*Motion graphics were created by Mac Square.*

How do you make sense of reversing the inequality symbol when multiplying or dividing by a negative number? What activities or resources have you used with your students to investigate the why?

Frank Schorn provides an outstanding reflection on how he helps students understand the why.

I take two approaches and always use both. First, I have the students solve the inequality using addition to move the negative variable terms to a positive zone. (No division or multiplication is done.)

Students are asked to then solve the inequality like an equation. They always copy the existing inequality symbol.

I ask if we have obeyed a rules, Including Order of Operations. They say we did.

Next I have them solve the same inequality by moving the constants away from the (negative) variable term. They “solve” the inequality arriving at a wrong symbol. They will have divided or multiplied by a negative value.

We compare results and see that the two results have the inequality symbol pointing in opposite directions. Students see something is wrong.

Then I point out that when we divided both sides of the inequality by, say, negative 1 to isolate the variable, something else happened. What we did by dividing by a negative number is use the opposite of the positive value.

I advise that if we transform to the opposite of the left and right sides of the inequality we should also transform the inequality symbol. That takes a bit of time to seep through, but it usually gets some satisfied uh-huhs from the class.

Just a few years ago I wouldn’t have been able to give a satisfactory explanation to the class. But looking at a negative as an opposite is useful in this situation.

Matt Enlow provided a wealth of great feedback via Twitter. Check out the conversation here. Some highlights that led to changes in the original video including adding the number line:

I feel like something incorporating a number line might work better. Multiplying/Dividing by a positive number => stretching the rubber band, by -1 => a flip.

James Tanton is doing some great work in this area: youtube.com/watch?v=tJP61g9atfU

Larry Peterson shared a great method using Algebra transformations.

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I take two approaches and always use both. First, I have the students solve the inequality using addition to move the negative variable terms to a positive zone. (No division or multiplication is done.)

Students are asked to then solve the inequality like an equation. They always copy the existing inequality symbol.

I ask if we have obeyed a rules, Including Order of Operations. They say we did.

Next I have them solve the same inequality by moving the constants away from the (negative) variable term. They “solve” the inequality arriving at a wrong symbol. They will have divided or multiplied by a negative value.

We compare results and see that the two results have the inequality symbol pointing in opposite directions. Students see something is wrong.

Then I point out that when we divided both sides of the inequality by, say, negative 1 to isolate the variable, something else happened. What we did by dividing by a negative number is use the opposite of the positive value.

I advise that if we transform to the opposite of the left and right sides of the inequality we should also transform the inequality symbol. That takes a bit of time to seep through, but it usually gets some satisfied uh-huhs from the class.

Just a few years ago I wouldn’t have been able to give a satisfactory explanation to the class. But looking at a negative as an opposite is useful in this situation.

Wow this is great work! Thank you so much for sharing. I included it in the post. I really appreciate you taking the time to share your insight. This will really help.