Chapter 7 - Building Speed In Multiplication

You recall that we stated three basic secrets for speed in multiplication:

First, work from left to right (possible only with this system).

Second, "see" the result of each multiplication of two digits, rather than the problem.

Third, use the no-carry method.

The second point is the one that obviously requires the most practice. The foundation of all your speed is the easy, natural, painless use of the no-carry system—but the way to make it easy and painless is to make as automatic and unthinking as possible the process of "seeing" 8 x 7 as "50's" and "ends in 6."

Your job now is to go over these half-products enough times to make the automatic response a habit. Since you undoubtedly got far more drill in the multiplication tables than you did in addition and subtraction tables, learning to see each product as only the left-hand or right-hand digit is not really all that much more work. Once you become fully used to it, you will find it quicker and simpler than the old way.

Let's review for a moment what we mean by left-hand and right-hand digits in multiplication. Try to "see" the left-hand (tens) digit of 4 X2

If you fully mastered the last chapter, you answered, almost without a second thought, "zero."

What is the right-hand (units) digit?

If the digit 8 sprang into your mind with little or no effort, you are already well on the way to accelerating your multiplication with the no-carry method. If you had to stop and think, however—as most of us do at this point—then that is exactly what this chapter is for.

Your first exercise is to go through the following digit pairs with the object of "seeing" only the left-hand (tens) digit—the one we have been describing as "is in the 20's, 70's," etc. See and think, as well as you can, 4 x 4 as "1."
The first time you go through these, it might be wise not to try for speed. The first job is to begin training the habit of recognizing the left-hand digit automatically. Just as important, you should build the habit of thinking "zero" when there is no real left-hand digit (that is, when the full product is less than ten) because this is so important to accuracy in multiplying longer numbers.

Remember to see 4 x 9 as "3"—not "the left-hand digit of 4 x 9 is 3, because 4 x 9 is 36 and 36 is in the 30's"—just as you see u and p as "up."

Go slowly and carefully this first time, training your mind to see only the answer. Left-hand (tens) digits only:

That is enough for the first dose. You will go through every possible digit combination before you are through, but doing them all at once might become tedious.

Compare, if you will, the study of speed mathematics to learning any new skill. There is a specific objective in mind, of course—in this case, to solve problems more rapidly and easily. But there is also a helpful secondary objective: becoming fascinated with the process of doing and excited about your mastery of the technique. Just as a craftsman enjoys the actual process of making a perfect joint in a woodworking project because it is satisfying to do something skillfully, so can you become fascinated with the dispatch and accuracy of your working of a sample problem in a new way.

When we use them in business, numbers always stand for something. When we practice with them, however, they become an impersonal sort of puzzle. Look on them as a crossword puzzle, or a chess problem, or a brain-teaser. Just as satisfying as these and far more rewarding—because your growing skill at this type of puzzle will pay you solid dividends for the rest of your life.
 
Now, carefully rather than hastily this first time, continue working at the habit of seeing only the left-hand (tens) digits of these combinations:

By now you should find the habit beginning to take hold. Once the proper response starts becoming a habit, you can go back over the examples with the objective of speeding up your reaction time.

Make very sure at this point, though, that you work at giving your response in the right fashion, rather than giving a fast but improper one. Going reasonably slowly now will contribute to greater speed in the future.

Now finish your practice on left-hand (tens) digits with the rest of the basic digit combinations:

Stop and take stock of your technique now. Do you find that you are looking only for the left-hand digit as you glance at each pair? Have you schooled yourself to give only the answer? Are you always thinking "zero" when the product of the two digits is less than ten?

If not, make a point of going back over the combinations from time to time, working specifically to develop this habit. If you feel that you are making the proper responses a routine, then your next step should be to develop speed. Time yourself in completing the tables, and make a note of how long it took. Next time, see if you can shave a few seconds off the last record.

Right-Hand Digits

So far, you have worked at accuracy and speed in seeing only the left-hand, or tens, digit of each product. This is only half the story. The other half is to do precisely the same thing for what the product "ends in." Glance at this example:

6
x 7

What is the left-hand digit?

What is the right-hand digit?

There would be little point to repeating all the tables again just for the right-hand digit practice. Instead, use the same tables on the last few pages.

Keep in mind the important practice points mentioned in connection with left-hand digits. Go slowly the first time, consciously making an effort to "see" only the right-hand digit, rather than the problem. You may find it helpful to say the answer to yourself; if you do, be very careful not to say the problem.

After you have gone over the tables just a few times, you should begin to find yourself simply reading the answers— just as you read these words or phrases rather than the letters.

If you need proof of this, stop right now and try to recall whether there were any f's in the paragraph above. In all likelihood, you haven't the vaguest idea. You undoubtedly read the first word "after" without even noticing the f in it. In the same way, you can approach this "end-result-only" ability with digit combinations.

Go back to the tables and do your first right-hand digit practice now.

Work at the tables conscientiously, but I would suggest that you alternate practicing the digit combinations with some of the other practice to come. Avoid the stale, "overtrained" reaction of too much consecutive time spent at only one part of the whole.

Two-Digit Practice

The whole reason you practice the basic multiplication table with left-hand and right-hand digits is so you can multiply from left to right without carrying. Picking up the right-hand digit from one product and adding it to the left-hand digit of the product to the right is the secret that eliminates carrying altogether.

You do have to keep one digit in your mind for a moment, but this is considerably simpler than juggling three (and sometimes four) digits in traditional right-to-left multiplication.

Refresh your memory with this example:

See if you can anticipate each step of this review: Step one: 7 x 4 is in the 20's:

2

Step two: 7X4 ends in 8. 8 x 4 is in the 30's. 3 minus 2 (complement of 8) is 1, and record the ten:

2 1

Step three: 8x4 ends in 2:

2 1 2, or 312

The step two above is as complicated as no-carry multiplication can ever get. You have to remember the 8 while getting the 3. If you have learned to read answers, you would think only "8, 3, 1, record." The same point in schoolbook multiplication would involve these thoughts: "Carry the 3 from 32. 4 x 7 is 28. Add the carried 3 to 8, which makes it 11. Put down 1 and carry 1 to the 20. Put down 3."

This review is to encourage you to spend some of your practice time on the two-digit tables that follow. It would be impractical to include every possible combination (there are just under a thousand of them), but you will find a good spread of every type.

The first time you do this section, work slowly and evenly, disciplining yourself to think along the lines we have covered:
Read only the answer to each digit combination.
 
Work from left to right.

Think an initial zero if this is the left-hand digit of the first product.

Add the center digits of the answer with a complement if it goes over ten, and mentally record the ten by underlining the imaginary digit to the left in the answer.

Say aloud the answers to these problems:

The most important thing you undoubtedly noticed is that your ease with these problems is based very directly on your ability to read automatically the left- and right-hand digits of the products of each combination. If they pop into your mind without thought—as they will after surprisingly little practice—then expanding your practice to two-digit examples is almost painless. But if you have to stop and think hard to get each digit, then you will find this section much harder and slower than it should be.

If you experienced trouble in "reading" the left- and right-hand digits to make these problems easy, go back and review your single-digit tables once or twice before going on. Each time you do them, the answer will come a little more automatically.

Now read from left to right the answers to these problems. Make sure you are building the right habits as you do so. Make it a point to use the proper technique:
 
The final practice table of this chapter follows. You have already practiced all the essentials. If you can handle two-digit tables with snap and decisiveness, then you can keep on doing step two through twenty-digit problems. You already know how to line up your columns for multipliers of two digits or more, and you know how to add more effectively and quickly than ever before. The final section asks you to say aloud, from left to right, the answers to a variety of multiplications with single-digit multipliers but differing numbers of digits in the numbers multiplied.

Just as you did with both the one-digit and two-digit tables, work slowly and carefully the first time over this varied practice group. Get the foundation of proper habits firmly established. Say your zero first digits where they are required, think an underline to the left when you use a complement, and do your very best to read only the answer to each combination—not the combination itself.

For longer problems you may wish to write down your answer. Just put your pad under the problem and jot down the answer from left to right, as it develops naturally in your mind.

Spend several minutes at this:

The mixing of problems with one, two, three, and more digits in this section was intentional. This is the way problems are presented to us in business. They do not ordinarily come neatly packaged in orderly rows of similar problems. Your mastery of each method and technique always has to become adaptable as well as proficient.

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