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Topic: Developing the product of two negatives
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Jerry P. Becker

Posts: 13,744
Registered: 12/3/04
Developing the product of two negatives
Posted: Jan 3, 2000 1:50 PM
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att1.dat (4.4 K)

*********************************
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 100-pegboard).

Then, multiplying a 1-digit number and a 2-digit number, putting the
2-digit 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 1-digit and
3-digit number, two 2-digit numbers, a 2-digit and 3 digit number, two
3-digit 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 2-digit and 3-digit number, as well as two
3-digit 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
1-4).

************************************************************

Jerry P. Becker
Dept. of Curriculum & Instruction
Southern Illinois University
Carbondale, IL 62901-4610 USA
Fax: (618) 453-4244
Phone: (618) 453-4241 (office)
(618) 457-8903 (home)
E-mail: jbecker@siu.edu

mailto://jbecker@siu.edu



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