Mathematics for Teaching Algebra Making Sense of Equivalent Equations and Expressions

## Making Sense of Equivalent Equations and Expressions

Here are some questions your students have been wanting to ask you in your algebra class. Daniel Chazan and Michal Yerushalmy in their article On Appreciating the Cognitive Complexity of School Algebra posed these questions about equivalence of equations , solving equations, and equivalence of expressions for us teachers to ponder upon.  How often do we give students the  opportunities to think about these questions?

Questions:

1. If one can add x to both sides of an equation in one variable—and if multiplication is repeated addition—why cannot one always multiply each side of an equation by x? In what sense is the resulting equation not necessarily an equivalent one, and why?
2. When solving equations, we often undo one operation by doing its inverse. Why isn’t taking the square root the inverse of squaring? Whys isn’t √x^2=x? And, for that matter, why does taking the square root of each of the expressions in the equation 2x-3=x2-2x-3
not yield an equivalent equations?
3. When I solve 10x-45=5 by operating on both sides and get x=5, x=5 feels quite different from 10x-45=5. When did the equation become a solution set? Or was it one all along? Is 10x=50 all that different from x=5? If so, in what sense?
4. When I solve equations in one variable, such as 3x+7=2x-4, the solution set consists of no numbers, all numbers, or one particular number. How do I understand the equation √(3x-5)^2 = 3x-5, for which infinitely many solutions exist but for which there also exist numbers that are not part of the solution set?
5. When solving 3x2+3x+3=0, I can think of the task as finding the zeros of the function y or f(x)=3x2+3x+3=0. In the context of finding its zeros, I can divide the coefficients of the function by 3, but when I’m working with the function y or f(x)=3x2+3x+3, I cannot divide all the coefficients by 3. Why is that? How is equivalence of equations different from equivalence of functions?
6. If only some equations in two variables are functions (e.g., 3x+y=5 is an implicit function, whereas x2+y2=1 is not), and only some functions have an equation (e.g., f(x)=3x-5 has an equation, whereas {(0,5), (1,8), (2.3,-1.7)} does not, which is a subset of which?
7. From what is written on paper, how can I tell that 3(x+2)=3x+6 as an equation to solve different from “simplifying” 3(x+2) and writing 3(x+2) = 3x+6? Or is the difference between the two not possible to discern simply by examining the strings themselves? Why is the equal sign used in two such different ways?
8. If 25x+.05y=100 is a relation in two variables, how does the isolation of y, y=(100-25x)•20, turn it into a function of one variable? How can creating an equivalent relation somehow make a relation into a function? Or was 25x+0.05y=100 a function beforehand as well? If so, could I have written 25x+.05f(x)=100? How do I know by looking at a relation whether it is a function? And when can I use function notation?
9. How is the solution to the system x2+y2=1 and 5x+y=5 similar to, or different from, the solution y=5-5x? How is the solution to this system similar to, or different from, the solution to x2+(5-5x)2=1?

What explanations can we give  our students when they ask us these questions?  If you’ve been asked questions like these in your class, please share the question. Please use the comment section to share your views or explanations.
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