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## Class 8 (Foundation)

### Unit 1: Lesson 2

Multiplying and dividing integers

# Why a negative times a negative makes sense

Use the repeated addition model of multiplication to give an understanding of multiplying negative numbers. Created by Sal Khan.

## Want to join the conversation?

• • strange way to think about it... -2*-3=6
i have a large amount of warriors, each warrior can take out 2 scouts. an enemy has a camp and i send out three of my scouts -3, then they come back each having taken out two scouts -2. how many scouts did they take out total. -3*-2=6 does this make sense or did i not understand this well enough. word problems are rough haha • I believe it would help to think of 0 not as nothing but as an equilibrium or balanced amount, having the same on both sides. For example I spent a fair amount of time thinking about this and I came up with this holes and hills analogy.

A hill is positive, meaning a grassy hill and hole is negative, a hole I dug in the ground. Let say I have 10 hills and 10 holes to start so I'm balanced and I'm at 0 equilibrium.

1. If I then start and I make three more groups of three hills (3 x 3) I will get 9 extra hills. So I have +9 hills overall.

2. If I make three holes and make three groups of them (-3 x 3) I will get -9 or 9 more holes in the ground so I will have -9 or 9 more holes than hills.

3. If I take away 3 groups of 3 holes (-3 x -3) I will be removing a total of 9 holes therefore leaving me with, in regards to my example, 1 hole and 10 hills or 9 hills equilibrium so +9.

This example is the same as you might think about balance scales it's the same principle. It works the same with money and I believe it applies in possibly all areas. In money the money you no longer have to pay back in debt payment becomes yours and is therefore kept by you as a positive amount. In wood working if you no longer have to build and give 3 items to three people you can keep those items yourself and you'll be up 9 items. Ect... Ect....

What do you think? • Who is the ancient philosopher? By the way you are a really good artist. • This is a way I think of a negative times a negative.

I visualize the number line as a scale with a pivot on the 0. I think that the "0" always has to be at the center of the two sides (negative/positive infinity) in order for them to be balanced. If I see -3*-4 it means "start from zero" and take out 4 groups of negative 3s from the number line (make them disappear). At this point the center between the positive and negative infinity would be skewed to the left by 12, otherwise the two sides wouldn't be balanced.

This is the best sense I can make of it! • The idea here is like getting a debt. Imagine that you have a debt of 500 dollars, this represents a negative 500 digit. Multiplying your debt with a negative number is like multiplying the profit equivalent of debt which is positive 500.

You're turning the debt into an income if you're getting the opposite of debt. Imagine turning the debt thrice into a profit. It's like multiplying negative three and the debt of 500 dollars. Just imagine the negative as the opposite equivalent of something, and when multiplying it to something it is like passing the property of a negative number to the other. • If the negatives can be canceled out in this equation then how come the positives never get canceled out? • • If the negatives can be canceled out in this equation then how come the positives never get canceled out? • "Cancel out" may not be an appropriate method to describe why a negative times a negative equals a positive, and in a way even misleading. I would use "cancel out" when two values added or subtracted together become 0, or a value divided by another equals 1. 