I am currently working on the Algebra 1 standard "Equations and Inequalities" for my new math teaching website At the core of teaching Algebra is an understanding of equality. One of the major obstacles for students is realizing the differences between an equaiton and an expression. I usually try to make the connection by explaining that an equation is a comparison of two expressions. I emphasize that most equations are actually questions. What value of the variable(s) make the equation true?
The above is more of a conceptual understanding of equations. But just as importantly students must learn how to apply the properties of equations (and inequalities). I introduce them by setting up an equation of two numbers. For instance, 20 = 20. I then add a number to one side and ask them the question, "what must I do to the other side so the equation is still true?" I then use this general method to try and generate the properties of equations.
What are some ways that you introduce and reinforce the properties of equations to your students?
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I think understanding the identity and inverse properties is key to understanding how to solve equations. I have had them posted on my room for several years and I tell students that this is where they frequently make mistakes. I think another big challenge is helping students understand that it is usually better to add/subtract first and then multiply/divide to solve equations. I think it is good to allow students to attempt to solve equations by using a different sequence of steps, as long as they don't violate any algebraic rules, and they get to a point that they realize they are moving farther away from a solution.
Regarding expressions and equations, I finally realized that using units to help teach both of the these areas would be useful. Simplifying 3cm + 5cm is 8cm just as 3x + 5x simplifies to 8x. Students could simplify various expressions with units and then a follow up question could ask when 2 different expressions were equivalent. These questions could also be posed in the context of story problem so all 3 areas could be presented along with unit analysis.
Thanks for the thoughts. I hadn't emphasized those properties with respect to equations before. Obviously we discussed the components, but I didn't make that extension. It is good way to reinforce the properties as well.
As for the centimeters method, that reminded me of a friend from grad school. He was teaching a college algebra course and he liked to draw pictures instead of using variables. He would use pictures of hats or some other easy to draw objects as his variables. This would serve the same purpose as the centimeters.
Of course he was a little off in general:)
To see why the properties of equations are so special, you might contrast your example (with 20=20) with one involving an inequality.
Here is an example.
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