We are studying inequalities and I need to explain to my students WHY, when we multiply or divide an inequality by a negative, that we flip the inequality sign?

Imagine a number line on the board. Put your thumb on the 2 and your pointer on the 4.

Now, add 3 to both of these. Can you imagine your fingers sliding to 5 and 7—sliding together not changing their relationship to each other?

Now, subtract 1 from both (or add -1). Can you imagine your fingers sliding to 4 and 6—sliding together, not changing their relationship to each other? The pointer is still farthest to the right and they are the same distance apart.

Now multiply both of these by 2. Can you imagine sliding your fingers to 8 and 12—sliding them both in the same direction, but altering their relationship to each other. The pointer is still farthest to the right but the distance between them got bigger.

Now multiply both of them by 1/4 (or divide by 4). Can you imagine sliding your fingers to 2 and 3—sliding them both in the same direction, but altering their relationship to each other? The pointer is still farthest to the right but the distance between them gott smaller.

Now multiply both of them by -1. Can you imagine sliding your fingers to -3 and -2—sliding them both in the same direction, but altering their relationship to each other? The distance between them stayed constant, but the order of them changed.

You could then try combinations of these, like multiplying by -1/2 or -2.

What do you guys think? Anyone willing to try it and report back?