Don t Slow Me Down with that Calculator Cliff Petrak (Teacher Emeritus) Brother Rice H.S. Chicago cpetrak1@hotmail.com

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1 Don t Slow Me Down with that Calculator Cliff Petrak (Teacher Emeritus) Brother Rice H.S. Chicago cpetrak1@hotmail.com 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, the shortcut speed methods stand out far and above for objectives 2,3 & 4. Common Core Standards for Mathematics Common Core Standards for Mathematics: Standards #6,7 & 8 are those that make reference to mental/speed math methods which so often receive their derivation from either a formal algebraic proof or more simply a study of number patterns which works well for the younger students who have not yet studied algebra. Common Core Standard #6, namely, Attend to Precision, states that mathematically proficient students calculate accurately and efficiently. How much more efficiently can calculations be made than with a minimum of time and paper, often orally as well! Common Core Standard #7, namely, Look for and make use of Structure, states that mathematically proficient students look closely to discern a pattern structure, that is, a logical order and structure in mathematics. Along with more formal algebraic derivations and proofs, this is exactly what we do in arriving at speed/mental techniques and methods in arithmetic calculations. Common Core Standard #8, namely, Look for and express regularity in repeated reasoning, states that Mathematically proficient students notice when calculations are repeated and look for both general methods and shortcuts. Note:. 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 self-published book, Don t Slow Me down with that Calculator. It also contains algebraic derivations of every shortcut method. Because it is self-published, it is not available in bookstores although it may be found on Amazon for $18). 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 $4.00 shipping charge. To purchase multiple, discounted copies, contact author Cliff Petrak at cpetrak1@hotmail.com. 1

2 Shortcut Speed Method #1: (Addition) When adding a column of 2 or 3-digit numbers without an available 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, , 2429 and 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 sum of 9 the next two or more integers to the 10 or multiple of This will eliminate one of the additions. This speed tip is illustrated 5 twice in the example to the left. 1 st, note how the running total of 10(after adding 4+ 6) 8 is quickly added to 13 to arrive at 23 (think ) rather than adding in two 6 separate additions. 2 nd, note how the running total of 40 quickly becomes 51 by adding (think 10+1) rather than adding 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) Technique Shortcut (carrying) Technique In these examples, all digit changes and slash marks are used only as clarifying markers. They are time consuming habits to be avoided in the subtraction process. Why spend needless time subtracting when you could be adding? 2

3 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 15 2 and Step 2: The last 2 digits of the answer will always be 25 (the square of the unit s digit). Ex. Find 65 2 Step 1: 6 x 7 = 42 Step 2: 25 Answer: 4225 Ex. Find 95 2 Step 1: 9 x 10 = 90 Step 2: 25 Answer: 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 24 2 Step 1: 4 2 = 16 (Write 6, carry 1) Step 2: (2 x 4 x 2) + 1 = 17 (Write 7, carry 1) Step 3: = 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 = = =195 Ex. 79 X 81 = = = 6399 Ex. 23 x 21 = = 484 1=483 Ex. 24 x 26 = = = 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 x11 Step 1: 4 Ex. 78 X 11 Step 1: 8 Step 2: 1+4=5 Step 2: 7+8 = 15(Write5, carry1) Step 3: 1 Step 3: 7+1 = 8 Answer =154 Answer = 858 3

4 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) (1 st unit s digit). Step 3: To that total, add the product of (10) (2 nd unit s digit). Step 4: To that total, add the product of the 2 unit s digits. Ex. 13 x 15 Step 1 Step 2 Step 3 Step 4 Ans. 10X X X = 195 Think Ex. 19 x 14 Step 1 Step 2 Step 3 Step 4 Ans. 10X x9 + 10x4 + 9x = 266 Think 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 Ex. Find = = 46 = Ex. Find 2-6 = = 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. 4

5 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. A DVD version is also available for $15, or the book and DVD for just $ 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 = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² = ² =

6 To order 1 or more additional books or DVD 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 (or DVD version) 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 and thoroughly derived using elementary algebra. I would like to order copy/ (copies)of the.(book Version) disc/ (discs) of the (DVD Version) Send $15.00 for each copy or disc (or $25 for 1 of each) Add $4.00 for each book or $3.00 for each DVD for postage & Shipping = = SHIPPING INFORMATION: TOTAL ENCLOSED = Name: (Individual) Name: (School) Mailing Address: City, State & Zip: Address: Your order will be processed the same day that it arrives and autographed (if you make that request).(yes, autograph ) ; (No, don t autograph ) (Be sure to check either yes or no) Make out your check or money order to Cliff Petrak and send it to: Cliff Petrak S. Central Park cpetrak1@hotmail.com Chicago, Il

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