Technically, yes, that fully proves it to be linear. However, trying to shove the phrases “Cartesian coordinates” and “linear dependencies” down our students’ throats defeats the entire purpose of the intuitive demonstration. Instead, I would argue that, at this stage, the best answer to the question is that, looking at it in the picture, it *looks* a lot like a line.

Does this prove, without a doubt, that the method works? Of course not. But it creates a visceral, satisfying sort of understanding — one that relies not on manipulation of symbols we’re vaguely proficient with, but on what we can see with our own two eyes. The definitive proof can come later.

OK, I think I can answer this one: because the circumference (2″Pi”r) formula is a LINEAR equation, so the gradual change of the argument (r) will make a straight line in Cartesian coordinates. And curves, from our experience, are NOT linear dependencies.

Good, your illustration has made me think :))) thanks a lot.

I entirely agree with the idea of looking for an understanding instead of a “definition”. In reality, it turns out, mathematicians are akin to technical inventors, are they not? Well, this is my idea of it, anyway…

Back to your “calculus” illustration of the area of a circle. On the page it’s taken from it says, after all the circle dissection done and the straightened “rings” placed next to one another to form a triangle, as the illustration makes one believe:”We have a bunch of straightened rings that form a triangle, which is much easier to measure.”

Yes, it IS easier to measure, but before we do we must be absolutely sure this IS a triangle. I mean, the base IS straight because we made it so, and so is the height side of it because we’ve “straightened” the “rings”. All right. But what proves that the “hypotenuse” of that triangle must necessarily form a straight line and not a curve? Sure it would suit as best if it was, but is it?

Or is this precisely that “intuitive” point we’re talking about? Something we FIRST accept in order to let it lead us to a solution, and later on find the ways to prove it WAS actually true?

@Valery: Thanks for the comment. I agree — with children, we seem to focus on historical techniques that actually work (immersion, intuition-first, examples, etc.), vs. the over-formalization that we *think* works for adults.

@Lance: Great example! I never thought about TMNT but wow, even that complex scenario can be understood tacitly. Glad you’re enjoying the site :).