I was going over a written assignment: “Using a calculator, graph f(x)=(x-1)(x-2) and g(x) = |(x-1)(x-2)|. Describe and explain the difference.” Several students said there was no difference and produced these graphs copied from their calculators.
To demonstrate the error of their ways, I stated that Rock Hudson and I look the same from a distance (a very far distance) but, when you zoom in, we are different. What?! How would they know anything about Rock Hudson, a 1950’s movie star. So I changed it to Brad Pitt who probably is still just some old man to them. I went back to my point: different functions (algebraically) => different graphs. So look closely and explain the difference: particular character of the function expression (in this case, absolute value) => particular look of the graph (in this case, all on or above the x-axis, negative values reflected about the x-axis.)
I will be introducing the standard equation of a circle today asking, “What is a circle?” Usually some student will say it is round. I hope I can refrain from singing, “Round thing, you make my heart sing, ” with the appropriate phrasing and a slight twang. I think I have become quaint.