Today will likely be a two post day after too long of a layoff from this space. It’s funny, I know that this writing relaxes me and lets my brain breathe in important ways. However, I am too prone to let busy-ness in my life prevent me from taking advantage of that.
In Geometry we are examining relationships between angles and arcs. You know the drill, right? Central angles = intercepted arc measure. Inscribed angles are equal to half of the measure of their intercepted arc. Interior angles (other than those at the center) are equal to half of the sum of their intercepted arcs. Exterior angles equal half of the difference of their intercepted arcs. As I am explaining this yesterday in class I am emphasizing this half relationship over and over but I just kind of gloss over the fact that the central angle does not fit this pattern. I have gone through this explanation in the past but I do not think I was as explicit about the developing pattern with the one half scale factor in the past. So this morning I was thinking about how I can best help my students focus on this detail. I was motivated, in part, by a lovely blog post by Michael Pershan (@mpershan) which, in part, is about managing the processing load of our students. I was struck when reading by the fact that I was asking my students to juggle seemingly different rules yesterday. So I had this idea and it probably is not revolutionary in any way. But I think I want to simply say that central angles are interior angles. They happen to be interior angles that form two congruent arcs. This way I can ask my students to think about the one half scale factor for every angle/arc question that they think about.
Does anyone else out there do this? Do you agree with this idea? Do you think that there is a downside here that I am not seeing yet? I’d love to hear some thoughts in the comments or you can find me over on twitter @mrdardy
But central angles do fit the pattern if you continue them on the other side, don’t they? Just like other interior angles? It’s just silly to say 1/2 * (m1 + m2) when m1 and m2 are definitely equal. Easier to just say “m1”.
Central angles are special in that you don’t need to continue them to the other side to find them because the vertex at the center gives you additional information that lets you skip that step. But if you DID treat them as an angle made from chords rather than radii (diameters!) then they would still follow the half rule just fine.
Oh my gosh, I somehow glossed over the whole second part of your blog post. Not enough coffee. Yes, what you said!
The only downside I can see is definitional; technically, as far as I know, the definition of “measure of an arc” is simply “the measure of a central angle that intercepts the arc”. So going about finding central angles using the measure of the arc with a different formula is a bit circular. The other downside, of course, being that it makes finding the central angle slightly more work!
Interesting point here David but I am not sure at this level for the students how much importance to place on this. I would love some pushback and/or agreement, but here is my take. I emphasize the idea of arc length as part out of a whole. I do this in precalculus with trig applications as well. My students seem to accept as a definitional point that the entire arc of the circle is 360 degrees. So measures of smaller arcs are seen in relation to the whole or in relation to the angle that intercepts them. While it is true that your definition feels more technical than mine, I don’t think I am doing them harm with the point of view we are taking. Please feel free – anyone reading this! – to school me if necessary.
David – thanks for taking the time to jump in on this conversation.
Exactly how I think about it. In fact, inscribed angles fit this pattern, too, averaging with the other arc of 0. I teach the radial measure of an arc as length/radius, so portion of 2\pi.
Super sweet idea about the 0 degree arc. I like it!
Less rules, more sense. I like it.
I have a lovely pair of GeoGebra visuals for these topics that I designed to emphasize that the angle is always the average of the arcs, regardless of interior, central, inscribed, or even external — though with external you have to accept one of the arcs as being negative. http://tube.geogebra.org/material/simple/id/74968
Andrew – This looks terrific. I will definitely plan on sharing this with my colleagues and my students. Thanks for sharing.
Great conversation. Thanks for starting it! Guess we are all about in the same place. Doing different this year. Going for the equation of circles with similarity first, then angle stuff. Will let you know how it goes.