Note: What follows is likely completely obvious to anyone who has been teaching math longer than a year, but since I’m essentially teaching the subject for the first time this year, I thought I would write observations like these down.

My students have seen the three ways to solve systems of equations (graphing, elimination, and substitution) before, but when we were discussing these ideas, I noticed one of the things they didn’t seem to be so clear on was why these various methods worked for solving equations.

We started with the following system of equations:

$\begin{array}{rcl} 5x+3y &=& -2\\ 7x-5y &=& -11 \end{array}$

I asked students to solve the problem any way they could. Some chose to do substitution, but found it to be difficult because it involved a lot of work with fractions, but then another student explained how she began by multiplying the top equation by 5 to get

$25x+15y=-10$

At this point, I asked why this was allowed under the rules of algebra, and I got a response like “you’re doing the same thing to both sides.” I tried to elaborate on this, that the equation is telling us something, $5x+3y$, which we could also call fizz is equal to -2, which we could also call buzz. If 1 fizz is equal to 1 buzz, it makes sense that 3 fizz should be equal to 3 buzz.

The student also explained that we could multiply the second equation by 3 to get

$21x-15y=-33$

And this move is legal for the same reason as above.

But then the students next move, adding these two equations to get

$46x=-43$

seems entirely different. When I asked why this move was legal, one response was “because it cancels out the y‘s.

I then asked if it would have been legal just to add the first two equations to get

$12x-2y=-13$

and a few students thought this wasn’t legal because it didn’t “cancel anything out.”

So it seems like we need to explore this further and we tried to define two types of moves in algebra:

• legal moves: these are moves that follow the “rules” of algebra and preserve the equality of both sides of the equation, e.g. multiplying both sides of the equation by 3.
• smart moves: these are moves that, in addition to being legal, get you closer to the solution of the equation.

With this framework, students were able to see that adding the two equations initially, while legal, isn’t smart, because it doesn’t get you any closer to the solution.

Students were also able to see that doing something like simply subtracting $3y$ from the left side of the first equation isn’t legal because it doesn’t preserve the equality of the two sides of the equation.

This brought us back to the question of why it is legal to simply add the following two equations:

$25x+15y=-10$

and

$21x-15y=-33$

with some thinking, students saw that if we go back to how were thinking about this before the first equation his telling us that an expression which we can call A, is equal to another expression, B.

In the second equation, a difference expression, which we can call C is equal to another expression D.

So when we add the left sides of the equation we get $A+C$, and when we add the right side of the equation, we get $B+D$. But since C and D are equivalent, we are really adding the same thing to both sides of the equation $A=B$, and thus preserving the equality of the equation, while at the same time, eliminating one variable, a smart (and legal) move.

Similarly, substitution works because you’ve formed one equation of the form $x=$ and you can then replace any instances of $x$ in an other equation with this expression since they are equivalent.

I like this legal/smart framework for looking at steps in math problems and think this will be even more powerful if I start to ask some assessment questions that consist of very algebra steps and ask students to classify various moves as legal or not,and smart or not.