Before class, I run off one photocopy for each student. (There are two per page, and I cut them down the middle). I then have my students use symmetry/folding to notice that:
- Two congruent chords create two congruent arcs;
- The radius is perpendicular to the tangent at the point of tangency;
- Two tangents meet at a point equidistant from the two points of tangency;
- A chord perpendicular to a diameter is bisected by the diameter; that is, the diameter is the perpendicular bisector of the chord.
Back in the day, I used to try to give a thorough lecture on every topic I taught: I did at least three examples (a medium-difficulty problem first, then a simple problem to reinforce the concept, and then a more challenging problem), making sure to incorporate every conceivable 'hiccup' the students might stumble over in completing their book-work.
After years of doing this, I realized that there was always very little time left during the class for students to actually do the work, particularly more challenging work (since the lecture was so long on the days with the most challenging material), and only the best students would bother with the homework.
Furthermore, despite how diligently students took notes, it was rare that they would ever look at them.
Finally, I had to admit to myself that even though I was perhaps more interesting and engaging than the average teacher, my lectures were still pretty boring.
In the past few years, I've changed how I teach. I try to replace lectures, as much as possible, with worksheets. Instead of me talking the kids through the 'hiccups,' I simply wrote a worksheet to "hold their hands" through the spots that traditionally caused problems.
I open class (at the bell) with a brief 'mini-lesson' or sample notes, lasting no more than 15-20 minutes, and that's what this resource is: it's what I give the kids in my mini-lesson to convey to them the most basic of the information they will need for the day. By giving these directly to the kids, it saves the time they would otherwise take copying everything down, and as a matter of fact, notes have become optional in my class, as students who simply pay attention do much better: the bulk of the material is covered on the accompanying worksheet!
If you can project these on the board and photocopy them so kids won't have to hand-copy them, I think you'll find, as I do, that the lessons get under way much more efficiently, and there is much more time available during class for them to do the work during class, while I'm there to help them.
Immediately before this mini lesson, I use the worksheet named 'Circles and Tangents Spring 2014' (http://www.teacherspayteachers.com/Product/Circles-and-Tangents-Spring-2014-1269745
), and immediately after this mini lesson, I use the worksheet named 'Central Angles Arc Measures and Chords Spring 2014' (http://www.teacherspayteachers.com/Product/Central-Angles-Arc-Measures-and-Chords-Spring-2014-1269791