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.
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.
Larry Peterson shared a great method using Algebra transformations.