Tag Archives: multiply

Why Don’t They Teach Our Kids “Proof by Nine” As A Part Of Math?

by Anura Guruge

“Proof by Nine” (I have always called it (possibly incorrectly) “Proof of Nine”) is a clever and totally foolproof way to check your multiplication — without having to rely on a calculator or an electronic device. I was taught it, in Ceylon, at school, probably when I was around 6 or 7, as an integral part of learning to do multiplication. We are talking the early 1960s. We did NOT, especially in Ceylon, have calculators or for that matter even slide rules. So “Proof of Nine” was the ONLY way we had — and even our teachers — of checking the correctness of multiplication.

The 3rd image below (albeit purely scanned) in its first paragraph spells out EVERYTHING as it pertained to I. I would do “Proof of Nine”, by rote, as soon as I did a multiplication by hand. I still do it TODAY, for the fun of it. I have tried to teach the kids but they don’t fully grasp it.

The two articles I show here try to explain it very well. I did this diagram to show you the basic mechanics AND I KNOW that most of you are going to say that it is totally confusing.

Shame that they don’t teach it in school. Because all of us would know how to do it.

PLEASE look it up. It is very neat and awfully handy if you don’t have any other means to check a multiplication.

prrofby9anuMy diagram. So I want to check whether 7826 x 43 = 336518.

I start by drawing the slanted ‘X’. The I come up with the two vertical numbers, in this case 5 & 7. I do that by repeatedly adding the digits until I have but one digit. I ALWAYS throw out any 9 I encounter — and even digits that together add up to nine.

I multiply 5 x 7. That is 35. Add the digits. 8.

I add up the digits in 336518. I notice two obvious pairings that come to 9. 3+6 & 1+8. I throw those out. Actually it will still work IF I didn’t throw them out! All comes out in the wash. Well it adds up to 8.

The same ‘8’ I got when I multiplied 5×7 and got 35. The two ‘8’s match. The multiplication was RIGHT.


First article. Click to ENLARGE and read here. Use link below to access original.

Click here to access original.


Second article. Click to ENLARGE and read here. Use link below to access original.

Click here to access original.


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by Anura Guruge

Can YOU Subtract ’12’ From ‘9’ Using ‘Number Bonds’?


..by Anura Guruge

Related posts:
>> PI day — Mar. 7, 2013.
The Math Museum In
>> New York City 
Mar. 4, 2013.

If so, as the incomparable Kipling said: “You’re a better man than I am, Gunga Din!”.

I guess unlike me, an old man pushing 60, who learned his 3 Rs in the 1950s (in a third world country), you must know all about ‘Number Bonds‘. Just this year, looking at Teischan’s homework I became vaguely aware that they were doing this stuff called ‘number bonds’. Initially all I had seen was the two dangling ball efforts to deconstruct an integer and I had no problems with that since I do think that kids should appreciate how a number comes to be what it is. Now to be fair, ‘number bonds’ is part of the new, ‘new math‘ – the so called ‘Singapore Math‘ (and in case you don’t know, ‘Singapore‘ is a tiny Asian country, really best known for its infamous capture by the damn Japs during WW II, when it, like my home country was a British colony). Singapore Math is being taught all over the place, not just in Alton.

Click to ENLARGE.

Click to ENLARGE.

Then yesterday Deanna showed me this homework that Teischan was struggling with. It took my breath away. I never realized that they were going to use the two dangling balls to do arithmetic operations. So, have a look at this.

They had done the top 2 in class, on the blackboard. She had supposedly transcribed the method and answer from the board onto her sheet. Number 2 was wrong and we asume she copied it wrong.

Though I had never seen subtraction done this way, I could work out how they got ’14’ for #1 and the (correct answer) ’12’ for #2.

Then I noticed that we had a problem! Number 3 and 4 (’17.’ & ’18’ on the sheet) were very different to the other three, and the two they had done in class. Can you spot the difference? Yes, the ‘ones’ number of the second operand is BIGGER than that of the first operand. To use the same technique as for one & two, you would have to use a NEGATIVE NUMBER, in this case ‘-4′ which when added to the ’10’ will give you the right answer ‘6’! But even I, with my high expectations of kids, do not really expect 6 and 7 year olds to be that conversant with negative numbers. IF you don’t use negative numbers, then you have to use a DIFFERENT technique to handle numbers 3 & 4!

I had no idea what that different technique would be. So I did what I always do when I am stuck. I Googled. I found this excellent video tutorial, with exactly the right example, at ‘onlinemathlearning.com‘. Here it is. You have to watch it.

Click to access page. It is the 1st video of the three.

Click to access page. It is the 1st video of the three.

Notice the BIG ‘No!’. I was mortified.

There is an exception to the method. This is for 6 and 7 year olds.

I have two issues with using this strange, two dangling ball approach for teaching kids subtraction.

1/ This method does NOT ELIMINATE the need to do subtraction! Ah? Kids still have to do subtraction with this approach. So what is the gain. I would be all in favor if this method eliminated the need to subtract and said kids could do subtraction by just adding numbers. Now that isn’t as crazy as it may sound to the uninitiated. Logarithms. Now that is real math. We (as kids who didn’t have calculators) used logarithms because ‘logs’ allowed you to do complex multiplication and division using just addition and subtraction. That is neat and useful. You eliminate a complicated process with an easier, better mastered technique. Not so with the two dangling balls. You still have to do the damn operation — in this case subtraction. Plus, how do they teach subtraction. They count the difference between the two numbers. If so, why bother with the two dangling balls. Just count the difference to begin with!

Click to access article.

Click to access article.

2/ Having an exception to deal with a common occurrence is beyond unacceptable. The abiding, (to some of us sensually stimulating) beauty of maths is its predictability, its uniformity. You can’t have a so called ‘easy method’ that has exceptions to deal with common occurrences. This is plain crazy.

Yes, I am the first to admit that I am an old fashioned and stuck in my ways. But, I see no problems with the way we learned our arithmetic, algebra, geometry and trigonometry. We had no electronic calculators or even mechanical ones. We learned things by rote and repetition, over and over and over again.

This was the dedication in one of my recent books.

Click to ENLARGE.

Click to ENLARGE.

P.S.: I collect old logarithmic/trigonometric tables (i.e., the  so called ‘log’ books) and old slide rulers. Send me pictures and quote me a price. Yes, every once in awhile, late at night, when I feel that I am due a treat, and have a few dollars stashed away, I log onto eBay and see what they have. Got a real beauty of a slide rule, cheap, very cheap, a couple of months ago — making use of the eBay, ‘make an offer’ feature.