Learning your multiplication facts does not have to be hard. In fact, depending on how you look at it, this feat can be quite fun. Repetition does play a role, but knowing some cool tricks can be the difference between success and failure. So why struggle with multiplication when conquering this arithmetic operation can be yours easily? Here you get the key to mastering the most important table you will ever learn.
If you read Teach Your Kids Arithmetic - Multiplication Shortcuts - Part I, then you know that mastering the Fundamental Multiplication Table is the first building block to effective multiplication skills. This table gives the product of any two numbers between 1 and 9. Knowledge of this table is probably one of the most essential things you will ever learn in school, and something that you will use for your entire life. Indeed calculators are omnipresent, but there is no escaping that knowing how to multiply the numbers 1 through 9 will help you function well in life. Imagine not knowing how to figure 8 x 3 or 4 x 6!
Learning the table is actually quite easy if you have some insights into its composition and the nature of multiplication in general. First of all, you really do not have to know all the products from 1 x 9 through to 9 x 9. There are some beautiful-and for our sakes-convenient properties of this operation that obviate this necessity. Multiplication is commutative. Remember that one from the days of your learning the Commutative Property of Multiplication? (And you thought the teacher was just trying to torture you with this stuff. )
What this means is that the order in which we perform this operation does not matter; the result, or product is the same. That is 3 x 4 = 4 x 3 or more generally a x b = b x a, where a and b are any two numbers.
Since there are nine numbers across the top and side rows of the multiplication table, there are eighty-one products that can be formed. Because of the wonderful commutative property, we do not have to learn this many. In fact, we only have to learn forty-five, of which another twenty-two are rudimentary. That leaves us with twenty-three measly multiplications, no grand feat. And with what you learn in this article, you will soon see that these twenty-three products are nothing to be afraid of.
Okay let's get to work. Of the eighty-one possible products, thirty-six can be eliminated by the commutative property. That leaves forty-five. Of these, the 1-times table is simple because the number 1 is the multiplicative identity. This means that 1 times any other number yields that number. Case closed. This eliminates another nine products, and now we are down to thirty-six. The 2-times table is quite easy and never created much trouble for students. This is true of the 3-times table as well. Early on, most children have learned to count by 2's, 3's, even 4's and 5's. Yet without taking anything for granted, the 2-times table can be looked at through addition. For example, 2 x 9 is the same as 9 9. This addition can be obtained by doing a “quick-add" on 9 9 to get 10 8 = 18. (For more on this see my articles on addition. Go to the link at the end of this article. ) Knowledge of the 2's eliminates another eight products, and this brings us down to twenty-eight.
On to the 3-times table, and we notice that these products can be handled by viewing them as a result of the 2-times table with an additional addend. In other words, 3 x 7 can be thought of as
(2 x 7) 7, or 14 7. This addition again can be handled as a quick-add on 14 7 to convert it to 20 1 = 21. Having dispatched with the 3's, we eliminate another seven products, and we are now down to twenty-one.
As you may have guessed, the 4-times table can be looked at through the eyes of the 2-times table. For example, 4 x 8 can be thought of as 2 x 8 doubled. Since we already know 2 x 8 = 16, we need only add 16 16. But by now, you are a master of addition and know this is 32. Mastering the 4's eliminates six more products, and we are down to fifteen.
The 5-times table is almost not worth mentioning as I can see and hear memories of my youth when I counted, “5, 10, 15, 20, 25, 30, 35, 40. . . " I am sure you can also. This notwithstanding, let's look at the 5-times a little more closely. For instance, learning 5 x 6 is easy because we know our 4-times table. (If you have not already noticed, we are learning new material based on existing material. Now you see why it is hard to progress in mathematics, if you have a weak foundation. ) Thus 5 x 6 is (4 x 6) 6 which is 24 6 = 30. This is how we build the table in our mind. Once the neural circuits are hard-wired, we will never have trouble with this table again.
This eliminates another five products, and we are now left with ten. On to the 6-times table. You got it. We learn this as a doubling of our 3-times table. Thus 6 x 8 is double 3 x 8 which we know is 24. So 6 x 8 is 24 24 which is 48. Aha! I can see those light bulbs going on all around.
We are now coming to the home stretch. Having rid ourselves of the 6's, we are left with only six products: 7 x 7, 7 x 8, 7 x 9, 8 x 8, 8 x 9, and 9 x 9. How do we master our 7's? Yes, through our 6's. Thus 7 x 7 is (6 x 7) 7, which is 42 7 = 49. How do we do our 8's? Exactly. Double our 4's. Thus 8 x 9 is double 4 x 9, or 36 36 = 72. Finally, we are left with 9 x 9, which is 81.
Looking at the table in this way, you now see that learning your multiplication facts is not so daunting after all. What is more, you have been given another perspective on learning something mathematical. Although some mental energy was required to accomplish what we have done here, you can be assured that the effort will pay dividends for the rest of your life. Indeed, having mastered the multiplication table, you have taken the first step toward conquering mathematics-all of mathematics. Welcome to my world.
See more on my articles page Articles Page and get some mental exercise here with these cool problems Problem of the Week and Cool Brain Teasers .
Joe is a prolific writer of self-help and educational material and an award-winning former teacher of both college and high school mathematics. Joe is the creator of the Wiz Kid series of math ebooks, Arithmetic Magic, the little classic on the ABC's of arithmetic, the original collection of poetry, Poems for the Mathematically Insecure, and the short but highly effective fraction troubleshooter Fractions for the Faint of Heart. The diverse genre of his writings (novel, short story, essay, script, and poetry)-particularly in regard to its educational flavor- continues to captivate readers and to earn him recognition.
Joe propagates his teaching philosophy through his articles and books and is dedicated to helping educate children living in impoverished countries. Toward this end, he donates a portion of the proceeds from the sale of every ebook. For more information go to http://www.mathbyjoe.com