Dad’s Strategy for Learning Multiplication

Rocket Math Rocket

Learning multiplication facts is a challenge because it’s the first math operation where your child needs to contend with relatively large numbers. Two digit addition and subtraction is squarely in the realm of numbers less than 20, which is familiar territory. There’s something concrete about 12 or 15 or similar numbers countable on fingers and toes, but 73 really is a big step out of the pond.

There’s two ways to approach this. One is just brute force memorization. I remember endless flash card drills after school, the timed tests in the classroom and the gradual accumulation of resentment towards anything with that little ‘x’ attached to it. While we love the Rocket Math program the schools use here, it is largely just memorization and could use something to back it up.

The other alternative is to make multiplication something of a game, with systems for some of the numbers. There still an inevitable amount of memorization that goes on, but by getting 90% of the multiplication table down to a few simple rules, the goal is suddenly within everyone’s reach. Split second, memorized results are still going to come, but having some means to reach incremental (albeit slower) success takes the fear and dread out of the process.


The place to start is understanding that multiplication is just repeated addition, and then using a few tricks to make the addition go faster. For example, if we need to multiply 4×6, an easy explanation is that this is just four copies of six added together, or six doubled twice. This “Double-Double” rule works for anything multiplied by four, and is easy to apply if basic addition has already been covered. A handful of these rules cuts 90% of the multiplication table away so that memorization is only required for ten facts.

Even if you ignore the ‘tricks’ in the rules below, the brute force way of multiplying by breaking it out into a bigger addition problem may seem like more work than just memorizing the facts, but it helps build some understanding of what multiplication actually means and provides a way to find the answers if they haven’t been memorized. Knowing what multiplication is, how multiplication works and (worst case, without any other tricks) how to solve a multiplication problem makes the whole process tangible. Also, this builds up some of the thinking processes used to multiply larger numbers where memorization isn’t possible.

Our focus for now is the core, so we’ll start with the 100 basic math facts (1×1 all the way through 10×10) and cut them down to size. The rules are ordered so that the easiest ones to memorize and use take the biggest chunks out of the table. If you learn them in this order, you cover the facts in the table in the fastest possible way.

Here are Dad’s eight simple rules for learning the multiplication tables.

Dad’s Eight Simple Rules for Mastering the Times Tables

Rule #1:

First Number Times Second Number is the Same as Second Number Times First Number

This rule, more formally known as the commutative property of multiplication, just means that A x B = B x A. If you can teach your child that 6×7=42, they should be able to remember that 7×6=42 as well. This should be the first question you ask if your child is stuck on a problem. If your child doesn’t know the answer to a multiplication math fact, swap the multiplicands and ask the question again. When you factor in the effect of perfect squares, this one rule cuts the number of facts we need to memorize almost in half to 55.

Rule #2:

Any Number Times One is that Number.

If multiplication is just instructions for addition, multiplying a number by one just means to add a single instance of that number up. The result is always that number. That takes 10 problems out of our remaining list of facts, dropping us already to 45. See how fast we’re moving?

Rule #3:

To Multiply by Ten, Attach a Zero.

Even if concepts about place value and shifting decimal places are new at this point, memorizing that multiplication by ten means just attaching a zero to the number is an easy rule to remember. The zero on the end of the ten should serve as a trigger, “Ten ends in zero. What do you attach to the other number?” Given the focus on reusing addition facts in our multiplication odyssey, I recommend avoiding the phrase “Add a zero” or you may garner some initial confusion. Multiplication by ten removes nine more problems from the grid and gives us 36.

Rule #4:

To Multiply by Two, Double the Number

This rule leverages facts learned during addition. 2×7 = 7+7 = 14. All of these facts should already be memorized, but even if they’re not they’re still in the range where counting on fingers and toes gets rapidly to a solution. Because we already crossed off 2×1 for Rule #2 and 2×10 for Rule #3, we only get to knock eight more off our list, but that still drops us to 28.

Rule #5:

Multiplying by Four is Doubling Twice (Double-Double Rule)

When my daughter pauses on a times-four problem, all I have to do is say “Double-Double” and the answer comes right back. 4×6 = 6+6+6+6 = 12 + 12 = 24. For numbers five and lower, the four double-double rule will work with addition math facts and should be performed in memory. If your child can do simple two digit addition without regrouping in memory, six and seven work as well. It’ll take a while, but eventually 4×8 and 4×9 aren’t too hard but you may find those facts get memorized before “carry the one” starts happening mentally. However you get there, we get to cross off seven more facts (skipping 4×1, 4×2 and 4×10 from the rules above), which puts us at 21 left!

Rule #6:

Multiplying by Five is Just Counting by Five

Your child should already know how to count by fives by the time they’re in multiplication land, so a quick short-cut for solving a 5 times problem is just to skip-count by fives up to the number. There’s other more complex strategies for fives (if the number is even, divide it by two and add a zero, so 8×5 = (8/2) * 10 = 40) but these are typically a bit complex when making a first pass here. The “Count by Fives” rule drops us down to 16 remaining facts.

Rule #7:

The Nine Rule – Tens is Number Minus One, Ones is Nine Minus Tens

When you multiply a number by nine, the sum of the digits of the result is always a multiple of nine. For the basic math facts, the sum of the digits IS nine, and in fact it has some other interesting properties. The tens place value is always one less than the number being multiplied, and because of the nines rule the ones place is always the nine minus the value in the ten’s place. The basic script for learning this rule goes something like this: “Multiplying by nine? Okay, what’s one minus the other number? That’s the ten’s digit. Okay, what number plus that equals nine? That’s the one’s digit.” Again, this strategy just falls back on basic addition facts, and it cuts our total number of math facts to memorize down to 10.

Rule #8:

Memorize the Ten Remaining Facts

The first seven rules cut our list of facts down from 100 to 10, so all we need to do is memorize the 10 multiplication facts to have the whole table down. We eliminated any number times 0, 1, 2, 4, 5 and 9. So here’s the multiplication facts that are left with a few rhymes to help remember them:

3 x 3 = 9 Three times three is so fine,
three times three is nine.
3 x 6 = 18 Three times my bird ate six beans, three times six is eighteen.
3 x 7 = 21 Three candies each for seven days, that would be fun, three times seven is twenty-one.
3 x 8 = 24 Three boys on skates fell on the floor, three times eight is twenty-four.
6 x 6 = 36 Six dogs with six sticks, six times six is thirty-six.
6 x 7 = 42 Sticks from heaven, stuck in glue, six times seven is forty-two!
6 x 8 = 48 What do we appreciate? Six times eight is forty-eight!

Flight Six Times Eight! Don’t be late! Leaving at gate forty-eight!
7 x 7 = 49 Seven kids in seven lines, add ‘em, up its forty-nine.
7 x 8 = 56 Five – six – seven – eight, Fifty-six is seven times eight.
Seven packs of gum, each with eight sticks. Can you chew fifty-six?
8 x 8 = 64 Eight times eight is sixty-four, close your mouth and shut the door!
Had two eights, dropped them on the floor, picked them up, had sixty-four.

The first four facts are all from the three-times table, and they’re fairly easy to calculate using addition or find by skip-counting by threes. The remaining six are the nasty ones. If you really look back, you can probably remember struggling with one or more of the remaining ones as a kid. This is a link to practice worksheets specifically for the ‘Rule #8′ facts here.


So that’s it, multiplication in eight rules built on top of basic addition. If we count rule eight as ten facts, it really means the whole multiplication table is wrapped up in only seventeen pieces of knowledge. Easy!

You will still find conventional worksheets and flash cards to be a powerful way to reinforce the strategy presented here, but by quickly learning these few simple facts, your child will immediately have a solid and successful grasp of multiplication. Good luck and see you at Division!

A Summary of the Eight Simple Rules for Mastering Multiplication

22 Comments »

  1. [...] Eight Simple Rules for Mastering Multiplication: The Full Strategy! [...]

    Pingback by DadsWorksheets.com » Dad’s Eight Simple Rules for Mastering the Times Tables — October 3, 2008 @ 10:29 am

  2. That was a great lesson!

    Comment by shayleece — February 22, 2009 @ 4:10 pm

  3. Thanks, this is the first time I have been able to find a method for memorizing times tables other than just plain memorization…I am thrilled

    Comment by Teresa — September 28, 2011 @ 4:20 pm

  4. “This rule, more formally known as the *COMMUTATIVE property of multiplication, just means that A x B = B x A.”

    *- fixed your post

    Sincerely,
    An algebra teacher & dad

    Comment by JP — October 21, 2011 @ 10:04 am

  5. Thanks JP! Updated!

    Comment by admin — October 21, 2011 @ 4:18 pm

  6. It helps me many times in my tutorial class the worksheets you prepared. Every time I give my students quizzes and math practices, it feels me no worries anymore, for i know your website is there and printable worksheets are ready to print. Thank you very much! God Bless you!

    Trixie

    Comment by trixie — January 3, 2012 @ 1:16 am

  7. My children were more visual and sometimes need to literally see that success is possible. So we looked at a 12 by 12 multiplication table and colored in all the things we knew. I see now that we were basically apply the rules that you have so well stated. We colored in everything below the diagonal because 2×1 is the same as 1×2 and we decided to always put the smaller number first. Then we colored in the 1x, 2x, 5x, and 10x rows an columns (your rules 2, 3, 4, and 6). Next we colored in 1×11 through 9×11 because x11 is the number twice. That left us 23 things to learn, we didn’t figure out the 9s rule. I found your site while looking for graph paper templates but am impressed with all the worksheets.

    Comment by EdS — January 7, 2012 @ 6:15 am

  8. Hi EdS -

    Seeing the patterns is a great tool. There are so many other concepts (not just in math… We’re doing some chemistry work with my older daughter…) where there’s a pattern that if you can make visual some how, it really makes a huge difference. It sounds like you really made this click for multiplication for your kids.

    There’s probably a good tie in with the concepts on this page and the Multiplication Grid worksheets doing something similar to your process… I’m inspired after your comment and I’ll add this to my list of things to bounce around in my head a bit…

    Glad you like the site!

    Dad

    Comment by Dad — January 8, 2012 @ 8:07 am

  9. Just wanted to say that this is by far my favorite-ist web site for math worksheets for my two young kids. We just started homeschooling in the fall (2011) and every time I’ve needed a worksheet to drill something, you’ve got exactly what I’m looking for. Thanks for all the work, it’s much appreciated!

    Comment by Liz — February 1, 2012 @ 11:06 pm

  10. Thank you so much for your multiplications Rule especially the nine rule’s. At first I didn’t get it but then I figured it out. It’s so much easier. My son will be in 2nd grade next fall and he already mastered his multiplication.

    Thanks a lot

    Comment by Rany — June 11, 2012 @ 4:42 am

  11. My 10 year old students really struggle with Times Tables as they are not learnt by heart any more in the UK. It’s tough convincing them that they are VITAL to good Maths. Great hints here, I’m putting a link to you – THANKS! Jane

    Comment by Jane Powell — September 14, 2012 @ 3:46 am

  12. Dad,
    I think you are a genius! Are you a member of Mensa?
    These rules are awesome. I have been teaching 4th and 5th grade for 18 years. i am so sad when my students struggle with the multiplication facts and I tell them to practice at home. Most do not have someone at home to practice with and the idea of memorizing all of those facts just shuts them down. I love your list of rules and all of your zillions of worksheets. Your daughters are probably geniuses too!

    Thanks!

    Comment by Tammy — November 15, 2012 @ 12:08 pm

  13. For practicing the x5 facts I have kids think of the clock and what the minute hand would say when pointing to that number. We start by looking at a clock face and I point to the different numbers and the kids shout out the minute. Kids then practice looking at a clock when completeing the facts on paper. Eventually they picture the clock in their heads and can solve the x5 facts without looking at the clock.

    Comment by Amy — November 28, 2012 @ 5:57 pm

  14. [...] Read on for this inspiring article, taken from the website http://www.dadsworksheets.com (Post taken from http://www.dadsworksheets.com/2008/09/02/dads-strategy-for-learning-multiplication) Dad’s Strategy for Learning Multiplication Learning multiplication facts is a challenge because [...]

    Pingback by Sage advice for learning the times tables… even a poetry connection?!! : Mrs. McGrath's Grade 3 class — November 30, 2012 @ 10:01 am

  15. I have a trick for multiplying any two digit number by 11.
    So lets say the problem is 34*11, basically you take the number in the tens place (in this case the 3) and that becomes the hundreds place or with some numbers the thousands place and the 4 becomes the one place and then you add the two digits together and that becomes the remaining digit(3+4=7). So 34*11= 374.

    Comment by Chloe — January 14, 2013 @ 3:12 pm

  16. Great resource, thank you!!!

    I wanted to add the rule about ’9′. In the table of 9 , the sum of each answer = 9. It is a good rule to cross check. Example, 9X2=18 which is 1+8 = 0. So when it gets tricky, like 9X8=72 so 7+2 = 9 and you can be sure of your answer!

    Comment by Divs — March 11, 2013 @ 9:50 am

  17. Also a trick with the 9s, and particularly for visual kids:

    Put your hands flat on the table in front of you. Starting with your pinky on your left hand, number your fingers. The left pinky is ONE, left ring finger is TWO, etc. and when you jump to your right hand, the right thumb is SIX and the right pinky is TEN.

    Any time you have a nines problem fold down the appropriate finger for the answer. Examples:

    9 x 2 … fold down finger number TWO, your left ring finger.
    How many fingers do you have on the left of the folded finger? (1).
    How many fingers do you have on the right of the folded finger? (8).
    So with a (1) and an (8), you have 18.

    9 x 7 … fold down finger number SEVEN, your right index finger.
    How many fingers do you have on the left of the folded finger? (6).
    How many fingers do you have on the right of the folded finger? (3).
    So with a (6) and an (3), you have 63.

    9 x 4 … fold down finger number FOUR, your left index finger.
    How many fingers do you have on the left of the folded finger? (3).
    How many fingers do you have on the right of the folded finger? (6).
    So with a (3) and an (6), you have 36.

    Comment by K — April 12, 2013 @ 9:40 am

  18. Thank you so much for paving the way for us. I love this site, it was recommended to me by a friend who homeschools her kids. My son goes to a parochial school that definitely leans toward rote memorization, but I am going to give your different approach to learning multiplication a try. Looks like way more fun! Thanks again for all your hard work creating this site.

    Comment by H — June 11, 2013 @ 7:16 am

  19. The trick I used to remember my 9s is as follows:

    9×1=09 90=9×10
    9×2=18 81=9×9
    9×3=27 72=9×8
    9×4=36 63=9×7
    9×5=45 54=9×6

    The simple visual trick of knowing that the 2 numbers in the answers of 9×1 to 9×5 are simply inverted (and in reverse order) in the answers for 9×6 to 9×10. Somehow the 9x were one of the easiest for me to learn and whispered all sorts of number magic into my (then) very young ear. My munchkin will start to learn multiplication next month! I’m so excited for her!!

    Comment by DG — July 4, 2013 @ 7:47 pm

  20. Can’t wait to introduce my grandchildren to this site.

    Comment by Wendy Fox — August 18, 2013 @ 6:42 pm

  21. [...] whole lot of worksheets and flash cards for math that can be printed off as well as a break down of multiplication strategies that are quite [...]

    Pingback by Interesting Multiplication website | 5 Chiasson's website — September 22, 2013 @ 5:42 pm

  22. This is the most useful and practicing website for children I have ever seen. Answers available for the entire worksheet in a single click…Amazing…

    Comment by Sujatha — February 16, 2014 @ 12:01 pm

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