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How to learn and teach multiplication

Updated on February 8, 2013

Using advanced thinking methods to "trick out" ways to learn math.

This isn't just a collection of silly tricks, like, "Take a number, multiply it by nine, add your age, divide by the number of socks in your sock drawer, subtract your grandmother's birthday, and I'll tell you some meaningless number that will bore you to tears."

Arthur C. Clarke once wrote, "Any sufficiently advanced technology is indistinguishable from magic." And that is how I view the Idea of magic. It is a sufficiently advanced method of doing something which is not comprehended by the masses. Or, as I like to say, "That which can't, and does."

Math Mojo's goal is to help people who have not mastered the boring, drudgery way of school math, by introducing them to the "magical" ways of advanced thinking. It happens to be easier than the drudgework, and it is certainly more fun.

So we won't be learning how to make other people look foolish, we'll be learning how to make ourselves actually smarter. Now that's magic!

Thinking about how to learn to multiply

Starting out

The standard method in western countries is to insist that you stare at some page filled with "the multiplication table" and "Just shut up and memorize it!"

Forget that, entirely! First of all, we are usually not taught any meaningful way to memorize things. The default method that most people have is untrained, and pretty shakey, at best.

My Idea is to use an active method that forces you to use your brain a bit. When you do this, your brain (which is smarter than your teacher) creates new mental pathways. When it does it often enough, those pathways are "etched" deeper into your brain, so they become "highways." Remember, though, that this is just a metaphor, and isn't quite the way your neurology works, but it makes sense to us as a metaphor.

If you practice something enough, it becomes easier. If you stare at something passively long enough and often enough, it will create pathways. But if you actively create the pathways, they will be made sooner and deeper.

And if you use a bunch of different ways, you will create wider, deeper paths.

So if you find ways to calculate the multiplications, instead of using passive memorization, you will be more successful.

The simplest (not necessarily the easiest or the best) way to calculate your multiplications of whole numbers {1,2,3...}, is to perform repeated addition on it. In other words 6*4 becomes 4+4+4+4+4+4.

(It is important to know that although multiplication of whole numbers can be accomplished by repeated addition, repeated addition is not the definition of multiplication. Like most things, the more you learn, the more you can understand, and once you learn basic multiplication and arithmetic you can go on to learn very interesting properties about multiplication that go beyond simple repeated addition. But when you are starting out learning with whole numbers, "multiplication can be accomplished by repeated addition" is a good rule of thumb.)

If you do your multiplications that way often enough, your brain starts thinking, "Hey, this is a pain in the neck. I can do it, of course, but there must be a better way. So the next time we do this, I'm going to store the answer in my memory, so that after that I won't have to go through this tedious process anymore."

That all happens on a subconscious level, you understand. It's a great talent, that brain of yours.

If you need an illustration of how this works in other fields, take someone who is learning to shoot baskets. If he has a good coach, the coach tells him all the things he needs to know, like how to bend at the knee, use the wrists, etc.

Now, the player can just listen to the coach, make notes of what he's saying, and repeat those instructions to himself over and over, with no physical effort.

But you and I know that at some time he's going to have to go out and shoot some hoops. And only by repeated trial, does his body memorize the actions, and streamline the process. That's because the body and the brain don't want to work too hard forever.

How to learn Basic Multiplication of single-digit-numbers

These are often called the "multiplication tables" or "multiplication facts" but both of those terms are misleading.

Somehow the term "multiplication tables" gives one the impression that there are some "magical tablets brought down from the mountain," like they are the only way to learn.

The "tables" are simply a list of what happens when you multiply a set of numbers by each other, they are not the multiplications themselves. (Give that a chance to sink in - it's deeper than you may think.)

The "tables" are like a roadmap, but they are not the road. If you teach a kid that the tables are the only way to learn, you are depriving him from experiencing real multiplication. Multiplication is about manipulating numbers, or amounts. Deep understanding of simple multiplication is a necessary part of developing numeracy (the ability to understand and work with numbers).

The term "multiplication facts" is also a really bad Idea to inculcate young minds with. Not only does it suffer from the problem mentioned above, but it also gives the impression that they just happen to be "facts" that exist outside of you. Yes, they are facts, but they exist inside of you. You can reconstruct them simply by using your mind. They are not like facts like, say, historical facts like "The Magna Carta was signed in 1215." That is something you cannot directly experience, and must simply learn as a "fact." (Actually, that is not strictly true either, just ask a real historian.)

See, learning something strictly as a "fact" makes you feel like a victim of that fact, rather than a participant. Direct, or indirect experience with something makes it so much more alive, real, and actually fun.

So the best ways to learn math are not with flash cards, silly rhymes, obnoxious cartoon characters or other methods that distance a child from direct experience of numbers and how to use them. (Don't even get me started about the evils of calculators...)

The best ways to learn math are ways that have you use actual math. Imagine that! Learning how numbers relate to each other is the way to go. I'm biased, of course (because I wrote it) but I think "Numbers Juggling - Times without the Tables" is the best explanation of how to learn basic multiplication that works for most people, especially the ones who think the "tables" are a pain.

I created this e-book because I was fed up with the way we are normally taughtmultiplication. This booklet will open your eyes to a new world. You will learn a way to teach any child basic multiplication of single-digit numbers (what we normally call "the "multiplication facts" or times tables") in just a couple of minutes."

The booklet then goes on to help you "lock this knowledge in."

It also includes seven e-mail lessons that will explain the math behind the method, so you will actually understand it and be able to show your child why it works!

And here's an important part: I've just added over a dozen videos to help you turbo-charge your practicing and learning. They are easy to do, and fun as well. I practically "hold your hand" as you learn, until you can do over dozens of multiplications in a less than a minute.

I truly believe that every parent and teacher should know the material in this booklet, so that every child can have a helpful, meaningful strategy for mastering this important subject.

You can order "Numbers Juggling - (Times without the Tables)" here.

How to mentally multiply any whole number by a repunit (Part 1 of 3)

What the heck is a repunit? A repunit comes from the words "repeating unit," in other words, it is a number made up of repeating ones, like 11, or 111,111.

Just as a refresher, a whole number is zero and all the counting numbers from 1 on, forever. The whole numbers are 0,1,2,3,4,5... (The three dots mean "and so on).

Now, there is a really good interactive lesson on how to multiply any whole number by 11, mentally, at this page at You should learn that before reading further.

How to mentally multiply any whole number by a repunit (Part 2 of 3)

Back already? Good, now let's try any number times 111. Let's use 674 * 111

  1. Start at the left of the multiplicand (that's the 674) and pretend there are two zeros behind it and in front of it, making it 0067400. Now add the final three digits, (in 0067400 that would be 4+0+0) and write the sum (4) beneath the 4 in the 0067400 . That's the digit in the units column of the product. (The product is the answer.)

  2. Next, add the three digits that start second from the end of the multiplicand (in 0067400 that would be 7+4+0 = 11) and write the units digit(1) of that number to the left of the last number you wrote, giving you 14 so far. Keep the tens digit of that number in your head - you are going to carry it to the next addition.

  3. Now add the three digits (plus the carry) that start third from the end of the multiplicand (in 0067400 that would be 6+7+4+1 = 18) and write the units digit of that number (8) to the left of the last number you wrote, giving you 814 so far. Keep the tens digit of that number in your head - you are going to carry it to the next addition.

  4. Now add the three digits (plus the carry) that start fourth from the end of the multiplicand (in 0067400 that would be 0+6+7+1 = 14) and write the units digit of that number (4) to the left of the last number you wrote, giving you 4814 so far. Keep the tens digit of that number in your head - you are going to carry it to the next addition.

  5. Now add the three digits (plus the carry) that start fifth from the end of the multiplicand (in 0067400 that would be 0+0+6+1 = 7) and write the units digit of that number (7) to the left of the last number you wrote, giving you 74814 so far. There is no tens digit to carry this time. You are finished.

The Math Mojo Multiplication Master Course

Learn Focus, Concentration and More Through Mental Math

The main reason I created this course is to help people who feel that they are not great at math. I hope they find themselves more comfortable with it after learning one of the basic operations so well that they gain confidence in their ability to learn anything.

You, your child,or your students will be taken through problems like 78*9, up to huge problems l like 367,938 * 765 (or larger) and be able to do them without pencil or paper! (Of course, they

can still use pencil and paper if they need to, like for a test to show their work.)

This course is not only about multiplication. It will improve your math skills in general, but more than that, it

is a way to improve focus, attention, accurate thinking, patience and perseverance. The lack of these traits is what causes the most "math anxiety," and this course takes aim at enabling anyone who takes it to develop those traits to a high degree. I should know. I have been diagnosed with ADHD, and have used the practice methods for this course to develop my own "bear trap" concentration.

The course is called the "Math Mojo Multiplication Master Method", and it is perfect for students, parents, teachers,home-schoolers and unschoolers. It is for anyone who already knows the times tables, but wants to be able to do larger multiplications, mentally.

The material in the course covers:

  • Mental multiplication of huge numbers
  • Left-to-right multiplication
  • An amazingly easy way to check your answers so that you'll

    never have to hand in an answer that you are not sure about

    ever again again.

  • Ways to absolutely master multiple choice tests.
  • Ways to estimate a multiplication product instantly and

    very accurately that are much better than anything you

    learned in school.

  • An amazing method of mental addition, which will enable

    you to avoid doing "partial products" and getting lost in a

    mess of writing.

  • A fun (but not childish) and effective way of practicing

    without worksheets that will keep anyone wanting to practice


The course consists of over twenty-five online videos and

eight downloadable PDF documents, including:

"The Math Mojo Multiplication Master Method - The Basics"

PDF, which teaches:

  • -
  • What is Multiplication? (You may be surprised at the

    answer, especially if you are a teacher.)

  • The names of the parts of the multiplication problems.
  • What is a Digit?
  • What is a Whole Number?
  • The Properties of Multiplication The "standard" (school)

    way to multiply any multi-digit number by any one-digit

    number, and why it works.

  • The Math Mojo way to multiply any multi-digit number by

    any one-digit number, from left to right, in your head, and

    why it works.

  • And much more.

The course also includes:

"The See-Say-Write Method of Addition" (with workbook). This e-book is usually sold separately, but is included as a bonus to help you turbo-charge your addition.

This is not just a collection of tricks for different numbers. It is ONE method for ALL whole numbers

This method is a "Learn-Once-Remember-Forever" method.

Of course the entire course comes with a no-risk guarantee.

To gain immediate access to all the material, or to get more

information, go to:

Unique Math Newsletter for all Ages (FREE!)


Get Free Instant Access to


("Comes out Semi-Annually, Mostly!")


Lots of methods to help you understand math, written in plain English.

It's FREE and it's FUN!

As a Bonus for subscribing,

you'll receive a Free Copy of:

"10 Things You can do Right Now

to Help your Child Learn Math"

Ten nothing-to-buy tips you can

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Click here to start getting these great math methods for free!

How to mentally multiply any whole number by a repunit (Part 3 of 3)

If the multiplicand had been longer, say, 876,346,974, it would still work the same way. You would just continue on until you ran out of digits.

If you learned this, and the original lesson for multiplication by 11 at, you must see a pattern by now:

If your repunit has four digits, you put three imaginary zeros in front and behind, then, starting at the end, add four digits each time.

If your repunit has five digits, you put four imaginary zeros in front and behind, then, starting at the end, add five digits each time.

and so forth.

P.S. Are you aware that squaring any repunit (at least up to nine digits) will give you a palindrome? Try it.

Some people like to complain that math is so uninteresting because "all you do is the same thing over and over, and get the same answers, etc." I feel bad for them. They don't get it that although, in arithmetic at least, any problem has only one answer, there are lots of ways to get there, and only the one you learned in school is boring.

What are you interested in?

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Great Math Books for Every Home - Personally Recommended by Professor Homunculus

The Number Devil: A Mathematical Adventure
The Number Devil: A Mathematical Adventure

From the review: "...introduce basic concepts of numeracy, from interesting number sequences to exponents to matrices. Author Hans Magnus Enzensberger's dry humor and sense of wonder will keep you and your kids entranced while you learn (shhh!) mathematical principles."

Professor Homunculus sez: "If you are helping a child learn math, this book is a MUST!"

Playing with Infinity: Mathematical Explorations and Excursions
Playing with Infinity: Mathematical Explorations and Excursions

From the a reader's review on "This book explains the why's behind math from principles as basic as counting to as complicated as series, geometry, and even some calculus principles. It is written in a conversational tone with lots of pictures (yes, and numbers). Each chapter builds upon the last, and it is easy to follow (though sometimes dense). It was my first "fun" math book and is still by far my favorite."

Professor Homunculus sez: "I couldn't agree more!"

Mathematics for the Million: How to Master the Magic of Numbers
Mathematics for the Million: How to Master the Magic of Numbers

This book is a classic. It was the book that launched the intellectual self-improvement movement in America. It is perhaps the best book on general math ever written. Hogben takes you from the birth of mathematics to calculus in a lucid, human way. Even if you use only this one book, you will not only understand everything you will need for all of high-school math, and most of college math, but you also have a good basis for understanding the spirit of mathematics. H.G. Wells called it, "A great book, a book of first-class importance."


Let me know your experiences or concerns about learning multiplication.

Reader Feedback

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    • profile image

      anonymous 5 years ago

      thanks for this page, please keep up the good work. It is so hard to help children become enthusiastic about maths as it is, but this could really make a difference!

    • profile image

      YourFirstTime 5 years ago

      Great tips!

    • profile image

      anonymous 6 years ago

      @anonymous: truth

    • profile image

      bilbo959 6 years ago

      Wow, there is a lot of stuff to get my head around in here. Thanks for all of this. I'm going to show this to my nephew, Frodo

    • marckq profile image

      marckq 6 years ago

      very informative lens, thanks for sharing

    • profile image

      anonymous 7 years ago


      Here's another book recommendation: Introductory Mathematics: Algebra and Analysis by Geoff Smith, published by Springer ISBN 3-540-76178-0. To my mind it has an accessible style, beneficial end of chapter exercises and its mathematical level on the whole does not demand maths-whizz ability. I'd say it's more for the interested amateur or student rather than a parent that wants to help their children (a young maths prodigy might like it perhaps).

    • Homunculus profile image

      Homunculus 7 years ago

      Missy, as far as basic multiplication, you really can't do better than theâNumbers Juggling - Times Without the Tablesâ booklet and course "above. For division, check out the page at

    • profile image

      anonymous 7 years ago

      I home school my daughter. She's in 6th grade, and her reading is at 7th grade. Boy O Boy where do I start ! MATH - the HORROR ! Teaching basics - Multiply / Divide has been a nightmare ! Literally ! She just can't get it, and I don't know why. I'm excited about your teaching technique. I've tried it ALL. I bought Brainetics "thinking" that would help ($250.00) waste of money !

    • Homunculus profile image

      Homunculus 8 years ago

      [in reply to anie] Anie,

      It shouldn't even take 10 minutes. Seriously. Get the booklet mentioned above , "Numbers Juggling (Times without the Tables)" and you will have it down in minutes.

    • profile image

      anonymous 8 years ago

      do magic to me so i could know my multiplacations in 10 minuts please !!!!!!!!!!!!!!!!!!!

    • profile image

      anonymous 8 years ago

      do magic to me so i could know my multiplacations in 10 minuts please !!!!!!!!!!!!!!!!!!!

    • profile image

      bilbo959 8 years ago

      This is a truly excellent site. My daughter actually thought it was "awesome." I homeschool her and can use any help I can get with math. I'll be checking out your other sites. Thanks so much!

    • profile image

      chetbyles88 8 years ago

      I bought your "Times without the Tables" booklet, and the videos were a great idea. My students love them. Thanks!

    • profile image

      anonymous 8 years ago

      [in reply to madoc]

    • madoc profile image

      madoc 8 years ago

      Please add this great lens to my Mathematics Education Headquarters group!

    • dustinmarx10 profile image

      dustinmarx10 9 years ago

      Do you teach any speed-multiplication or any other speed-math? Card games are great for that. Blackjack is one of the best games to teach addition (as well as other things.)

    • teamlane profile image

      teamlane 9 years ago

      I'll show this one to my daughter the teacher!

      Blessed by a Squid Angel today! :)

      Colleen ~

    • profile image

      nadiasorkin30 9 years ago

      I really like your suggestions for math books. I intend to pick one of them up today. Thanks for the hints!

    • profile image

      anonymous 9 years ago

      Thanks for sharing your gift of knowledge with us...

    • profile image

      PotPieGirl 9 years ago

      5* - As a mom, this is GREAT information! As a lensmaster, this is a wonderful lens!

      Nice work =)

    • profile image

      sidvicious2008 9 years ago

      Nice lense. I appreciate your effort.

    • profile image

      anonymous 9 years ago

      Check out my workbook, Teach Your Child the Multiplication Tables, Fast, Fun & Easy with Dazzling Patterns, Grids & Tricks. My son had difficulty with rote memorization, so I developed a method based on patterns. Patterns enhance recall. Eugenia

    • evelynsaenz1 profile image

      Evelyn Saenz 9 years ago from Royalton

      Thank you for sharing such a great lens. Playing games is a great way to practice and learn math. Using a Hands-On Approach to learning is the best way for children to truly grasp mathematical concepts. I look forward to reading more of your lenses.

    • profile image

      SteveRogers 9 years ago

      Damn, this makes so much sense. Why don't schools "get it?"

      Best of luck with your lens and mission,


    • profile image

      Max_Nix 10 years ago

      Hey, that's pretty good! Got more tricks (I mean "magic")?

      Max Nix

    • profile image

      anonymous 10 years ago

      Cool, cool, cool! I never knew that magic was really a serious art!

    • profile image

      puppetess 10 years ago

      This looks like a great site to send homeschool kids to. I'll recommend it to my friends. Thanks for the good tips.

    • profile image

      anonymous 10 years ago

      Great lens, Brian! I'm going to teach my kids using this stuff. Thanks!

    • profile image

      anonymous 10 years ago

      Interesting. Do you also have hints for division? I think I can use this stuff to help my middle-school students who didn't "get" multiplication in elemementary school.

    • profile image

      klapskie 10 years ago

      thanks for all the tips you've written, this can help people of all ages.Giving you 5 stars. :)

    • profile image

      CateSanderson 10 years ago

      Wow Brian! I'd say you certainly have put a lot of work into your lenses. How do I see my own lens?

    • profile image

      dwkinney 10 years ago