has two decimal places (5 and 7). We multiply 35 and 27 which gives us 972, we then add those two decimal places back again, leaving us our final answer of 9.72.

Oh, and Dean, have fun in a non-calc paper :)

Frankie Roberto 23 Sep 06

I think this system has been taught, alongside the traditional method, in British schools for quite a while.

Vishnu Vyas 23 Sep 06

Hmm.. seems like the same number of multiplications and additions overall.. and the carry-over seems counter-intitutive..

Doesn’t seem like a big advantage over the conventional method either..

On the other hand using algebraic identities around helps multiply a lot faster (like z’s trick.. which is basically splitting it on the nearest 10).

speedy 23 Sep 06

numbers that are so close to eachother can be calculated much faster and without using paper.

so it’s 27 * 36. So first we calculate 27 * 33 (where the average of the multiplicators is an easy 30), and then add the last 3*27.

27*33 is 30 * 30 - (30-27)*(30-27) = 900 -9
3*27 is 60+21 = 81

900+81-9= 972

sure this method has somekind of name. since it’s so simple it just appeared to me one day.

and yes I know it’s not the best description of what is going on…

Anonymous Coward 23 Sep 06


I learned basically the same technique in elementary school in the 80’s. They didn’t use a grid, but I can see the algorithm is the same. It was called the ROIL method, for Right, Outer, Inner, Left. You multiply the right hand digit of both, then the outer digits, the inner digits, the left digits… no grid need be involved, just an acronym.

Mark 23 Sep 06

Reminds me of Vedic maths.

Which is a facinating way to approach maths, basically from ancient India. They have sixteen “sutras” which are simply algorythmns that enable just about any maths problem to be solved.

A very easy example is 98x98 - find the difference between it and the nearest base -> 100 - 98 = 2 -> 98 - 2 = 96 which is the first part of the answer and then we do 2 squared, so the answer is 9604…

This is just a simple example, my 5 year old can now work out squares upto 16 in his head and work out the change from a tenner etc….

http://www.vedicmaths.org/Introduction/Tutorial/Tutorial.asp

ted 23 Sep 06

We were taught this in Primary school, it’s called Napier’s Bones…

http://www.bvisual.net/examples/Napiers%20Bones.htm

sammy 23 Sep 06

Coincidence: a coworker of mine was telling me just yesterday about his daughter, who is a freshman at a local university. Apparently, she’s been having trouple with her first year maths. In part, that’s due to the fact that her professor is just an outright putz (“You don’t know what mod means? Go find a tutor and stop wasting my time.”).

However, he told me that his daughter also learned a system for arithmetic when she was in elementary school called the “Touch point system.” Essentially, it adds a tactile component to basic arithmetic operations - it can be useful if a student has trouble memorizing, say, multiplication tables.

The problem is that his daughter never really memorized those tables. Or even some basic addition or subtraction. And as a result, every time she’s called upon to do something slightly more complicated - like, say, long division - she winds up having to do the method to do even the incidental subtraction steps. It eats up a lot of time, and tends to put her at a disadvantage.

Not that I’m saying that the system (which you can find here) is necessarily bad. But… um, my friend sure seems to think so.

Paul O'Shannessy 23 Sep 06

This reminds me of when my youngest brother was having trouble with his math homework. My mom volunteered me to help him, but I had no idea what he was doing because he had been taught a different way. Maybe it was better, maybe not. I guess it’s good that change is happening, even if it’s in ways that don’t exactly make sense to us “old” (20 years old) people initially.

Anonymous Coward 23 Sep 06

Congratulations, Mark!

Steve 23 Sep 06

When I took math education - almost 20 years ago - we called that “lattice multiplication”. Easier for a younger child than standard multiplication, but it does use more paper. The idea was that if you were working with kids just learning 2-digit (or larger) multiplication, you could start them with lattice multiplication, and then transition to standard multiplication as they got used to it.

Yes, it has been around that long. If I’d been teaching grades 3 and 4, I might have used it myself. But as I taught high school and college, I never really used it (except in a couple of math classes intended for teachers).

Will 23 Sep 06

Takes up way too much paper and time gridding. Super Lame.

Phydeaux53 23 Sep 06

I could see advantages to this method for multiplying arbitrarily long numbers on a computational grid. Not your everyday computing for sure …

sloanNYC 23 Sep 06

Neither way is usually explained properly though. They are taught as “tricks” and never really give the student a deeper understanding of the principles of numbers to begin with. The stacked method really is JUST the distributive property. The basics of how multiplication and division work are often glossed over. Case in point, how many of you can actually divide 2 fractions? And no, don’t flip the second fraction and multiply, that’s another trick! :-)

Dan Boland 23 Sep 06

That grid just seems bizarre to me. But if it helps a kid learn multiplication better than the traditional method, good.

Tim Hill 23 Sep 06

I watched my kids use this method in elementary school and frankly I don’t like it. It’s too much like a “magic trick” and doesn’t, imho, help teach the underpinnings of multiplication as an extended form of shorthand for addition. The traditional method helps reinforce the idea of multiplication as a sequence of partial sums, and also helps underline the positional model (10’s, 100’s columns) of numbers which is so important later when decimals and division are introduced.

Masud 23 Sep 06

It’s nothing new: it’s been known for years, called the Gelosia method, and we are taught about it in Europe. See for example: http://faculty.ed.umuc.edu/~swalsh/Math%20Articles/GelosiaMultiply.html

There’s nothing in it new over the traditional method, of course. It just automatically does the “shift one place to the left” that we do usually. In fact, in the traditional method, some students are taught to shift one to the left and write an additional zero. Which makes sense, since it teaches them that you are individually multiplying out the unit term, then the tens term, then the hundreds term etc. i.e. the distributive property that 854 = 800 + 50 + 4.

Linden 23 Sep 06

Isn’t this basically Napier’s bones? I did some research on that. Merchants back then didn’t have access to paper and pen, so they used rods called Napier’s Bones (which had diagonal notches and products), and it’s the exact same as this diagonal multiplication.

Jesse 23 Sep 06

Of course, the drawback to this approach is not the ease of calculation but that drawing the grids is time-consuming. Perhaps everyone should do these on graph paper?

pjm 23 Sep 06

Speedy: the method you’re using is pretty much “the difference of two squares”.

(a+b)*(a-b) = a^2 - b^2

So you could also do the sum as 27*36 = 27*35+ 27*1 = 31^2-4^2+27 = 961 - 16 +27 = 972. But “centering” about a multiple of ten —when feasible— leads to a much nicer first multiple.

Jim C. 23 Sep 06

As previously said, it’s closely related to Napier’s Bones http://www.cut-the-knot.org/blue/Napier.shtml

A variation is Genaille’s Rods (pdf, page 18ff) http://www.projects.ex.ac.uk/trol/trol/trolha.pdf

shari 24 Sep 06

Different people learn using different methods. The more methods that are presented in school, the more the fringe students will learn, too. I think this is an excellent alternative.

Smitty 24 Sep 06

So many math geeks. Honestly it’s all very interesting but at the same time very funny how strong some feel about different methods of getting to the same result.

Ericose 24 Sep 06

The thing that makes this so interesting to mathematicians is not so much the fact that this method gets a result as the insight that it gives to the nature of the number system itself.

Brian 24 Sep 06

“…was called the ROIL method, for Right, Outer, Inner, Left. You multiply the right hand digit of both, then the outer digits, the inner digits, the left digits…”

I went through elementary school in the late 90’s, and when we started working with polynomials in middle school we used the “ROIL” method. Only it was called “FOIL” ( front instead of right, and last instead of left). But I can’t say I’d ever use it for anything larger that 2-digit constants (ie: 27 x 34), it just gets too messy. The grid method at least lets you keep track of things. I’m taking Pre-Calc in college now and I may start using the grid system for the larger polynomial multiplications, trying to foil them can get awfully confusing.

Kiran Garimella 24 Sep 06

Pretty cute, but seems more cumbersome. There is simply no way to multiple single digit numbers and add in both methods. The diagonal method takes up more space. None of the methods makes intuitive sense, and kids are not interested in understanding it anyway. Here is my decision tree:

If (calculator or computer is handy)
then [use it, for crying out loud! - unless they are obviously easy to do mentally, like multipying by a factor of 10;]
elseif (numbers involved are small, say 2-3 digits)
if (paper and pen handy)
then [use traditional method;]
else /* no paper and pen handy */
[factor to estimate;]
else /* numbers too large, no calc or computer nearby */
then [bite your lip and wait it out; in a pinch, call your kid’s math teacher;]

Most of the time, we never multiply large number together; spending too much time learning to that efficiently is a total waste of time. These days, a calculator or computer is always nearby, even in cell phones, or the other guy in the transaction (say, the cashier) has it anyway. Super numerical ability is totally overrated.

Kiran Garimella 24 Sep 06

*Fixed some errors - pl ignore my earlier post*

Pretty cute, but seems more cumbersome. There is simply no way to avoid multiplying single digit numbers and adding numbers in both methods. The diagonal method takes up more space. None of the methods makes intuitive sense, and kids are not interested in understanding it anyway. Here is my decision tree (easier to read if indented):

If (calculator or computer is handy)
then [use it, for crying out loud! - unless they are obviously easy to do mentally, like multipying by a factor of 10;]
elseif (numbers involved are small, say 2-3 digits)
if (paper and pen handy)
then [use traditional method;]
else /* no paper and pen handy */
[factor to estimate;] /* best to factor to nearest ten */
else /* numbers too large and/or no calc or computer nearby */
then [bite your lip and wait until you find one; in a pinch, call your kid’s math teacher;]

Most of the time, we never multiply large numbers together; spending too much time learning to do that efficiently is a total waste of time. These days, a calculator or computer is always nearby, even in cell phones, or the other guy in the transaction (say, the cashier) has it anyway. Super numerical ability is totally overrated. If your kids have learnt the idea behind multiplication, know the basic math facts completely, know how to multiply 2-3 digits, then leave them alone!

dgsaspen 24 Sep 06

The benefit might be for the kids. Since you don’t have to hold numbers in memory. The numbers are all there in front of you. Incase you get distracted like most kids do. You wouldn’t have to start the problem over. No more “carry this to the left” Breaking it down to simple problems I guess.

George Bailey 24 Sep 06

A new way to solve math problems.


Sure, if your math problem is multiplication.

“The toaster - a new way to prepare food.”

Will 24 Sep 06

In old elementary school, this was known as the lattice method. While it is simple and breaks the problem up into small steps so an eight year old can do 2+ digit multiplication, it puts these children at a disadvantage when they get to middle school and cannot understand how to do multiplication in the traditional manner. This alternative method also uses much more paper and requires a grid and diagonal lines to be drawn, which makes it less efficient than standard multiplication.

anonymous 25 Sep 06

interesting

Luke Noel-Storr 25 Sep 06

I was taught a form of this method in 3rd year at Junior School (that was age about 9, I think), and I have used it ever since. Like others have mentioned, we were taught that it was based on Napier’s Bones.

The method we were taught had the second number on the left, rather than the right, but the method shown above looks even better.

I was the only person from my Junior School at my secondary school, and everyone was always very impressed at how fast I could do long multiplication using this method, and several other pupils adopted the method from me.

I’ve never understood why the method isn’t more widely taught, as it has always seemed much simpler to me, and to anyone else I’ve explained it to.

The only times it ever caused me confusion, was when multiplying to numbers that included a decimal part.

matt m 25 Sep 06

In the above example, both cases require 4 multiplication operations to be calculated and 3 addition operations.

From a pure speed of calculation method, the primary difference is that the traditional stacked method requires less drawing… From an instructional perspective, the “new method” seems much less instructive in terms of reinforcing what they used to call place value…1s, 10s, 100s, 1000s, etc.

The people who are saying you don’t carry in this “new method” are wrong. Try 77 x 88. You end up with 516176 as the answer, so the instructions above are obviously incomplete. You have to start at the lower right and carry. In this particular example, the trad method seems much faster.

If you want to speed things up, check out a site like this: http://www.mathforum.org/k12/mathtips/mathtips.html
I used some things like this in math competitions when I was younger to great effect. Unfortunately those weren’t the sort of trophies that improve one’s social life. :-/

jenn.suz.hoy 25 Sep 06

Ahh yes, the multiplication grid. Any kid who lived through the horror of trying to multiply numbers this way will have nightmares with this being dredged up again.

Thanks for the memory, ha!

Frazer 25 Sep 06

For somebody who might have trouble viewing numbers and/or letters on mass, this is an incredibly simple solution to visualise the relationships between numbers and their results. Something which I have always found differcult.

Thanks for the pointer, I wish I knew this back at School!

Frazer 25 Sep 06

For somebody who might have trouble viewing numbers and/or letters on mass, this is an incredibly simple solution to visualise the relationships between numbers and their results. Something which I have always found differcult.

Thanks for the pointer, I wish I knew this back at School!

Roman 26 Sep 06

Sammy (and everybody else):

Steer clear of touch points. I accidentally learned it while attending an inner city school and now I can’t even do simple addition without it. The tactile element is helpful, but tremendously slow and not well-suited for higher maths.

It’s ruined me.

Gregg Irwin 27 Sep 06

Did anyone ever learn the Trachtenberg system in school? I didn’t, but am trying to find time to spend on it now, and encouraging our kids to as well (we homeschool).

Paul Phillips 28 Sep 06

Interesting - but looks quite wierd at first. We were taught vedic maths at school and I still use the ‘vertical and crosswise’ method to do multiplication. Often its so simple you can do it in your head!. With this example you can write the answer straight down: so
36
27
(Vertically, starting from the right handside. 6x7=42, write down 2 carry the 4. Crosswise, 3x7+2x6+4 carried = 37 write down 7 carry the 3. Then vertical again) (3x2+3 carried=9) = 972

a 29 Sep 06

I give it two thumbs up. There’s no point in using it for something like 15*120, but I tried with 8679*6478, and it really saves time. After carrying 16 digits in the traditional way, then having to add up four rows, and carry all those digits, its pretty clear the grid is faster, neater, and doesn’t take much more space.

Post a comment

(Basic HTML is allowed)

NOTE: We'd rather not moderate, but off-topic, blatantly inflammatory, or otherwise inappropriate or vapid comments may be removed. Repeat offenders will be banned from commenting. Let's add value. Thank you.