How does substitution work as an alternative to the balance model in introducing solving equations? My biggest worry is that, lacking a concrete representation, is too abstract for middle schoolers. However, the balance model has enough limitations that I want to pursue using teaching students how to solve equations using solely substitution. Maybe someone here can stop me from making a horrible mistake!
The balance model has two main limitations
- It presupposes that there is only one possible value for $x$.
- It masks the most fundamental meaning of the $=$ sign: $a=b$ iff $a$ and $b$ are are different names for the same object.
- It only works when the objects on both sides of the equation are numbers.
Albeit more computationally intense, Substitution allows one to use language that makes clear to students that there could very well be more than one possible value for $x$. This allows the equality in $y=2x+4$ to be instantly fused with the equality in $x+4=2x$, instead of being treated separately, to be fused at a later date (more often never).
I am thinking of starting by using teaching students to evaluate an expression by substitution of a variable, and slowly introduce larger substitutions (as in, first evaluating some $f(a)$, and then using the result to evaluate some $g(f(a))$ where $f(x)$ appears in the definition of $g(f(x))$ multiple times. We would spend some time establishing what equivalence means when dealing with expressions with variables (that they have the same value no matter what $x$ is.
One and two step equations would be introduced purely as a guess and check / undoing procedure, without a formal way to write out steps. This is purely so that students understand that finding $x$ means ending with a statement $x=\_\_$ We will only require that for tackling equations for which the undoing model does not make sense - particularly equations with variables on both sides.
When given an expression such as $x+1=2x$ and asked to solve for it, guess and check is still an available method. We would use this until positing an equation such as $10x+4=2x-1$ where guessing and checking is unlikely to help one arrive at an answer. Problem: there is an $x$ on both sides! We will explain that a purposed substitution can take care of this problem.
We can point out that $2x- 1 -2x = 2x -1 -2x$. We can combine this with the premise that $10x+4=2x-1$ to substitute $10x+4$ for $2x-1$ on one side of the equation, giving us $10x+4-2x=2x-1-2x$. We then can write $10x+4-2x=8x+4$ and $2x-1-2x=1$. Substituting back into both sides, we have reduced the problem to a previously solved problems.
As students get used to this, we could introduce shorthand notation of performing an opertation to both sides under the equation. The end result of the process of solving an equation would look the same, but students would come away with an exact understanding of what all these variables and equations mean.
My own evaluation:
Teaches students the full meaning of the equal sign.
Exposes students to more rigorous mathematical thinking.
Allows teachers and students to freely use the powerful tool of substitution in other contexts, such as geometry.
Well unified progression of concepts
"This thing is the same as this thing, so I can interchange them" is an easier concept to grasp than "This thing has the same value as this thing, so if I add the same thing to both of them, their values will stay the same".
Gives students the opportunity to problem solve when asked to solve an equation ("What substitution should I use"), rather than having to be told that one is allowed to divide both sides of an equation by the same number
More computationally intense.
Relies on many abstract concepts.
Requires building a strong foundation on equivalence in expressions.
Takes more classroom time to build up to.