James Key uses a function monster to illustrate transformations:

f(x) is a function monster, and it can only *eat* numbers between -2 and 4. Now we define g(x) = f(x-3). We know that f eats numbers from -2 to 4. What numbers can g eat?

Cathy Yenca uses a number talk to draw out the distributive property:

I love how this scenario never fails me. Inevitably when I ask — not for the final answer — but the process and thinking that students used to find the answer, someone shares that they thought of “outfits” … 3(20 + 25) … and someone else shares that they thought of shirts and jeans separately … 3 – 20 + 3 – 25.

In the middle of a lengthy and fun post describing his first day of school, Andrew Knauft asks his students which number in the set {9, 16, 25, 43} doesn’t belong and why:

Here was a student, on the first real day of class, evaluating an argument independently of her own person bias, without forgetting that bias! (She believed her reason, for a different number, was more convincing, so the argument she read, although good, wasn’t good enough to sway her off her choice.)

Also in the vein of constructing and critiquing arguments, Andrew Shauver asks which image-preimage reflection is “best” out of a set of imperfect reflections rather than which one is “right”:

That question opens up a lot of potential thought-trails to wander down. As I did this activity with students today, the class settled on three criteria for rating these reflection attempts. The first is that the image and pre-image should be congruent. The next thought was that the image and pre-image should be the same distance away from the line of reflection. Finally, the students thought that the segment connecting the image/pre-image pairs would should be just about perpendicular to the line of reflection.