Dad's Strategy for Learning Multiplication
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 4x6, 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 (1x1 all the way through 10x10) 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
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 6x7=42, they should be able to remember that 7x6=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.
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?
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.
To Multiply by Two, Double the Number
This rule leverages facts learned during addition. 2x7 = 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 2x1 for Rule #2 and 2x10 for Rule #3, we only get to knock eight more off our list, but that still drops us to 28.
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. 4x6 = 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 4x8 and 4x9 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 4x1, 4x2 and 4x10 from the rules above), which puts us at 21 left!
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 8x5 = (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.
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.
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 multiplication charts, 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!