**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.