So here's an example: the discriminant in Algebra 2, or said another way "What kind of roots does this quadratic equation have?" We're not particularly concerned in Virginia with them being able to define the word "discriminant," but they should be able to recognize whether the solutions are rational or irrational, real or non-real. Given a quadratic equation, they should be able to figure out what kind of solutions it has, and know how to describe those kinds of numbers. Here is a sample released item (question 50 of 50 in this document). (Which, you may notice, has exactly zero correct answers. Good job, Virginia. (Poor Virginia. Maybe she'll marry well.))
You can find all kinds of MTDS/MTNS lessons about the discriminant. Here and here, for example. But here's what I came up with to turn it into MTFSO. We had already spent a day on simplifying radicals (including with imaginary results), a couple days on solving by undoing and solving with the quadratic formula. Then we made a big map of different kinds of numbers with special attention to recognizing rationals vs irrational and real vs non-real.
For this particular lesson, they first solved four equations using the quadratic formula: x2 - 10x + 9 =0, x2 - 6x + 9 = 0, x2 - 7x + 9 = 0, and x2 - 4x + 9 = 0. When we debriefed their solutions, we spent time describing the types of solutions, but we did not belabor the point about why the roots came out each way.
Then, they sat in groups of 2-3 at big whiteboards and got one of these sets of questions:
A1) Come up with a new, original quadratic equation whose roots are real and irrational. Demonstrate that your equation works by using the quadratic formula to solve it.
A2) Come up with a new, original quadratic equation whose roots are real, rational, and unequal. Demonstrate that your equation works by using the quadratic formula to solve it.
B1) Come up with a new, original quadratic equation whose roots are imaginary. Demonstrate that your equation works by using the quadratic formula to solve it.(In the first class to do this, I gave out the tasks haphazardly. But from that experience, learned that "rational" and "equal" are harder to find than "irrational" and "imaginary." So I adjusted accordingly for the second class. Each set consists of one of the easier ones, and then one of the harder ones.)
B2) Come up with a new, original quadratic equation whose roots are real, rational, and equal. Demonstrate that your equation works by using the quadratic formula to solve it.
I saw different approaches... start with a desired answer and try to work backwards. Write out the quadratic formula with blank spots and repeatedly fill in and try values. Start with an equation very close to one we had just worked with and see how it worked out. And of course the bane of all group work reared its head - one kid grabs a marker and takes over while everyone else is happy to let them.
After groups had a chance to figure stuff out and explain how they did it (it all came down to paying attention to what kind of number was under the radical, of course), we summarized with some notes:
And that is that. A general understanding of what was going on persisted to the next day... We'll see how Monday goes.