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Developing the product of two negatives
Posted:
Jan 3, 2000 1:50 PM



********************************* Developing the product of two negatives. *********************************
Assume that elementary students understand place value and the basic multiplication facts and then work towards an algorithm as follows:
3_ 9_ 9_ To begin 3  9 , 6  54 , 9  81 , etc, writing answers this way. Meaning can be given to multiplication through the use of rectangular arrays of dots (e.g., like a 100pegboard).
Then, multiplying a 1digit number and a 2digit number, putting the 2digit number in 'expanded' notation:  10__3_ 2 x 13 = 2 x (10 + 3): 2  20  6  = 20 + 6 = 26
30  8_ 7 x 38 = 7 x (30 + 8): 7  210  56  = 210 + 56 = 266
Then, let us assume that students know how to handle addition of integers like 3 + (1) , 5 + 3 , and so forth, which would be taught before the product of two negative integers.
Now, since 9 = 10 + (1), 9 x 9 could be represented like
 10  1  10  100  10  1  10  ? 
We know that 9 x 9 = 81, and above we have 100  10  10 = 80, so doesn't (1) x (1) = +1 , giving 81?
Similarly, for 7 = 10 + (3), 7 x 7 would be represented like
 10  3_ 10  100  30  3  30  ? 
We know that 7 x 7 = 49, and above we have 100  30 30 = 40, so doesn't (3) x (3) = +9 , giving 49?
Similarly, one can show the results for (a + b) x (a + b) and (a  b) x (a  b) in this 'grid' format.
Following this, if we choose, we can represent the product of a 1digit and 3digit number, two 2digit numbers, a 2digit and 3 digit number, two 3digit numbers, and so on.
Another thing: Would this approach also facilitate developing mental computation and number sense? Is the student always adding 'nice' numbers in the grid, doing it mentally, and also getting practice with addition/subtraction, previously learned, while learning multiplication?
Beyond the above, could a transition be made to doing multiplication completely in one's head, writing down the product one digit at a time? [See Jagadguru Swami Sri Bharati Krsna Tirthaji Maharja, (1997). Vedic Mathematics, Delhi, India: Motilal Banarsidass Publishers Private Limited  41 U.A. Bungalow Road, Jawahar Nagar, Delhi 110 007 INDIA]
12 x 13 (3 x 2) or 6 156 (3 x 1 + 2 x 1) or 5 (1 x 1) or 1
so, 156.
36 x 27 (7 x 6) or 42, so write down the 2 and remember the 4. 2 972 (7 x 3)+ (6 x 2) or 33, now add in the 4 to get 37, write down the 7 and remember the 3 7 (2 x 3) or 6, now add in the 3 to get 9 and write it down 9
so, the answer is 972.
Similarly for multiplying a 2digit and 3digit number, as well as two 3digit numbers.
Also, using a sort of m x n pegboard, multiplication can be visually represented as the sum of the four 'partial products' and the distributive property is learned and used along the way and used quite a lot. Maybe there is a great deal more that can be done handling multiplication in this manner.
All good wishes.
Jerry
P.S. The approach outlined above is the approach that is used in a new primary school mathematics series of books in Germany (primary = grades 14).
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Jerry P. Becker Dept. of Curriculum & Instruction Southern Illinois University Carbondale, IL 629014610 USA Fax: (618) 4534244 Phone: (618) 4534241 (office) (618) 4578903 (home) Email: jbecker@siu.edu
mailto://jbecker@siu.edu



