On Tuesday a question bubbled up in me for no apparent reason, saying, “What *does* happen when you vary the side lengths of a rectangle but keep the perimeter constant?” To others in the math teaching blogosphere, I’m sure this question is old hat, and even I recognize it from the days of my own high school homework. I can picture myself reaching a question about varying side lengths at the end of an assignment (you know, in the word problem section), weighing the point value of the problem against the inconvenience of figuring it out, and opting to not worry too much about leaving it half done.) But on Tuesday I flipped over the answer key in my hand, made a table of values, and graphed area vs. height.

After noting and confirming that a perfect square yields the maximum area, another question struck me and I thought, “I wonder if I can write the equation for that graph.” It looked similar to a y=–x^{2}, so I wrote in the adjustments that would put the max of that graph in the same location as my max, (6.5, 42.25). I expected to need to make additional adjustments to fit my curve, but what do you know, it was exactly right!

So, said I, I bet I can do that for right triangles with the same give and take between height and base (for every increase in height, with a max height of 12, the base decreases by the same amount). Looking at it now, of course it makes sense that you’d divide the rectangle equation by two, but I went through a process of fitting the curve to see how it would turn out.

With success number two in the bag, I went on to holding the area of those right triangles constant at 50 and relating the base length to the height.

Then, finally, using the same data about right triangles with area set at 50, I related perimeter to height and truly offered myself a challenge. I quickly pegged it as developing from y=1/x, but had no clue how to make it *fit*. After a fair bit of graphical exploration, it suddenly dawned on me: all I have to do is write the process of finding the perimeter as a function of height! Connection made!

What a triumph. This is what I want for my students, and how to “give” it to them is clearer to me now. My students know how to use area and height to solve for a right triangle’s base (b=2A/h); they know how to use base and height to find the hypotenuse (a^{2}+b^{2}=c^{2}); they know how to use all three sides to find the perimeter. But they may not know that they can use these geometric pieces to explore totally cool graphical relationships.