It’s a simple rule: When multiplying two numbers, if the signs are the same, the product is positive, if the signs are different, the product is negative.
But why does it work?
In the spirit of Khan Academy’s new and improved video, here is a lesson, in a context familiar to students, that builds upon the Distributive Property, multiplying by zero, sums of opposites, and substitution, in order to explain why the rule works.
Have students answer the following questions:
1) You are going to a new restaurant with your family. Tonight’s special is $7 for any dinner on the menu. Dessert is not included and costs $2 for each item ordered. If there are five people in your family, and each orders dinner and dessert, how much will the meal cost (ignore tax and tips)?
2) The waiter brings you your bill. You pay it. What is your balance now for the meal?
3) If you decide to eat at home instead tonight, what will your cost be for dinner from the restaurant?
Invite a student to share their strategy for question 1. Then ask if anyone solved the problem differently. Typically, some students will add to get the total cost of the dinner first and then multiply by 5 people, while others will first multiply to find the cost of 5 dinners and 5 desserts and then add. Discuss how this is an application of the Distributive Property. Use an array to show that, when multiplying two numbers, you can decompose one of the factors in any way and you still have the same number of total items no matter how you break them up. Practice this by challenging students to mentally calcuate, for example, 6 x 45, by decomposing 45 and writing 6 x (45) = 6 x (40 + 5) = (6 x 40) + (6 x 5) = 240 + 30 = 270.
Next use question 2 to review that opposites sum to zero. The situation can be described by writing – 45 + 45 = 0, and you owe zero dollars after you pay.
Use question 3 to discuss that multiplying by zero yields a product of zero. The meal costs $9 but zero people purchase it, so 9 x 0 = 0 cost for the restaurant meal. Also, since 9 x 0 and 0 are equal, we can just say 0 instead of 9 x 0.
Finally, use these concepts to show how the rules for multiplication can be generated:
We know that any number times zero equals zero.
So, for example, 5 x 0 = 0
We also know that opposites sum to zero. So we can replace the factor of zero with any sum of opposites:
5 x (-3 + 3) = 0
By the Distributive Property, this can be rewritten as
(5 x -3) + (5 x 3) = 0
Evaluating 5 x 3:
(5 x -3) + 15 = 0
And since we know that opposites sum to zero, then the addend (5 x -3) has to be equal to the opposite of 15, or -15.
Since we could have used any pair of opposites, our procedure is general, works for all numbers, and therefore the rule that a positive times a negative is negative must always hold true.
Next, try -5 x 0 = 0
Repeating the steps from above, first substitute a sum of opposites for zero:
-5 x (-3 + 3) = 0
Distributing, (-5 x -3) + (-5 x 3) = 0
We saw before that -5 x 3 = -15 so substitute
(-5 x -3) + -15 = 0
Since the addends sum to zero, -5 x -3 has to be the opposite of -15 or positive 15 and, since we can generalize with any pair of opposites, we have shown that the product of negatives must always be positive.