I like to teach mathematical concepts via problem solving. It right away engages students mind. It creates a need for learning a more systematic way of doing things and hence a reason for learning the concept. It provides a context for making connections. Most important of all, it gives students opportunity to learn before they are taught.
One of the ways of creating a problem solving task for this kind of lesson is to start by giving the solution/answer. For example, a standard textbook task is to solve for x in an equation. Why not do it the other way and ask students to find the equation given the solution?
Here is my favorite problem for introducing the properties of equality:
Write three equations in x with solution x = 5.
Fresh from their elementary school math experience of solving equation of the type 15 – ___ = 20 or x + 12.3 = 20, students will generate equations by trial and adjust. That is, they think of an equation then check if it gives x = 5 by substituting the value of x to the equation. It will not take long for them to realize that this is something very tedious.
Other students will verify their equations by expressing the terms into equivalent structure. I explained this method in my previous post. This method is not also very efficient for some equations. But students have to experience these so that they will be able to appreciate the efficiency of solving equations using the properties of equality.
Usually when they already have at least two equations I will extend the task to:
Two of the equations should have x on both sides of the equal sign .
Challenge them further to think of a more systematic way for making the equations. You will not hear the bell ring before students will realize that all they need to do to have an x on both sides is to add or subtract x on both sides of the equal sign! (Multiplying or dividing by x is a different matter as there is a possibility that x can be zero). Now they know that those ubiquitous equations in their algebra textbooks did not just drop from they sky. They can even create one themselves. The students are applying the properties of equality before they even know what they are! Speaking of constructivist teaching here.
Teachers I shared this with were also very happy because they no longer have to do trial and adjust when they make exercises for solving equations.