Mathematical induction and blank space

I tried out a one-liner in class that I’d been itching to try all summer.

I was introducing my students to proofs by mathematical induction; my example was showing that

$1 + 3 + 5 + \dots + (2n-1) = n^2$.

After describing the principle of mathematical induction, I wrote out the $n = 1$ step and the assumption for $n = k$:

$n=1$: $1=1^2$, so this checks.

$n =k$: Assume that $1 + 3 + 5 + \dots + (2k-1) = k^2$.

Then, for the inductive step, I had my students tell me what the left- and right-hand side would be if I substituted $k+1$ in place of $n$. I wrote the answer for the left-hand side at the top of the board, the answer for the right-hand side at the bottom of the board, and left plenty of blank space in between the two (which I would fill in shortly):

$n = k+1$:

$1 + 3 + 5 \dots + (2k-1) + (2[k+1]-1) =$

$~$

$~$

$~$

$~$

$~$

$~$

$= (k+1)^2$

So I explained that, to complete the proof by induction, all we had to do was convert the top line into the bottom line.

As my class swallowed hard as they thought about how to perform this task, I told them, “Yes, this looks really intimidating. Indeed, to quote the great philosopher, ‘You might think that I’m insane. But I’ve got a blank space, baby… and I’ll write your name.’ “

The one-liner provoked the desired response from my students… and after the laughter died down, we then worked through the end of the proof.

And, just in case you’ve been buried under a rock for the past few months, here’s the source material for the one-liner (which, at the time of this writing, is the second-most watched video on YouTube):