In continuation of my series on arithmetic, I present here a topic that was one of the cornerstones in my book “Arithmetic Magic. ” To fully understand how this concept aids one in arithmetic operations, we need to lay some foundational ideas first. The “Quick-Add” is an enormously valuable tool to help children master quick arithmetic, particularly applied to summing numbers. Today the calculator has crippled even the ablest students. Hardly a one knows his fundamental multiplication facts, as the omnipresent calculator does this operation for him. This situation is understandable, and a comparison I can make is one regarding remembering telephone numbers. Since the ubiquitous cell phones store numbers, I no longer memorize people’s numbers as I have no need to. Analogously, students no longer can add or multiply because the calculator does it for them. This is a problem for the following reasons: 1) not mastering arithmetic leads to problems in mathematics down the road; 2) not being able to add or multiply engenders frustration when doing basic homework assignments; and 3) lack of doing = future lack of doing, which further increases the chance of mathematical illiteracy.
The Quick-Add method gives students a viable alternative to performing quick sums without the aid of calculators or pencil and paper. This method is based on the idea of “complements. ” The word “complement” means “to complete, ” and this is exactly what these numbers do. A “10-Complement” completes the 10; a “100-Complement” completes the 100, and so on. Why this idea is so useful is that it aligns itself with the simplicity inherent in the metric system, in which all units and measurements are based on the number 10 and its multiples. To begin to understand this idea, let me present the following scenario: If I said to a child, “What is 8 + 9?”, and wanted a fast answer, the child would probably start and stumble, resorting to counting on his fingers or trying feverishly to reckon the sum. Granted, there are those children who are quick with this type of thing and, rather fast, can come up with the answer of 17. My focus, however, is not on these children. The healthy have no need of a doctor. My focus is on the children who struggle with basic arithmetic operations and experience tremendous frustration: which when germinated, leads to negative attitudes toward mathematics and ultimately crystallizes into self-doubt, fear, and dread of this most wonderful subject. The consequences are truly disastrous as many students I have worked with realize—after I healed them of their mathematical ills—that they were actually good at math. Imagine what better problem solvers we would be in general if we had math on our side rather than against us!
Let’s return to the idea of complements. In the 8 + 9 example, we see the sum is 17. How much faster would a child come up with the answer 17, if I said “What is 10 + 7?” Now the careful analysis of the difference between 8 + 9 and 10 + 7 reveals some very interesting things, and shows how the circuitry of the brain capitalizes on some very important mathematical facts. Let us examine these. It is indeed true that 0 and 1 are two very special numbers, but for addition, 0 is the number whose special property applies here. The number 0 has the “Additive Identity Property. ” This simply means that 0 plus any other number yields the given number. That is 0 + 5 = 5; 0 + 4 = 4, etc. (From an addition perspective, I guess one could say that 1 is special in that adding 1 to any number is quite intuitive as we are only incrementing said number one unit: thus 8 + 1 = 9—you get the idea. )
Now complements of a number are those numbers, which when added to the given number, yield a sum of 10. For example, the 10-complement of 8 is 2, since 8 + 2 = 10. The 10-complement of 3 is 7, since 3 + 7 = 10. How we tie the concept of complements to the Quick-Add is as follows: in analyzing 10 + 7, we rewrite this example as 10 + 07. We insert a 0 in front of the 7 as a placeholder for the empty “tens column, ” and to bring the numbers into parallel structure. Now let us examine how the brain circuitry works in doing 10 + 07. The brain performs 1 + 0 in the “tens column” and 0 + 7 in the “ones column, ” thus capitalizing on the “Additive Identity Property” of 0. This is in fact a “no-brainer. ” Therefore, our strategy tool for addition will be to convert addition problems into their associated “Quick-Adds. ” Once done, this simplifies additions enormously.
Stay tuned, as in Part II I will go into much more detail about this whole procedure.
Joe is a prolific writer of self-help and educational material and an award-winning former teacher of both college and high school mathematics. Under the penname, JC Page, Joe authored Arithmetic Magic, the little classic on the ABC’s of arithmetic. Joe is also author of the charming self-help ebook, Making a Good Impression Every Time: The Secret to Instant Popularity, 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 educcational 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 www.mathbyjoe.com .