## IM Commentary

The purpose of this task is to provide students practice in drawing conclusions about expressions they might encounter in classes outside mathematics, by parsing them in terms of their algebraic structure. Teachers might stress the subtle difference between "seeing" why the expression must be zero and the more mechanical process of algebraically simplifying the expression upon substituting $v=c$. Although part (b) might initially be though of as an optimization problem in a calculus course, some elementary reasoning with the structure of the expression gives the same answer in a more fluid and conceptual fashion.

For reference, this formula is for the length contraction of an object travelling near the speed of light, denoted $c$. While the constant $c$ in the problem is indeed a positive constant, its positivity is irrelevant to the task at hand since only its square, $c^2$, appears in the formula.

## Comments

Log in to comment## mprice says:

almost 4 yearsThere is a meaningful discussion to be had when students see the statement "v=c". In a computer science class, this would be read "v is assigned the value of c", so the logical replacement would be to take all instances of v in the equation and replace them with c. Of course, mathematically either substitution accomplishes the goal (and the solutions posted here don't specify either strategy, just that the ratio must be 1). I can definitely imagine a student asking which value "should" be replaced, though.

## Cam says:

over 3 yearsGood point. I think from a mathematical point of view, it's preferable to have equality be symmetric, so that $v=c$ and $c=v$ are equivalent expressions (which, I agree, differs from the computer science interpretation). Under this interpretation, I'd encourage students not to think there is a "should" involved at all -- there often several algebraic deductions for the same fact, and while some might be more direct than others (not that this is the case here), that doesn't mean any are wrong.

## Ashley Williams says:

almost 6 yearsIs it possible to have a solution posted?

## Noah Heller says:

almost 6 yearsIs it important to say v<c or v=c, or at least make mention real solutions?

## Kristin says:

almost 6 yearsThe task statement does say that v=c. Did you have something else in mind?

## Ellen Whitesides says:

about 6 yearsWhy does this task not have a published solution?