How well do you know the multiplication table?

[MadAtom]

I'm now a 19 years old undergraduate physics student. I've a reasonable proficiency with calculus (well, no really...) but if you ask me 6*7 (unlike my mother and many other older people I know) I'll take awhile until I come up with an answer (I'll ask myself 6*5 and then add 12 to that result).

I've never recognized the value of multiplication table memorization. Instead, I have a very poor "multiplication database" on my brain and so I've to do some tricks to get to the results out of my database (like the exemple above). This can take awhile sometimes (most of the times actualy). Other people, intead, have a specific location for 6*7 on their brains and, therefore, they are faster on this.

What do you think of this? To which group do you belong? Do you value the hability to do mental math (multiplication and sum)?

 

[phinds]

I very much value the ability to do simple math quickly in my head. I can't imaging having gone through the computer-based and technology-based career that I've had without having it.

 

[zoobyshoe]

When I was in elementary school, we drilled and drilled the multiplication tables for hours. (We only went up to 12. I wish we'd gone up to 15 like some schools.)

So, for years it was automatic. Then calculators became cheap and ubiquitous and I found myself pulling one out automatically when I had any math to do whatever. The result was a terrible erosion of my ability to do any math in my head.

 

[MadAtom]

I can't imaging having gone through the computer-based and technology-based career that I've had without having it.

So, it is the ability per se that helped you or by practicing those techniques you've gained some general ability in math and problem solving?

 

[DivergentSpectrum]

odd question lol. i remember that 42 is 6*7 because that's the meaning the life, the universe, and everything (hitchhikers guide to the universe)
i still struggle with 6*8 and 7*8, sometimes its pretty frustrating.

 

[phinds]

So, it is the ability per se that helped you or by practicing those techniques you've gained some general ability in math and problem solving?

The ability itself is what I have found extremely useful, and not just limited to my professional life. For example, I often find that statistics given even in reputable publications like Time Magazine (just as an off-the-cuff example, they are not, by far the worse) are obviously wrong and don't pass even the most simple smell test. I can SEE very quickly that they HAVE to be off because simple in-my-head rough calculations that give me estimates for what they would imply for the underlying facts show clearly that they just can't be right.

I'm old enough that when I was in school we drilled arithmetic seriously including memorizing the squares up to 15 (and when I got into computers, 16 became ingrained).

 

[Matterwave]

I have multiplication tables memorized up to 9x9=81. 10 is super easy so there's no need to memorize up to 10x10, and 11x11 is also quite easy and no need to memorize. 12 I know somewhat well due to there being 24 hours in a day, but something like 12x7 or 12x11 would be problematic for me.

 

[Pythagorean]

I've pretty much got 12*10 nailed down.

 

[DiracPool]

What do you think of this? To which group do you belong? Do you value the hability to do mental math (multiplication and sum)?

That's interesting. These are things I don't typically think about. I always just assumed every (especially physics undergrad) body had essentially the same single digit multiplication table memorized. It's just automatic for me, but maybe if it hadn't been hammered into my brain in grade school I wouldn't have the same "knee-jerk" capacity I have now. I don't know. Maybe there's a window for learning it so you can bypass the calculation process. After that, you have to do things like add 12, etc., although my guess would be that with focused training, you should be able to memorize these, and even 2 digit numbers up to say, 20 or more. Whatever the case, at the end of the day it's simply a time saving tool, that's all, kind of like memorizing the power rule in calculus.

 

[Medicol]

...
What do you think of this? To which group do you belong? Do you value the hability to do mental math (multiplication and sum)?

It's fine to do computation whatever way you'd want to. I do all 2 digit multiplications mentally and am always slower than others too. Why do you want to do this faster than others ?

 

[Pythagorean]

I always do 7*3 = 21, then *2 = 42 for 7*6. I hate that one. And 7*8 .

 

[WWGD]

How about Roman numeral multiplication : (XVI ) x (ML) =? Or base 16 : (AB ) (3C)=?
For the first : Multiply L by I , carry a V... So quit complaining, you have it much easier
than Roman kids did ( and maybe Klingons or something use base 16 ).

 

[Pythagorean]

Alright, fine, fine, I'll get off your lawn, sorry dude.

 

[Rocket50]

I can do up to 20x20 or a little more. In India, in 5th grade (in CBSE), they have to know up to 20x20.

 

[WWGD]

There is a nice trick for squaring:

$a^2 = a^2-b^2+b^2 = (a-b)(a+b)$ , then, given an a, you choose the right b; say

a=28 . Then you can choose b=2 , and get : $28^2 = (28+2)(28-2)+ 2^2 = 30(26)+4$ . Usually

choose a to get a number ending in 0 ; you just need to be able to compute 26(3), which is not easy, and then just put a zero on the right, but not impossible. If you know the squares up to 20 , you can figure something like: $987^2 = (987+13)(987-13)+13^2 = 974(1000)+13^2$

And you can always reduce multiplication to a squaring issue when the two numbers being multiplied have different parity.

 

[Bacle2]

And how about the trick of dividing by 2 and appending a zero when multiplying by 5 ? This is just because 5 =10/2. Say you want to find 36 x 5 . Then 36/2 =18 and 18 x 10 =180. By the same trick, dividing by 5 is multiplying by 2 and appending a zero :

240/5 240 x2 =480 , 480/10= 480.

 

[lisab]

odd question lol. i remember that 42 is 6*7 because that's the meaning the life, the universe, and everything (hitchhikers guide to the universe)
i still struggle with 6*8 and 7*8, sometimes its pretty frustrating.

I always do 7*3 = 21, then *2 = 42 for 7*6. I hate that one. And 7*8 .

It seems many people are weak for 7*8 (including myself ).

Are there any elementary school teachers here (or lurking) who can explain why this is? Are we teaching multiplication tables wrong? Or is this a consequence of having a base-10 number system?

 

[Pythagorean]

Yes, it's probably a consequence of base-10 in addition to the fact that we only go up to 10 or 12 in the US (or did when I was a kid). The base-10 leave 7 and 8 the numbers with least obvious mnemonics to break it down quickly (in terms of multiplication). For instance, the sums of digits and how they relate to their multiples is seen here:

http://www.sjsu.edu/faculty/watkins/Digitsum0.htm

This accounts for why 3, 6, and 9 are easy. 2 and 4 are easy because it is just doubling twice (since we're staying on 10-12, all fairly easy to double twice, only leading up to 48 max. 5 and 10 are easy because their last digit is always the same as well as the division by two and multiplication by 10 above.

With 8, you have to double three times now, and that gets more tedious, but you can handle it for the most part on the smaller numbers, especially because you can use the mnemonics of the other number for 8x[1-6]. For whatever reason, we learn the diagonal well so we know 8x8, and 9 and 10 we're back to some of the simplest mnemonics. But that 7 is fairly large prime with no digit addition mnemonics, a sum of digits pattern that aren't easy to remember, and a pattern for last digit that's not easy to remember. So when 7 and 8 clash, our lowest hanging fruit is to to double 7 3 times or go 7*4*2 = 28*2, and we don't go up to 28 with the multiplication table, so we haven't memorize 28*2 and so we have to do the computation in our head the one time we need it every blue moon.

But that's just my personal theory

 

[ShayanJ]

I'm weak in remembering my past, so I don't know how I did it in elementary school. But I think practice, and not repeating it over and over and over, stuck into my mind, multiplication table till 9x9. For other more complicated multiplications, I have tricks.
For example for 34 by 62. It is (30+4)x62=3x10x62+4x62.
And that's easy to calculate in mind. But further simplification is still possible: 3x10x62+4x62=60+3x10x60+4x60+8.
This way, I can even multiply two 3digit numbers in mind. But for more digits, this becomes messy and actually you don't have to do that much calculation in your mind.
Also division can be simplified. If you want to divide a number with a zero at the end by 2(5), just throw out the zero and multiply the remaining by 5(2). You also can factorize the divisor(not to primes) and divide the dividend by one factor and divide the result by the next factor and so on.
I think I'm good at such calculations by mind. Sometimes I even differentiate or integrate in mind and not just for simple functions. I only need to be able to focus enough.

 

[MattRob]

Heh, I'm actually in the exact same boat as you, OP. Sophmore, undergrad, 20, taking calculus 2 and I don't have my memorization tables down.

Funny thing. I've always had a really hard time with rote memorization things. It's why I hate these calculus sections with trigonometry. Thank goodness for flash cards!

It's really funny, actually, I'm so huge into the sciences now and I profess to love mathematics, yet I'm not particularly advanced in them and I really wasn't the brightest kid in Elementary...

 

[jz92wjaz]

We did everything up to 12*12. 6*7 and 7*8 took a little more thinking when I first learned them, but once I got them I never lost them (or any of the others).

 

[zoobyshoe]

It seems many people are weak for 7*8 (including myself ).

Are there any elementary school teachers here (or lurking) who can explain why this is? Are we teaching multiplication tables wrong? Or is this a consequence of having a base-10 number system?

For me it's because 7 x 8 = 56 breaks a pattern that starts at 5 x 8.

5 x 8 = 40 is a very satisfying result. I always like it when the one's place is empty. (That's half of what makes the ten's table so easy.) The jump from 4 x 8 = 32 to 5 x 8 = 40 is, therefore, satisfying.

6 x 8 = 48 is even more satisfying because it's very easy to add 8 to 40, and 48 outright repeats the 8 from 6 x 8. (That last thing is half of what makes the 11's table so easy.)

I'm always hoping the next one continues the expectation of ease, repetition, but of course it dashes both, and I resent 7 x 8 for it. It becomes associated with a kind of 'party's over, back to work' disappointment, and we don't like to remember disappointments. Every time I come to 7 x 8, there's a pause. When it crops up outside the context of the table, the disapointment never-the-less comes along with it.

 

[leroyjenkens]

I have all of the single digit multiplications memorized. I never had to use any mnemonic or add numbers to the previous multiplication. The answer just followed. So when I say "7 times 8", the automatic continuation of that in my mind is "is 56."
I never remember the 9's multiplication tables, though. And that's because my cousin taught me a trick where you hold your hands in front of you, palms facing away, and whatever you multiply times 9, you pull that finger down, and the remaining fingers left standing are your answer, separated by the down finger. For example, with 9 times 3, you pull your left middle finger down, since it's the 3rd finger in the sequence, and you have 2 fingers left standing to the left of the downed finger, and 7 standing to the right of that finger. That gives you the answer; 27. Due to relying on that neat trick, I never memorized them. Even if I try to do it in my head without using my hands, I automatically picture virtual hands in my head. I'm stuck.

 

[Choppy]

I disagree that being able to perform mental multiplication is just a matter of time savings.

I think being able to do calculations in your head can give you a few advantages, particularly in a STEM career.

You see, if you know you NEED to perform a calculation, you'll do it. You'll find a calculator and punch in the numbers if you can't do it in your head (or don't trust yourself).

But what if you don't recognize that need? Or what about those scenarios where it might be nice to know what something works out to, but not nice enough to bother digging out your calculator (or opening up the app)? Or how do you know that the results that your calculator is give you are correct?

Some people are very good at ballpark estimates and I think that can be a huge advantage in error recognition, project planning, and resource allocation.

 

[MadAtom]

Funny thing. I've always had a really hard time with rote memorization things. It's why I hate these calculus sections with trigonometry. Thank goodness for flash cards!

This memorization part made me not like maths too much at first. Not because I was struggling with it, but because I didn't find it interesting. Only when we started doing algebra problems about coming up with an equation given some real life situation.

 

[SteamKing]

At my elementary school, to prepare us for learning the multiplication tables, they started to teach us how to count by 2's, 3's, etc., starting in the first and second grade, without mentioning the word 'multiplication' to us. By the time third grade rolled around, and we started to learn and drill the multiplication tables, it was old hat by then, because we had learned how to count by all the single-digit numbers.

'Two, four, six, eight, how do we multiplicate?" could be a cheer, I suppose, for all those who are somewhat intimidated by math.

 

[voko]

In India you are required to memorize up to 20x20? Wow. What would be a practical use of that? 9x9 has a simple explanation: you need that to multiply multi-digit base-10 numbers. 9x9 (and 10x10 because it is so easy) is pretty much automatic for me.

 

[SixNein]

I think memorizing the first 10 numbers is important (100 combinations). Afterwards, one can use algebra to break a multiplication problem down.

For example use the foil method, 25 * 21 = (20+5)(20+1) = 400 + 100 + 20 + 5 = 525.

There are other small tricks like with squaring numbers ending in 5.

5^2 = (0^2+0) * 100 + 25
15^2 = (1^2+1) * 100 + 25
25^2 = (2^2+2) * 100 + 25
35^2 = (3^2+3) * 100 + 25
45^2 = (4^2+4) * 100 + 25
etc...

 

[Char. Limit]

Alternately...

25*21 = (23+2)(23-2) = 23^2 - 2^2 = 529 - 4 = 525

Super easy.

 

[voko]

Super easy?

25 x 21 = 25 x 20 + 25 = 500 + 25 = 525.

That's what easy means.

Oops, I used 25 x 20 = 500, which is beyond 10 x 10.

 

[Char. Limit]

I'm... not exactly sure where you're disproving that differences of squares is easy. Do you not have squares memorized? I have squares memorized up to 25 at least.

 

[voko]

No, I do not have any squares memorized beyond 11 x 11, and I do not think that memorizing squares up to 25 x 25 is widespread. Even without that, I find decomposing 25 x 21 into (23 + 2) x (23 - 2) far less intuitive than into 25 x (20 + 1), but that may be because I do not have those squares memorized.

 

[reenmachine]

Guess I know them by heart if both factors are 11 or lower.If one factor is 12 I might have to do the calculation in a split-second, but that's still different than "knowing it by heart".After that I either do it in my head or use a calculator depending how hard it is and how sharp I feel.

 

[symbolipoint]

I'm now a 19 years old undergraduate physics student. I've a reasonable proficiency with calculus (well, no really...) but if you ask me 6*7 (unlike my mother and many other older people I know) I'll take awhile until I come up with an answer (I'll ask myself 6*5 and then add 12 to that result).

I've never recognized the value of multiplication table memorization. Instead, I have a very poor "multiplication database" on my brain and so I've to do some tricks to get to the results out of my database (like the exemple above). This can take awhile sometimes (most of the times actualy). Other people, intead, have a specific location for 6*7 on their brains and, therefore, they are faster on this.

What do you think of this? To which group do you belong? Do you value the hability to do mental math (multiplication and sum)?

What you are doing, the mental tricks to find the multiplication fact you do not know, is good. Yes, also memorizing the multiplication table is good. At least you understand multiplication and know how to access logical tricks to find what you have not memorized. Understand that hearing the words something multiplied by someother is not the same as seeing the written expression of something multiplied by someother. Mathematics, arithmetic, mostly works better for some people as a written language, and not as a spoken language. One supsects that you will improve with practice and experience, and restudying details of whole number multiplications. A cashier in a retail place should probably be very good at adding, subtracting, and multiplying whole numbers; but an engineer or physicist needs to UNDERSTAND these things far more than does the cashier.

 

[symbolipoint]

It seems many people are weak for 7*8 (including myself ).

Are there any elementary school teachers here (or lurking) who can explain why this is? Are we teaching multiplication tables wrong? Or is this a consequence of having a base-10 number system?

At a very young age, when the multiplication facts for 7 were taught, I memorized them fairly quickly, even though the sequence did not seem to show a pattern in them (for a your person still a child). This lack of apparent pattern was interesting and I paid more attention and memorized them from 7*1 up to the required 7*12.

We live in base-ten normally, and this is why we tend to know the basic facts in base-ten. The few typical facts we do not know, we can find through what facts we do know.

 

[voko]

I memorized them fairly quickly, even though the sequence did not seem to show a pattern in them (for a your person still a child). This lack of apparent pattern was interesting and I paid more attention and memorized them from 7*1 up to the required 7*12.

Interesting. By the time I was taught the multiplication table, we had addition drilled into us rather solidly, so it was plain to see the pattern.

7 * 1 = 7
7 * 2 = 14 (+7)
7 * 3 = 21 (+7)

And so on.

Likewise,

6 * 2 = 12
7 * 2 = 14 (+2)
8 * 2 = 16 (+2)

Etc.

 

[skeptic2]

As an RF engineer, multiplication is too difficult for me. Instead of memorizing the multiplication tables, I memorized the log tables (well, part of them). All I know is addition and subtraction.

 

[SteamKing]

A cashier in a retail place should probably be very good at adding, subtracting, and multiplying whole numbers; but an engineer or physicist needs to UNDERSTAND these things far more than does the cashier.

Some of the cashiers I've dealt with would be hard pressed to give the answer to 2+2, and any cashier under 30 will be more likely to not know how to count change without use of the cash register or calculator, whereas older cashiers could function perfectly well without such tools. People take mental arithmetic for granted, once it is acquired: it's a skill which is valuable to have because it trains the mind, if nothing else.

I think the UK finally succumbed to decimalization of the currency because their educational system could no longer teach its pupils how to manage pounds, shillings, and pence in this new, modern age, when earlier generations of shopkeepers could probably run rings around anybody in terms of applying mental math.

 

[Dembadon]

It's interesting to see the replies and all of the different methods for mental arithmetic. My best description for how I learned:

I made "visual" correlations with pairs of single digit numbers and their product. Whenever I see 6 and 8 together, I immediately "see" 48 (it also helps that 8 rhymes with 48). The result is involuntary and doesn't require any additional arithmetic or computation. This occurs for the entire table up to 12. I don't remember much about how arithmetic was taught when I was young, so I've no explanation as to why it's like this for me.

For other mental arithmetic, I usually employ some sort of reference point, as many other people do. How many times does 15 "go into" 75? Inner thoughts: I need four 15's to get to 60 (a reference I've memorized), which is only 15 away from 75. Really simple example but should illustrate the point.

As to whether I think it's valuable; it's really nice not having to pull out a phone/calculator if the situation calls for some simple multiplication. I'd be a little embarrassed if I had to do so. Also, as has already been mentioned, I believe it builds "math muscles" that can be used for quick sanity checks on mathematically trivial statements.

 

[symbolipoint]

Once in the past was taught, "eight times eight fell on the floor; pick it up and it's sixty-four".

 

[branwik1]

If you have some problems with multiplication for example you can always visit many of helpful sites like this - http://www.aztekium.pl/Master
I've use it to teach my kids how to multiplicate quickly :)

 

[Student100]

I think memorizing the first 10 numbers is important (100 combinations). Afterwards, one can use algebra to break a multiplication problem down.

For example use the foil method, 25 * 21 = (20+5)(20+1) = 400 + 100 + 20 + 5 = 525.

There are other small tricks like with squaring numbers ending in 5.

5^2 = (0^2+0) * 100 + 25
15^2 = (1^2+1) * 100 + 25
25^2 = (2^2+2) * 100 + 25
35^2 = (3^2+3) * 100 + 25
45^2 = (4^2+4) * 100 + 25
etc...

Seems like it would be easier saying 25 * 20 is 500 +25 = 525.

I never saw the importance of memorizing multiplication tables, after enough practice using them simplifying other mathematics they become burned into your brain anyway.

I actually think teaching kids tables is about as inspired as the FOIL method, which is also complete garbage.

 

[Bystander]

I never saw the importance of memorizing multiplication tables, after enough practice using them simplifying other mathematics they become burned into your brain anyway.

?

 

[Mark44]

Seems like it would be easier saying 25 * 20 is 500 +25 = 525.

Except it isn't true. This might have been a shortcut that cut too many steps. The original example was 25 * 21 = 25 * 20 + 25 *1 = 500 + 25 = 525.

I never saw the importance of memorizing multiplication tables, after enough practice using them simplifying other mathematics they become burned into your brain anyway.

I'm not sure I buy this. Maybe a few of them get burned in, but I doubt that all of them do.

I actually think teaching kids tables is about as inspired as the FOIL method, which is also complete garbage.

No, it isn't, provided that it is used only for what it's supposed to be used on: a product of two binomials.

Plain old arithmetic, to include what are called the addition facts and the multiplication table, are IMO important, and are necessary as the foundation on which more complicated mathematics are built. While it's important to be able to write down computations, having the ability to add numbers and do multiplications in your head are skills that can be used in estimation, another important skill.

 

[Student100]

?

Let me explain a bit better,when I was young we were giving laminated cards with both division and multiplication tables and told to memorize them, before we ever knew what multiplication or division was. I still don't see the benefit of this practice, I would have rather learned what division and multiplication actually was and used that knowledge to solve simple equations like 5*6 + 4 = 34. After you solve enough problems like that it's a natural consequence to "memorize" simple multiplication because you've worked it out so much.

I'm not sure I buy this. Maybe a few of them get burned in, but I doubt that all of them do.

That's how I learned the majority of them, I actually did quite badly at memorizing just the cards we were given.

No, it isn't, provided that it is used only for what it's supposed to be used on: a product of two binomials.

Why do that when you can just use the same multiplication format you've been using for the last 8-10 years, and apply it generally?

Plain old arithmetic, to include what are called the addition facts and the multiplication table, are IMO important, and are necessary as the foundation on which more complicated mathematics are built. While it's important to be able to write down computations, having the ability to add numbers and do multiplications in your head are skills that can be used in estimation, another important skill.

Sure, I agree with you. I just don't think targeted lessons that require memorization of tables are any better than the memorization that naturally comes from working problems. Maybe yours and others experiences were so vastly different than mine that we're actually talking about nearly the same practice. I'll always look back at being handed cards with tables on them and told to memorize the random numbers on them as a negative experience.

 

[Mark44]

No, it isn't, provided that it is used only for what it's supposed to be used on: a product of two binomials.

Why do that when you can just use the same multiplication format you've been using for the last 8-10 years, and apply it generally?

The people who learn about FOIL either haven't been doing this multiplication for the last 8 to 10 years, or maybe aren't able to generalize from, say, 23 * 15 to (a + 3)(b + 5).

Plain old arithmetic, to include what are called the addition facts and the multiplication table, are IMO important, and are necessary as the foundation on which more complicated mathematics are built. While it's important to be able to write down computations, having the ability to add numbers and do multiplications in your head are skills that can be used in estimation, another important skill.

Sure, I agree with you. I just don't think targeted lessons that require memorization of tables are any better than the memorization that naturally comes from working problems. Maybe yours and others experiences were so vastly different than mine that we're actually talking about nearly the same practice. I'll always look back at being handed cards with tables on them and told to memorize the random numbers on them as a negative experience.

And so what? Virtually any endeavor, if you want to get good at it, requires a lot of time on the basics. If you want to learn how to play the piano, you have to learn the names of the keys so that you can read music sheets. After that, there's a lot of time spent on practice, building "muscle memory" so that you can play a tune without having to think about each and every note. The same is true for sports of all kinds, with a lot of time spent hitting a ball, or throwing a ball into a basket, or whatever. The more you perform these actions, the more fluid and automatic they become. The same is true for arithmetic and mathematics at a higher level, I believe. If you aren't sure whether it's 6 x 9 = 63 or 54, it becomes much more difficult to do quick and dirty approximations as sanity checks on more difficult problems.

 

[Fervent Freyja]

Let me explain a bit better,when I was young we were giving laminated cards with both division and multiplication tables and told to memorize them, before we ever knew what multiplication or division was.

Nice to know, I was under the impression that you were a little boy...

 

[Student100]

The people who learn about FOIL either haven't been doing this multiplication for the last 8 to 10 years, or maybe aren't able to generalize from, say, 23 * 15 to (a + 3)(b + 5).

Does it require more thought to extend the multiplication in this way, than it does learning an mnemonic built around a very special case?

And so what? Virtually any endeavor, if you want to get good at it, requires a lot of time on the basics. If you want to learn how to play the piano, you have to learn the names of the keys so that you can read music sheets. After that, there's a lot of time spent on practice, building "muscle memory" so that you can play a tune without having to think about each and every note. The same is true for sports of all kinds, with a lot of time spent hitting a ball, or throwing a ball into a basket, or whatever. The more you perform these actions, the more fluid and automatic they become. The same is true for arithmetic and mathematics at a higher level, I believe. If you aren't sure whether it's 6 x 9 = 63 or 54, it becomes much more difficult to do quick and dirty approximations as sanity checks on more difficult problems.

I don't consider memorizing tables without motivation a viable part of the basics. The basics would be working out the arithmetic , then as you become experienced you'll naturally develop a working memory for these calculations.

And the "and so what" part I guess would be that the class time spent memorizing tables could be better spent working problems or motivating mathematics in general.

 

[Student100]

Nice to know, I was under the impression that you were a little boy...

Uh, thanks?

 

[PeroK]

Does it require more thought to extend the multiplication in this way, than it does learning an mnemonic built around a very special case?

In my head I would do 23 * 15 as (22 * 15) + 15 = (11 * 30) + 15 = 330 + 15 = 345

Or:

23 * 15 = (23 * 10) * 1.5 = 230 * 1.5 = 345