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Maths Café


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Gelosia/lattice multiplication

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Christine_Jones 04 March 2010 20:05
Posts735
Joined09/10/09
I've just finished doing some basic operation work with one of my Year 10 tutees. They were using the Gelosia or lattice method to multiply two numbers. I've never used this method but realised that the student wasn't carrying it out in a manner that would get the correct answer. I worked out what was needed to get the method to work but was completely lost when decimals were involved.

Does anyone know how this method handles decimals? Do you just do the calculation and then place the decimal point afterwards? My student came up with an answer, that the teacher apparently told them, but as it didn't give the correct answer I have discounted it.

I have also searched on-line to find more information about this process. There is a large amount of information which appears to suggest that almost anything will work. As I know this isn't true, because my students method failed to get the correct answer, I wondered why all the conflicting information? The most important item appears to be the way you draw in the diagonals, which makes sense as this is what basically ensures the method works.

Does anyone teach this as the method of choice for multiplication?
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stevencarrwork 04 March 2010 21:10
Posts665
Joined28/07/09
I teach that way.

I ignore decimal places, and put them in when I have finished.

And you do have to draw in the diagonals in the correct direction.
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sarahrichards 04 March 2010 21:24
Posts74
Joined22/12/07
Many of my students who have problems with the traditional method  of long multiplication take to this like ducks to water.  In my experience it's worth spending time on seeing how it works and relating it to other ways of multiplying, otherwise it just seems like magic.
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maggieh 04 March 2010 22:34
Posts67
Joined26/09/08
Yes I love the way this method just works (like magic!) for some students (although I don't like using it myself).

I don't see the problem in just putting the decimals back at the end - you have to do this in traditional long multiplication anyway? Or am I missing the point here?

Have never heard it called Gelosia method before. Must look up what Gelosia means!

Regarding getting the diagonals right etc. here's some paper I designed especially for the Lattice method a couple of years ago. I always have trouble with this myself - and blame it on being left handed!

http://www.skillsworkshop.org/n/e3l1l2latticepaper.pdf 

and here's a practice sheet and a PPT (not written by me) both very simple but useful, no decimals and only 2-3 digit nos
http://www.skillsworkshop.org/l1num/n1l1latticeworksheet.pdf
http://www.skillsworkshop.org/l1num/n1l1l2lattice.zip

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MarkDawes 05 March 2010 00:46
Posts291
Joined09/04/07
Christine_Jones wrote:  Does anyone know how this method handles decimals?

You write the decimal points in the usual place in the two numbers.  The decimal point in the number along the top slides vertically downwards, while the decimal point in the number down the side slides horizontally across to the left.  When the two decimal points meet they combine and slide diagonally down to the left until they reach the answer digits.  This is the correct place to put the decimal point.

Christine_Jones wrote:  Does anyone teach this as the method of choice for multiplication?

No - absolutely not.

sarahrichards wrote:  In my experience it's worth spending time on seeing how it works and relating it to other ways of multiplying, otherwise it just seems like magic.

This, I think, is the only reason to use it.  I am happy for people who can multiply using a different method to try this out and to explore why it works (including why the ridiculous "sliding decimal point" that I mentioned above gives the correct answer).  In a similar way we can explore Egyptian Fractions but wouldn't want to use them on a daily basis, or Russian Peasant Multiplication but wouldn't use it as our main method.  This exploration and explanation of why gelosia works is useful in that it illuminates and makes links with other methods, but there is too much that is unnecessary about it to make it worth using as anyone's "standard method".

I would very much prefer to use the box method.  The multiplications are actually the same as are required by the gelosia method, except that numbers that are "tens" are treated as such and not as individual digits.  For example, if we have 32 x 57 we multiply 30 by 50, rather than have 3 x 5 and some magic involving diagonal lines.

What are the benefits of the box method? 
  • There is a clear link with area, which you don't get with gelosia.
  • You can extend it to help multiply algebraic expressions, which you can't with gelosia.
  • You can multiply fractions using it, which you can't with gelosia.
  • The numbers all means something, rather than being part of a magic trick.
  • You don't have to worry about which way to draw the diagonal lines (as mentioned by stevencarrwork).
What are the benefits of gelosia?
  • You can use it (once) to explore the link between this and a sensible method (like the box method).
  • Er....
  • that's it
  • (no more gelosia, Ed).
[With apologies to those who don't read Private Eye, to whom the final three bullet points may not be of interest!]
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stevencarrwork 05 March 2010 08:13
Posts665
Joined28/07/09
We 'absolutely not' teach this method as the method of choice?

I see students who are taught to put in all the zeros.

They end up with a lot of zeros and can't line up them up properly to do the final addition.

So they get it wrong.

I introduce the Gelosia method by explaining that multiplication is working out the area of a rectangle.

I then do composite areas, to show we can split up areas into smaller areas and then add them up.

Which explains why the rectangle gets split up into smaller areas.

And I easily adapt the 'lattice' method to deal with algebra. You just drop the lattice lines, which I already explained were just a trick to help with the carrys.

Of course, the Gelosia system doesn't explain the 'place value' system.

You should write all the zeros of the 4 in 423 and the 7 in 678 and write 400 times 70 . Otherwise it is just a trick.

Yes, and if I multiply a^2 by a^4 I should write out the aa and the aaaa and count them all up.

After all, if I don't, and just add up the 2 and the 4, the students won't understand the index system.

Adding up the 2 and the 4 is just a trick. The 2 means aa and the 4 means aaaa , so that is what the students should write out.

Just like the 4 in 423 means 400 and the 7 in 678 means 70, so that is what the students should write out.

After all, if the students master the place value system, we don't want to let them build on that knowledge by abstracting it into a useful time-saving concept.

We want them to keep writing out all the zero's because they are incapable of abstraction.

Write out all the powers of 10.

Just like the way real mathematicians write out all the powers longhand when multiplying a^3 by a^5, and write 'a' out eight times and count them up.

 
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Christine_Jones 05 March 2010 08:43
Posts735
Joined09/10/09
Steven, you seem to use this method and I'm interested to know why you choose this over other methods. Were you ever formally taught this method? Do you find that this method achievs the most successful results for your students? Or is there some other reason behind your choice?

The student I mentioned had obviously been taught this method but had not remembered exactly what was needed. The question I was asking involved multiplying a 3 figured number and a 2 figured number. The student had written it as if to start long multiplication but then hestitated. I asked what they were going to do next and their explanations showed that they didn't understand how to proceed. I then asked how they would tackle the question in school and this lattice method idea came up. I let the student proceed with the calculation mainly to see what they were going to do and then discussed whether the answer was correct.

The student had clearly grasped the idea of producing the diagonals and then adding down those diagonals remembering to include any carries. The problem appeared to be where to put the diagonals. However the student seemed very clear about the direction they had to be in. I wanted to leave the numbers where they were and change the diagonals but the student said that wasn't correct. This meant I had to change the order the digits of the number, placed along the side edge rather than the top edge, were written so that I could make the calculation work. The student didn't like this, and neither did I, but accepted it in preference to changing the diagonals.

I see now that what I should have done was move the number on the edge over to the opposite edge and the method would have worked. I don't know why I didn't see that at the time but I shall talk about that next time with the student.
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Christine_Jones 05 March 2010 08:48
Posts735
Joined09/10/09
Maggieh, thanks for all the information you have given.

Interestingly the student I was working with insisted that the diagonals had to go in the other direction and would not shift from that idea. I suspect that the strong visual image this leaves you with tends to stick in the students mind more than where the numbers go on the outside of the figure. I intend to spend sometime today working through all the possiblities with this method so I'm better prepared for our next lesson.
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stevencarrwork 05 March 2010 09:04
Posts665
Joined28/07/09
One of my students taught me that method.

As it happens, if I had to multiply 423 by 654, I would probably do it in my head, for fun, and then check on the calculator.

I do it like this web page, except that I would write the 37 on the left hand side of the box - not that it matters.

http://bodmas.org/blog/maths/multiplication-different-methods-2/index.html

My students get it right more often with this method than with others, and it leads naturally into multiplying out algebraic brackets, and factorising algebraic expressions.

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Christine_Jones 05 March 2010 09:29 - Last edited by Christine_Jones on 05 March 2010 09:31
Posts735
Joined09/10/09

MarkDawes, your explanation of how to manipulate the decimal point makes sense of what the student was trying to do during the lesson. I'm afraid I was completely unable to sort out what they were trying to do so had to abandon that and resort back to the idea usually used of counting the number of decimal places. I shall try out a few decimal multiplications using the method use describe and hopefully I'll be able to use this method much more effectively.

This exploration and explanation of why gelosia works is useful in that it illuminates and makes links with other methods, but there is too much that is unnecessary about it to make it worth using as anyone's "standard method".

The thing I liked about this method is that the student only has to deal with multplications of numbers up to 9. This reinforces their 'time tables' and perhaps provides a reason for why it is important to know these so well.

As most of the students I work with have problems dealing with multiplying by powers of 10, using the box method cause real problems. For instance in the example I was working on one box involved multiplying 400 by 30. Most of the students i work with have problems with this. Since they are unable to multiply 400 by 10 correctly then changing the 10 to 20 causes lots of problems. This lattice method gets rid of those problems.

I'm going to think about the advantages and disadvantages of the two methods. I fully agree with your list of advantages of the box method and infact think it is a better method because of these advantages. However I do think there are disadvantages to this method such as:
 

  • Students can't multiply by powers of 10 so have problems with this method.
  • We only ever demonstrate breaking the numbers down into their place value components But the method is much more general than that but this fact is very rarely used.

If we routinely used different ways of breaking the numbers down to obtain the answers then the connection to other topics within maths would be clearer. Such as algebraic expansion etc.

Now consider the possible benefits of the lattice method:
 

  • It only involves times tables use so helps reinforce those facts.
  • Students can carry it out without worrying about the maths behind it.
This last point you may consider a disadvantage. However, consider long multiplication and the rigmarole involved with that. Although I knew to add an extra zero everytime I changed to the next line/ digit I never altered the idea of multiplying by single digits. So for instance when multiplying 436 by 34 my internal patter would go something like 4x6=24 so 4 down carry 2, 4 times 3 is 12 add 2 is 14 so 4 down carry 1, 4 times 4 is 16 add 1 is 17 so write 17 down. Put zero at end then 3 times 6 is 18 put 8 down carry 1 and so on.

This is fundamentally the same idea that is behind the lattice method. Most of the students in my generation had no deeper understanding of maths than either of these methods imply and yet 30 of us went on to do A level maths and then to university to study subjects that most would consider to have a high mathematical content. This 'lack of umderstanding' never really caused us a problem.

Whilst I'm not suggesting that understanding isn't important I think there are other issues that need to be considered before dismissing a technique. Steven has made several valid points that are worth considering and seeing whether they provide advantages for the students.


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