In the previous lessons, we used hangers to reason about ways to approach equations of the form \(px+q=r\) or \(p(x+q)=r\) (which can be summed up as “do the same thing to each side until the unknown equals a number”). Since the things we do to each side of an equation are just arithmetic operations, and the properties of operations extend to negative numbers, this method of solving equations also works when there are negative numbers, even though it doesn’t make physical sense to think about weights on hangers representing negative numbers. After a warm-up designed to remind students about operating on rational numbers, students are asked to solve some straightforward equations involving negative numbers. “Doing the same thing to each side” is presented as a valid method, even though negative numbers are involved. In the last activity, students do the same thing to each side of an equation and their partner tries to guess what they did. The purpose of this is to communicate that doing the same thing to each side maintains equality even when the moves aren’t intended to lead to the equation's solution.
- Generalize (orally) that doing the same thing to each side of an equation generates an equivalent equation.
- Solve equations of the form $px+q=r$ or $p(x+q)=r$ that involve negative numbers, and explain (orally and in writing) the solution method.
Let’s show that doing the same to each side works for negative numbers too.
- I can use the idea of doing the same to each side to solve equations that have negative numbers or solutions.