Don’t Slow Me Down with that Calculator
Cliff Petrak (Teacher Emeritus)
Brother Rice H.S. – Chicago
In any computation, we have four ideal objectives to meet:
1)Arriving at the correct answer
2)Using as little time as possible
3)Utilizing as little writing as possible
4)When possible, producing an oral answer without any writing
Any of three tools can be used to compute whole number operations:
1)the standard (or traditional) algorithmic methods learned in elementary school
2)the calculator
3)the shortcut speed methods
While all three tools will produce correct answers, differences are quickly seen in trying to meet objectives 2, 3 and 4.
Advantages and disadvantages of these three tools include:
Standard Method:
adv. – No calculator is needed
There is no need to “enter” the involved numbers as with the calculator since they are
already there on the paper.
disadv. - Often slow because of the many in between numbers that must be written (as
with multiplication & squaring & finding common denominators when adding &
subtracting fractions) before the correct answer appears.
Calculator:
adv. - Requires no in between numbers in arriving at the correct answer.
- Displays the correct answer upon the touch of one button following the entering of
the numbers to be operated upon.
disadv. - With addition, subtraction, squaring and multiplication, the numbers involved must be
entered beforethe answer can be found.
- Since almost all answers need to be recorded, time is needed to transfer the
displayed answer to paper.
- The intended user of this method must have a working calculator with him.
Without one, the user must be familiar with one of the other methods.
- With a standard calculator, working with fractions is not possible.
- For some tests, calculators are not allowed.
- It often produces answers more slowly than with the use of shortcut speed methods.
Shortcut Speed Methods:
adv - No calculator is needed.
- When applying these methods, the answers roll out without need for any in
between numbers.
- No transference of an answer is ever needed as with a calculator.
- Not only are fractional operations performed more quickly, but never require
finding an LCD when adding or subtracting.
- Many of the methods produce oral answers.
- It is “fun” for the user & usually leaves any audience in amazement.
- There is no need to “enter” any of the involved numbers as with the calculator.
- These methods are not only faster, but increase one’s math knowledge unlike the
calculator.
disadv. - A satisfactory speed method does not yet exist for long division.
- The required mental discipline and application of the steps involved when multiplying or
squaring integers of 4+ digits make this type of calculation more easily done with
either of the other two methods, especially the calculator.
- To use all of the speed methods involving multiplication and squaring, it is necessary to
memorize the squares of all integers up through 30 & have a firm foundation in the
multiplication tables.
In conclusion, if all three calculation tools are available,. I would recommend as follows:
Standard Method-Try to avoid this method whenever possible.
Calculator- Use this tool only for long division as well as for multiplication &
squaring of number containing 4 or more digits.
Shortcut Speed Methods -Use this tool for all additions and subtractions of numbers of any
size.
-Use this tool for all multiplications and squaring of numbers
containing 2 or 3 digits.
-Use this tool for all operations involving fractions.
While I would like to explain and demonstrate all 59 of my favorite shortcut procedures, the time that we have today will confine us to a limited number of them. Nine of these methods are detailed in this handout. All 59 may be found fully described in my recently self-published book, Don’t Slow Me down with that Calculator. It also contains algebraic derivations of every shortcut method. Bcause it is self-published, it is not available in bookstores although it may be found on Amazon.Use the attached order blank to order a copy in the future. However, copies will be available for sale after this presentation. The cost is $15 per copy. Buying a copy today would also save you the $4.00shipping charge. To purchase multiple, discounted copies, contact author Cliff Petrak at .
Shortcut Speed Method #1: (Addition)
When adding a column of 2 or 3-digit numbers and being without the use of a pencil, add left to right to;
1) come up with the answer orally and
2) to eliminate the need to memorize the digits as they are arrived at in the usual right to left fashion.
For example, in adding 349, the running totals (if we travel up, then down and then up again)
895 will be 900, 1100, 1900, 2200, 2240, 2330, 2340, 2410, 2416,
218 2424, 2429 and 2438.
976
In traveling right to left, we would have to remember the unit’s digit
to be 8, the next digit a 3 and so on, all while trying to concentrate
on the addition process.
Shortcut Speed Method #2: (Addition)
2 Lets assume that we are adding this column from the bottom up.
9 When we find the running total becoming 10 or a multiple of 10, immediately add the next
9 two or more integers needed to produce a teen sum that can then be easily added to the
8 10 or multiple of 10. This will eliminate one of the additions. This speed tip is illustrated
5 twice in the example to the left. 1st, note how the running total of 10(after adding 4+ 6)
8 is quickly added to 13 to arrive at 23 (think 10 + 3) rather than adding 10+8+5 in two
6 separate additions. 2nd, note how the running total of 40 quickly becomes 51 by adding 11
4 11(think 10+1) rather than adding 40+9+2 in two separate additions.
51
Shortcut Speed Method #3: (Subtraction)
Because of the necessity for borrowing in most subtraction problems, the typical subtraction problem contains a number of both mini-additions and subtractions. The trick is to convert as many of the mini-subtractions as possible into additions. If we can add faster than we can subtract, we will complete the problem more quickly. The method is as follows: Whenever the borrowing step is needed, add 10 to the minuend (upper number) as usual, but instead of subtracting 1 from the minuend digit to the left, add 1 to the subtrahend below. The answer will be the same, but with far more additions employed and far fewer subtractions.
Performed with the traditional (borrowing) technique, the following subtraction problem includes only 7 additions, but 15 subtraction steps. However, on the right side, where the borrowing technique is converted to the speed method (carrying) technique, the numbers change dramatically to 14 additions and only 8 subtractions.
Traditional (borrowing) TechniqueShortcut (carrying) Technique
Shortcut Speed Method #4: (Squaring a 2-digit Integer ending in 5)
Step 1: The first 1 or 2 digits of the answer are found by multiplying the ten’s digit by the next
larger digit. Only a single digit product will result in the case of 152 and 252.
Step 2: The last 2 digits of the answer will always be 25 (the square of the unit’s digit).
Ex. Find 652Step 1: 6 x 7 = 42 Step 2: 25Answer: 4225
Ex. Find 952Step 1: 9 x 10 = 90Step 2: 25Answer: 9025
Shortcut Speed Method #5: (Squaring ANY 2-digit Integer)
Step 1: Square the unit’s digit, writing down the unit’s digit of this product as the unit’s digit of
the answer. Then, mentally hold any “carried” number into Step 2.
Step 2: Multiply the product of the ten’s and unit’s digits of the 2-digit number by 2, adding any
carried number from Step 1. Then, write down the unit’s digit of this product as the ten’s
digit of the answer. Again, mentally hold any carried number into Step 3.
Step 3: Square the ten’s digit of the 2-digit number and add any carried number from Step 2.
This sum will complete the answer, serving as the hundred’s and any thousand’s
digits in the final answer.
Ex. Find 242 Step 1: 42 = 16 (Write 6, carry 1)
Step 2: (2 x 4 x 2) + 1 = 17 (Write 7, carry 1)
Step 3: 22 + 1 = 5
Answer: 576
Shortcut Speed Method #6: (Multiplication of 2-Digit Integers Differing by 2)
The only step requires squaring the integer midway between the 2 numbers and subtracting 1. The greater the number of squares that have been memorized, the greater the number of such specialized multiplications can be solved mentally.
Ex. 13 x 15 = 142 - 1 = 196 – 1 =195Ex. 79 X 81 = 802 – 1 = 6400 – 1 = 6399
Ex. 23 x 21 = 222 - 1 = 484 – 1=483Ex. 24 x 26 = 252 – 1 = 625 – 1 = 624
Shortcut Speed Method # 7: (Multiplication of any 2 Digit Integer by 11)
The digits of such products can be found almost instantaneously from right to left in 3 steps.
Step 1: The unit’s digit of the product will always equal the unit’s digit of the 2-digit integer.
Step 2: The ten’s digit of the product is found by adding together the ten’s and unit’s digits of the
2-digit integer. If this sum is greater than 10, carry the “1” into Step 3. When the number
being multiplied by 11 contains more than 2 digits, just continue this pattern, adding
together the hundred’s and ten’s digits along with any carried number from the previous
step.
Step 3: The hundred’s digit of the product (which may contain 1 or 2 digits) will equal the ten’s
digit of the 2-digit integer plus any carried number from the previous step.
Ex. 14 x11Step1: 4Ex.78 X 11 Step1: 8
Step2:1+4=5Step2: 7+8 = 15(Write5, carry1)
Step3: 1Step3: 7+1 = 8
Answer =154Answer = 858
Shortcut Speed Method #8: (Left to Right “Teen” Multiplication) for Integers 10 through 19
This procedure is recommended for the multiplication of any 2 integers in their “teens.”
Step 1: Mentally, multiply the 2 ten’s digits, always yielding a product of 10 x 10 or “100.”
Step 2: To 100, add the product of (10) (1st unit’s digit).
Step 3: To that total, add the product of (10) (2nd unit’s digit).
Step 4: To that total, add the product of the 2 unit’s digits.
Ex. 13 x 15Step 1 Step 2 Step 3Step 4 Ans.
10X10 +10X3 +10+5 +3X5
100 +30 +50 +15 = 195
Think100……………..130…………….180………………195
Ex. 19 x 14Step 1Step 2Step 3Step 4 Ans.
10X10 +10x9 +10x4 +9x4
100 +90 +40 +36 = 266
Think100…………….190………………230…………….266
Shortcut Speed Method #9: (Adding & Subtracting Fractions with Unlike Denominators)
Called the “Smiling X Method,“ this is a simple 3-step procedure that can usually be performed without paper or pencil. The finding of a least common denominator is never needed.
Step 1: Find the sum or difference (as indicated by the sign) of the 2 cross (diagonal) products.
Each cross product consists of one of the numerators multiplied times the opposite denominator. With subtraction problems, the first cross product found must be the one running from top left to lower right.
Step 2: Place the number found in Step 1 over the product of the 2 denominators.
Step 3: Reduce the fractional answer, if possible.
Ex. Find 2 + 4 = 14 + 20 = 34
5 7 35 35
²
Ex. Find 1 + 9 = 10+ 36 = 46 = 23
4 10 40 40 20
Ex. Find 2 - 6 = 14 – 18 = -4
3 7 21 21
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This concludes a small sampling of my 59 super-shortcut calculation methods that are out there waiting for your mastery. They are all fun, fast and easy to learn. All of the speed techniques discussed in today’s presentation can be performed mentally. The only need for a pencil is for the recording of your mentally arrived-at answer. I only wish that more time was available to explain the many other shortcut methods that exist. However, they all may be found in my book.
Use the attached order form to purchase one or more copies of Don’t Slow Me down with that Calculator if you wish to purchase a copy by mail.In this 162 page user-friendly book, each technique is discussed and explained in great detail. Several step-by-step solved problems are presented along with several additional unsolved problems for your practice. Algebraic derivations of every procedure are also included. Answers are provided.
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For many of the multiplication shortcut techniques, it will prove very helpful to memorize as many of the 2-digit squares as possible, but for certain those through 30. All are provided below.
2 – Digit Squares
10² = 100 20² = 400 30² = 900 40² = 1600 50² = 2500 60² = 3600 70² = 4900
11² = 121 21² = 441 31² = 961 41² = 1681 51² = 2601 61² = 3721 71² = 5041
12² = 144 22² = 484 32² = 1024 42² = 1764 52² = 2704 62² = 3844 72² = 5184
13² = 169 23² = 529 33² = 1089 43² = 1849 53² = 2809 63² = 3969 73² = 5329
14² = 196 24² = 576 34² = 1156 44² = 1936 54² = 2916 64² = 4096 74² = 5476
15² = 225 25² = 625 35² = 1225 45² = 2025 55² = 3025 65² = 4225 75² = 5625
16² = 256 26² = 676 36² = 1296 46² = 2116 56² = 3136 66² = 4356 76² = 5776
17² = 289 27² = 729 37² = 1369 47² = 2209 57² = 3249 67² = 4489 77² = 5929
18² = 324 28² = 784 38² = 1444 48² = 2304 58² = 3364 68² = 4624 78² = 6084
19² = 361 29² = 841 39² = 1521 49² = 2401 59² = 3481 69² = 4761 79² = 6241
80² = 6400 85² = 7225 90² = 8100 95² = 9025
81² = 6561 86² = 7396 91² = 8281 96² = 9216
82² = 6724 87² = 7569 92² = 8464 97² = 9409
83² = 6889 88² = 7744 93² = 8649 98² = 9604
84² = 7056 89² = 7921 94² = 8836 99² = 9801
To order 1 or more additional copies:
Order Form
Copies of Don’t Slow Me Down with that Calculator by Cliff Petrak are available from the author by mail. Being self-published, the book is not available in bookstores. Its 162 pages contain 59 shortcut methods and speed tips dealing with addition, subtraction, multiplication, squaring and fractional operations. Each unit contains step-by-step, easy-to-follow directions with both solved and supplementary problems. Every shortcut method is explained or thoroughly derived using elementary algebra.
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