I just figured out a way to quintuple a square using only a straight edge and compass.
That is, you are given a square and you have to draw a square with five times the area of the original square.
How hard is this problem?
What grade-level student could I give this to and have it be difficult but solvable?
So starting with a unit-square, you’re able to construct a square with sides 2.236… times as long as the original?
Wikipedia says SQRT(5) figures prominently in geometry/trigonometry, but it’s been a long, long time since I played with geometric constructions. Geometry was a tenth-grade class for me.
Care to share your procedure?
I can do it, so it can’t be too hard. 
I’ve no experience as a teacher, but I wonder if grade level has little to do with it. Many 9th-graders could do it, assuming they’ve learned plane geometry, but I’d bet many 12th-grader with B grades would be stumped unless you gave a big hint. (The easy solution is a very straightforward application of Pythagorean Theorem.)
Extend one side of the square. Measure out two side-lengths on the extension. Draw a line from there to the adjacent corner, to make a right triangle with legs of length 1 and 2. Use the hypotenuse as one side of the new square. Done.
It’s trivial to extend this method to construct a line interval which is the square root of any positive integer time a given line interval, and hence to construct a square with an area of any integer multiple of a given square.
Yes, that’s the way to do it, but the question was what grade-level student (assume an A student) could solve this?
e.g. when do kids learn how to make a square using the hypotenuse they just drew, using only a straight edge and compass?
In principle, anyone who’s been through a high school geometry class should be able to solve this. However, there’s variation in when exactly kids take geometry, and how much time they spend on constructions, so it’s hard to give an exact grade level.
I would say grade 9 or 10. I’m pretty sure we learned a^2+b^2=c^2 in 9th grade, but I don’t remember doing anything with straight edges and compasses until geometry class in 10th grade. I may be wrong–it’s hard for me to remember what was learned when–but 9th or 10th grade should be a safe estimate.
And people complain that Trig has no use in everyday life…
Let’s see - he’s from Alaska, he’s got Downs, and he’s less than 2 years old. What good would he be?
Will you be here all week?
Which is exactly why Palin warned us about death panels.
That depends on what toys the kid played with in pre-school. I had a set of blocks with some squares and some 1-2 triangles, which I knew were the same area, and noticed that I could fit one of those squares and four of those triangles together into a larger square, and that’s really the crux of the method. From there, it’s just details like learning how to do constructions with compass and straightedge, and the (fairly straightforward) method of using those to construct a square.
When do kids learn to make a square using a given side? I wager every kid who has ever bothered to learn the game “Straight Edge and Compass! ™” can carry out this move. The only question is do kids learn this game and, if so, when. The answer to that, I suspect, is highly variable.
(For example, I have almost no experience playing the game myself (but then, I never took a geometry class). I always found it rather silly an object of such enthusiasm and (apparently?) pedagogical attention. [But even I can see how to carry out a 90 degree turn…])