First you need to establish that –a × b = –ab:
One method to show this is using red and black counting objects. Suppose the red object represents –1 and the black object represents +1.
Then 4 × –3 can be represented by 4 rows of three red objects:
The product (total) is –12.
Suppose you need to represent –5 × 2. For students who understand the commutative property, then can represent 2 × –5 as 2 rows of five red objects:
The product (total) is –10.
After –a × b = –ab is established, then have students find a pattern such as:
3 × –3 = –9:
2 × –3 = –4
After a pattern is formed there are two ways to complete the patterns for the product of two negative numbers:
Place the above pattern into a table
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The students should be able to recognize the pattern in the last column. The pattern is to add three to the number in one cell to obtain the result in the cell below. Completing the table:
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The pattern shows that –a × –b = +ab!
It has been shown that 4 × –3 can be represented by 4 rows of three red objects:
The product (total) is –12.
And 3 × –3 = –9:
And 2 × –3 = –4:
And 1 × –3 = –3:
The students should be able to recognize the pattern as removing 3 red objects (i.e., subtracting –3). The next picture in our pattern is obtained by removing 3 red objects. The result is:
no blocks representing 0 × –3 = 0.
There are no more blocks, so to continue this pattern we can add 3 red and 3 black objects (the sum of which is 0).
So, the next picture (–1 × –3) in our pattern is obtained by removing 3 red objects. The result is (+3):
There are no more red blocks to remove, so to continue this pattern we can add 3 red and 3 black objects (the sum of which is 0).
So, the next picture (–2 × –3) in our pattern is obtained by removing 3 red objects. The result is (+6):
The pattern shows that (–a)(–b) = +ab!
We have shown two ways to demonstrate to students that –a × –b = +ab.
John Weber
Denise Mewborn