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Posts Tagged ‘Multiplication’

Another Multiplication Technique

June 20, 2016 2 comments

I’ve written a few posts over the years on different multiplication techniques (see this and this), there are many and each has its own appeals and pitfalls.

Today I discovered another. I was looking over Q D1 exam paper and came across this flowchart:

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The questions were all fairly reasonable and one of my students was completing the question to see if he had for it right. Afterwards I asked if he knew. What the algorithm was doing, he wasn’t sure at first but when another student explained it was finding the product of x and y he realised.

Then he asked, “but why does it work?

I looked at the algorithm and initially it didn’t jump out at me. I tried the algorithm with 64 and 8.

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I could see it worked through mocking factors of 2 from the left to the right but this time there was no odd numbers, so I picked some other numbers:

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And that’s when it all made sense. Basically, what’s happening is you are moving factors of 2 from x to y thus keeping the product equal. When x is odd, you remove “one” of y from your multiplication and put it in column t. Your product is actually xy + t all the way down, it’s just that until you take any out your value for t is 0. T is a running total of all you have taken from your product.

The above becomes:

40 × 20
= 20 × 40
= 10 × 80
= 5 × 160
= 4 × 160 + 160
= 2 × 320 + 160
= 1 × 640 + 160
= 0 × 640 + 160 + 640
= 0 × 640 + 800
= 800

I tried it out again to be sure:

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This is an nice little multiplication method that works, I’m not sure it’s very practical, buy interesting nontheless.

Have you met this method before? Have you encountered any other strange multiplication techniques?

Vedic Multiplication

November 30, 2014 2 comments

On Friday, timehop told me it was a year since I wrote this post on multiplication methods. I’d forgotten about the post, and tweeted a link to it. A number of people commented and tweeted about it and a nice discussion ensued. Part of the discussion moved to Vedic Multiplication.

I know a few Vedic Multiplication methods, and believe there to be many more. Most of the ones I know link to the most common algorithms but there is this curious one used for numbers close to one hundred.

Start with a multiplication problem:

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First, take bother numbers away from 100:

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Multiply those numbers and make them your last two digits:

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Take one of the differences from 100 away from the other number (it should be the same):

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That becomes your first two digits:

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It’s an interesting little trick. I don’t see it as something that there is any reason to teach, and I don’t think it promotes understanding at all, but it’s interesting nonetheless. I think it may have a use in lessons, as an interesting introduction to algebraic proof.

How would you prove it?

First, consider the product ab, and apply the same steps:

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The last two digits are (100-a)(100-b). The first two are a-(100-b) which equals a+b-100. Or b-(100-a) which also equals a+b-100. To make these the first two digits of a four digit number we need to multiply the expression by 100.

This gives:

(100-a)(100-b)+100(a+b-100)

Which expands to:

10000-100a-100b+ab+100a+100b-10000

Which cancels to:

ab As required.

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A nice, accessible, algebraic proof that proves this works. It works for all numbers, not just those close to 100, but if your product (100-a)(100-b) > 99 (ie more than 2 digits) you need to carry the digits.

Bizarre multiplication

June 17, 2014 4 comments

A while ago I wrote this post about multiplication. In it I explored a few different ways of multiplying and what I felt worked best and why. I covered the main ones, and I realise there are many different methods.

However, today, while marking my year 8 books I discovered a method I had never seen before:

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I looked at it and thought, “bizarre”. I asked the girl in question who taught her it and she said she’d didn’t know, but that it made sense to her but grid and column methods don’t.

I like the method, it reminds me of expanding a pair of binomial brackets, but it the numbers were 19 + 4 etc. It’s fantastic for 2 digit by 2 digital or 2 digit by 3 digit and I think it shows the distributive property quite well. There are major limitations though. It looks really neat in these two examples, but the questions that were 3 digit by 3 digit were a little spaghetti like and the ones with 4 digits were totally illegible.

Have you seen this method before? Do you like it? I’d love to know!

Multiplication Methods

November 28, 2013 7 comments

 

Last night I saw an intriguing tweet from @mr_chadwick . Mr Chadwick is a primary teacher and he was worried that his daughter who is in year 8 was still multiplying using the grid method. This caused a fascinating conversation regarding the different methods of completing multiplication tasks and the pros and cons of each and it got me thinking quite a lot about the subject.

 

There are many ways to complete multiplication problems, the main ones being Grid, Column (aka Standard or Long), and Chinese Grid (or Spagetti Method or Lattice Method). (Also, I have recently been shown this ancient method by an A Level pupil who grew up in the Democratic Republic of Congo).

 

I was wondering if any one was particularly faster than the other, so I tried some out:

 

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Each method took me a little over a minute to compute the product of 2 three digit numbers. (Except the grid method which took about 15 seconds longer). From experience I know that people tend to prefer the method they were shown first, and I don’t see a problem using any of them as long as there is an understanding of the concepts and that the person in question is proficient at using the method they have chosen.

I did think my prefered method for teaching someone who didn’t know any methods would be the Lattice Method, as I see a much larger potential for silly errors in the other methods than i do for this, but last nights discussion has got me thinking a little differently. The discussion moved onto applications in algebra. I know a lot of people prefer to use the grid method to expand double brackets, I personally prefer crab claw method, but i teach both and allow my students to decide, and some much preefer the grid. It also works quite well for larger polynomials, as shown here (in a video which rather confusingly calls it the lattice method!). The grid method can also be applied to matrices, as I have written about previously.

I’m still unsure as to which i prefer. The Lattice method gives a far lower chance of making silly errors, and I think it is the best one fro ensuring the decimal point ends up in the correct place when multiplying decimals, but the grid does have the benefits of being applied to much higher levels! I’d welcome your views on the subject.

I feel that i should also include another multiplication method which I discovered a few years ago, I was introduced to it as Japanese multiplication, but I’ve recently heard of it referrred to Gorilla Multiplication. I think it may have its origins in india and if you want to know more then Alex Bellos has written about it in his book Alex’s Adventures in Numberland (Another on my christmas list… and a book who’s american title is the most amazing title for a maths book I’ve ever heard: “Here’s Looking at Euclid”!)

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