*From P.J.:*

I don't have a simple definition of an "authentic" problem, but here are two examples from the last couple of weeks. Though quite different from each other, both problems generated intense discussion and a variety of ideas.

Students entering my Geometry classes on Tuesday were confronted by a string running from the top-north-west corner of the room to the bottom-south-east corner, bearing a pink sign with the words "How long is the string?" Even students who ordinarily sit back were up and about, measuring, arguing, and figuring out a way to determine the length of the string. They extended the Pythagorean theorem to find diagonals of three-dimensional boxes, discussed their formula's validity, and had a great time.

The week before, we started our lesson with a problem I found in a samizdat geometry book from Germany (any reference would be appreciated--my copy of a copy of a copy has no title page or publication markings of any kind). Compute the area of the trapezoid, if

*A*is the center of the circle.Although totally abstract, this problem was also fruitful. It took students a while to draw in the radii:

Then more questions started: what's the length of

*EF*? Does it matter? Is*EAF*a right triangle, and how can we be sure? Ultimately, our discussion led us to Garfield's proof of the Pythagorean Theorem, #5 here.These stories have two morals. First, to grab students' attention, problems must be challenging and doable. But (second), those problems don't need to be "real-world" or "applications:" kids can get excited about figuring things out, even when what they're figuring out is "just" a math problem.

What do you all think? What are your favorite problems? What made them valuable? Share links in the comments or email them to us directly.