First, I was hoping we could develop the definition of a tangent line in our shortened class period. It's a discussion I've done every year, and it can go very quickly. I give the girls these 6 pictures of tangent lines, carefully chosen so their previous definition of "touches a function at only one point" will not hold, and tell them that they are all pictures of tangent lines. Their job is to come up with a definition of what a tangent line is.

Graph A |

Graph B |

Graph C |

Graph D |

Graph E |

Graph F |

The second problem is that when a student offered up "It touches at only one spot, it doesn't go through..." and trailed off, I jumped in with, "What about graph C? Or E and F?" Centering myself in the conversation seems to be the fastest way to kill it.

I tried asking a question and just waiting. And waiting. And waiting. Then, I'd cave, and ask another question (or a variation on the last one). I still got blank stares. Thank God for the three girls who were willing to eventually speak up.

On the bright side, one student asked if I could give them a definition of a secant line, so they could work from there. Here's what I gave them: "A line that cuts through two points on a curve and whose slope equals the average rate of change of the curve between those two points."

Eventually, I told them to think about these bullet points we'd developed and come up with a statement about what makes a tangent line different from just a secant line:

- Touches the curve at at least one point
- Touches but doesn't go through (what about E & F?)
- Some tangent lines can also be secant lines (B, C and E, but not F)
- Slope?
- Terminology: point of tangency (See D)

I had to end the discussion a little early, because they have a take-home assessment on proving functions are continuous that is due tomorrow, and they asked for time to ask questions on it.

Enter frustration #2. On both sides.

They're somewhat upset that they don't have any examples just like these problems to refer to in their notes or practice problems. No, they don't. Instead, they have lots of examples and practices problems about different aspects of continuity that they can pull together to develop an understanding of the functions they have to prove are continuous. They get frustrated because I'm not pointing them to an exact template to follow. Instead, I'm suggesting that they think about what they know about what continuity and think about how it can be applied to these unusual functions.

Who said take-home tests were supposed to be easy?

Here's what I want to say to them. I have the email written, but I'm not sure yet I'm going to send it.

Struggle is good. Embrace it. Work through it.So the question of my day is, how do I teach them how to think and synthesize information? How do I get them to engage in the activities that will allow them to practice those skills instead of begging to be told "the answer" or "the method"?

You have all the pieces you need. Now you have to put them together. Think about what you know about these functions. Think about what you know about continuity. Think about what you know about how properties and theorems work: conditions to be met, results you get when they are.