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numbers. So the trick is, if I have two numbers to add, let's say 8 + 7, and don't remember what they add up to, and don't want to count on my fingers, all I have to do is ask my self, "W hat do I have to add to 8 to make it 10 ? Answer: 2 . OK; add the 2 to the 8 t o make 10, then subtract 2 from 7 to make 5, and add that to 10 to get the answer, 15."
8 + 7 =
10 + 5 = 15.
You can use th e same trick wi th bigger numbers, thus: 87 + 56 =
90 + 53 =
100 + 43 = 143
for ways to rearrange the factors to make the work easier. For example, suppose you want to multiply 15 by 24 . Well, 15 is the same as 3 x 5, and 24 is the same a. 2 x 12, so 15 x 24 is the same thing as 3 x 5 x 2 x 12. Now you can multi ply th ose in any order, and the easi est would be the 5 x 2, which is 10 . And the 3 x 12 is 36. So now all you have to do is multiply the the 10 x 36, which is easy - just tack a zero onto the end of the 36, and the ans wer is 360 . Another example : 36 x 75
4 x 9 x 3 x 25
From the New York Times, 10/4/80 :
100 x 27 = 2700 You can use this trick to do addition; o r to check an additi on that you have already done "the regu lar way." Here's a trick for subtraction. As an example, take the fact that 16 - 7 = 9 . If we add the same number to the 16 and the ~ it does not change the-ans wer . Thus, 17 - 8 = 9, 18 - 9 = 9, 19 - 10 = 9, and so on . Again, it's easy to see why this muse be true of all numbers . If we have t wo pieces of wood and lay them side by side, matching the ends, the "differenc e" (what you get by subtrac ting one length from the other) can be seen as the leng t h of the part that "sticks out." Adding the same quantity t o both numbers can be shown as putting two sticks the same length at the foot - whic h wouldn't change th e "d ifference" at all.
So let's say I have two numbers to subtract, say 15 - 8, and again, I don't remember a nd don't want to count on my fingers, I say to mysel f, "What do I have to add to that small er number to make i t 10 ? Answer: 2 . So I add 2 to both numbers, which makes my problem 17 - 10 = ? , which is easy. " 15 - 9 =
16 - 10 = 6
11 - 4 = 17 - 10 = 7
With bigger numbers: 122-87 =
125 - 90 =
135 - 100 = 35
Again, you can use this trick to get you r answer, o r if you do your subtracting "the regular way," you can use this to check your answer . You may find thi s hard to be lieve, but years ago I taught in a supposedly "very good" private school where teachers had been arguing pass ionately for years about wh e ther children should be "allowed" to do subtraction this way' Finally, a related trick for mul tiplication. Try breaking either or both numbers down into the product of several numbers - this is called "fac t oring the number" - and then look r.RnWINr. WTTHOIIT SCHOOLING #21
By the way, I think people who never had piano lessons at all could still benefit from this book. As the au~hors say in the first section, "Who Is Allowed To Read This Book," all you have to know is where middle C is, and "you should know the names; of the lines and spaces of the tr eb le clef so that you can pick out a . Single-note, one finger tune - no mat ter how badly. If you've forgotte n everything you ever knew about that or if you never knew anything about it at all - take a look at the back of any music paper notebook. It's all said right ther,?"
Unlike the other tricks here, this procedure won't work on every problem; it's best with "nice, round numbers" like 36, while . prime numbers like 37 can't be factored at all. Nobody says you have to do these steps in your head, of course; even if 'you write them all out, it will probably still be faster than the "regular" multiplication procedure. And it's more fun, because you have more choices, and you can be more con fident that each step is correct. -- JH ' & DR PIANO DESPITE LESSONS
[DR: 1 A boo k that many GWS read ers will find useful and enjoyable is HOW TO PLAY THE PIANO DESPITE YEARS OF LESSONS by Ward Cannel and Fred Marx. It's full of sensible, practi cal advice on how to play songs by ear; it debunks a lot of the myths that have mast people convinced they're not smart enough to make music; it provides some sound, under standable ' explanations of such things as chords; and, it is fun to read. For example, from one of the opening chapters: [Myt h: 1 Harmon is. very mathema tical. Yes, thatelief is true. In order to understand harmon y , you must be 3ble to count to t welve . You can perform this scientific e xper iment at hom e to d e~ermine whether yo u hav e the neces sa ry math, ematica l ability. Just look a t a clock or a wrist watch . Can you tell what time it is? If so, yo u will be able to understand harmony. If not, then wait for the seque l to this book: How To Tell Time From A Clock and Wrist Watch .
S
Unfo rtunat ely, the book is no longer available to the general pub lic, o r we would have added it to ou r booklist . You may still be able to find copies in stock in your l oca l bookstore, since it only recently went out of print. The book is still being printea in order to be-Used for courses that are offered around the country . These cou r ses are arranged by the PIANO CON SORTIUM (6t h Fl oor East, 200 Par k Av, NY NY 10017) through a numb er of col leges and universities. Elaine Apple of the Piano Consortium sent some art icles about the courses which we quote below, and s he said anyone want ing to find out if the courses are offered in their area should write to the Consortium . But for the time be ing, at least, th ey wo n't sell the book separately. Maybe if e noug h peo ple write and ask them to, they'll change their minds.
Phoebe Epstein, one of six women enrolled in a course entitled "How to Play the Piano Despite Years of Les sons" ... said "I love to listen to Alicia de Larrocha play Bach, but I don't expect to play that way my- . self." What she does expect is: "When members of my family get ~ogether and sing, I will be able to accompany them on the piano. Knowing how to play Chopin waltzes just doesn 't do that." On her registration card, Miss Epstein had written, "I had seven, (!) years of less ons between the ages of 7 and 15. I can't playa thing ." What might superficial l y seem to be fai lure - having tried and given up on th e piano - is, in fact, the only prerequisite for the course, pro viding the candidate can play the treble clef o r right-hand melody part . The course, first offered four years ago a t five locations in New York and New Jersey, r ef lects a be lief that increasing numbers of adults are taking up the piano . The response to initial advertisements was so great that the course was imme diately expanded to 19 sites by the Piano Consortium, a nonprofit edu cational organization that now spon sors iX in about 75 colleges as well as in other settings . Ward Cannel, a former newspaper man ... began to ask questions about h ow people organize tones co make music . As a result, a group of musi cia n s ... conducted experiments on ways to t ea'c h the piano based on the premise that understanding and not rote was the key to playing. ... "There are 30 million pianos in the United States ,and of those, we think that 90% are not being played they just sit around collecting du s t," sai d Shelby Nee ly, education administrator of the consortium. The approach is to make playing the piano pure fun, and that is accom plished by the unorthodox method of dispensing wit h all but the princi ples necessary to play popular songs right away. "They already k'now the right h a nd part, so we show them what t o do with the l eft hand, which has e luded them all their lives, " said Dr. Myro n Leshowitz, teacher of the course at th e New School ... The class encour ages improvisation in the chords and rhythms provided by the left hand. Although the New S~ h oo l class takes 10 sessions, most institutions complete it in eight wee ks ... LETTERS ON MUSIC
Manfred Smith (Md. ) writes: ... 1 have started t o play the electric guitar. I had a lways wanted to play the guitar, but beCame easily frustrated because of the awkw ard