# How well do you know the multiplication table?

SteamKing
Staff Emeritus
Homework Helper
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.

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
Gold Member
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...

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Char. Limit
Gold Member
Alternately...

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

Super easy.

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
Gold Member
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.

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
Gold Member
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.

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symbolipoint
Homework Helper
Gold Member
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.

• Medicol
symbolipoint
Homework Helper
Gold Member
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.

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.

• atyy
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.

• diogenesNY
SteamKing
Staff Emeritus
Homework Helper
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.

Gold Member
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.

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

• Medicol
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
Gold Member
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
Homework Helper
Gold Member
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
Mentor
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.
Student100 said:
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.
Student100 said:
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
Gold Member
???
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.

• Fervent Freyja
Mark44
Mentor
No, it isn't, provided that it is used only for what it's supposed to be used on: a product of two binomials.
Student100 said:
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.
Student100 said:
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.

• russ_watters
Fervent Freyja
Gold Member
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
Gold Member
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
Gold Member
Nice to know, I was under the impression that you were a little boy... Uh, thanks?

• ProfuselyQuarky and Fervent Freyja
PeroK