# How to teach the properties of equality through problem solving

July 28, 2010 6 Comments

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. 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? I found this kind of task a very powerful context for introducing a concept.

Here is my favorite problem for introducing the properties of equality:

**Write three equations in x with solution x = 5.**

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. 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 “put” the *x* on both sides of the equal sign! Examples:

1) *x* + *x* = 5 + *x* => 2*x* = 5 + *x*

2) *x*•*x* = 5•*x* => *x*^{2} = 5*x*

3) *x*-2*x* = 5 – 2*x* => –*x* = 5 – 2*x*

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.

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It also allows you to differentiate the task per pupil

Say u give them 7/5 as the answer

Unknowns on both sides

6 or -6 as the answer

I use this method a lot and you can specify how many steps / methods a kid can think of to get the same answer

yes, there’s no limit for any kind of answer you want, isn’t it? thank you.

what i love about it, each student is given an opportunity to contribute an answer because there is no limit also as to the number of equations with the same answer. our class size here in the Philippines in 50-60 students in public schools so it’s kind of hard for teachers to give each student an opportunity to contribute to the lesson. open-ended tasks such as these will give them that.

I like the combination of the approach in your previous posting and this one. I often use a somewhat similar approach to get students comfortable with how all those problems in the book were developed:

http://mathmaine.wordpress.com/2010/01/11/how-problems-created/

What a fantastic blog. Thank you! I shall certainly be trying this out with my own classes. The idea of using problem solving to engage and introduce a concept before teaching it is brilliant.

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