A-REI.1

[tomergal]

This is an extremely important standard in my opinion. However, I find it hard to implement without getting into quite meticulous logical considerations. I wonder if that’s expected, if you have any helpful tips for implementation, or whether you can refer me to good sources on this subject.

Furthermore, I wonder what you think about reasoning with *inequalities*. This is absent from the standard although it seems that, like more complicated equations (e.g. quadratic), inequalities can shed more light on the importance of reasoning with algebra.

[Bill McCallum]

Yes, I agree it is an important standard. One way not to implement it would be to get too bogged down in formality and terminology (like insisting that students keep referring to the properties of equality by name, for example). I would have students get in the habit of talking through their solutions:

“If $x$ is a number such that $x^2 – 3x – 4 = 0$
then $(x-4)(x+1) = 0$ because $x^2 – 3x – 4 = (x-4)(x+1)$ no matter what $x$ is.
for all $x$ (by the distributive law). This means that either $x-4=0$ or $x+1=0$, so $x =4$ or $x=-1$. ”

At first I would want students simply to understand that solving an equation is a flow of if-the statements; then I would start asking why each step was correct (distributive property, zero-factor property). And then I would raise the question of the converse: you’ve shown me that if $x$ is a solution to the equation it has to be 4 or $-1$, but does that tell me that 4 and $-1$ have to be solutions? How do I know they are solutions?

Maybe one of these days I will write a blog post on this.

[Author]

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