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Why does the inequality reverse when dividing by a negative number?
Posted by Mary Manske on January 13, 2020 at 12:48 pmDoes anyone have any good activities for students to discover WHY this happens? I have an investigation where they see that it happens and therefore can generalize it to be a rule, but not really understand why. The best we can come up with is that since you are taking the opposite of both sides, the operation has to the opposite as well.
I do have and use a double number line so I feel like there could be something there but I’m not sure what.
Mary Manske replied 2 years, 10 months ago 4 Members · 7 Replies 
7 Replies

hey @marymanske
I think a number line is an excellent way to model this! As an example you can show visually on the number line 5 < 9 and then if when you divide both sides by 1 and you didn’t switch the inequality then you would get 5 < 9….. show this on the number line. Is this true? Kids will say nope becuase 5 is NOT ‘more’ to the left than 9 is.
What are you thougths?

@jon we worked with the number line some (we use a number line a lot so its a familiar context). I don’t think I’m confident enough in showing how dividing by a negative number works on the number line. It just sort of reverses so then it doesn’t work. Any ideas for how I can be clearer than that?

Hey @marymanske
I think showing them with a few examples that when you divide by 1 on both sides that the inequality is not true until you do switch the sign should convince anyone.
sorry I don’t have a clearer example here. 
@jon its such a weird thing–they totally *believe* me but I think they believe me very much in a memorizewhatshesaid why and cannot figure out why it happens at all. Ah well, hopefully they at least remember to do it (we focus on using a test point anyhow, so hopefully not too big of a problem)

I’ve struggled with the WHY on this myself. I think Jon had some great thoughts about looking at patterns with the numbers / equations. For example, I used to start with equations: if x = 4, then it makes sense that x = 4 (mult. prop. equality). BUT, a true statement like 1<3 is not true when using that property: 1 </ 3. Gives an opportunity to discuss how negative numbers are inverses of positive numbers so using an inverse inequality when a sign change occurs makes some sense.
This Dr. Math website explanation also looks pretty clear to me on first glance: http://mathforum.org/library/drmath/view/53142.html
Good luck!

Nice approach, @patrickkosal!
Going concrete and visual is so important, so like @jon mentioned, a number line is helpful. It could also be helpful to do so with colour tiles…


Thanks @patrickkosal –I like the idea of reversing the number line (and the explanation of why) as well as the scale idea (in another post there). I think that makes it a lot clearer as opposed to just “that’s what happens!” even when I can prove it.