A Quick Geometry Snippet

So, we were reviewing this morning in Geometry getting ready to finish up our circle unit. I was reminding the kiddos of the angle and arc relationships we have been discussing. I had been writing things like x=frac{1}{2}left ( a+b right ) where x is an angle formed by the intersection of two chords and a and b are the measures of the intercepted arcs. However, today as I drew my diagram for it one of my students suggested I mark is as seen below. Interior Angles

I think I am delighted by this and will write it this way from now on. It feels to me that it is a more natural way to think about this relationship instead of having a coefficient of 2 or a coefficient of 1/2 that might not seem at all intuitive. I then drew the following Exterior Angles

I’d love to hear from some other teachers about whether this seems at all like an improvement over the more standard way of writing these equations.

That’s all for now, just needed to get that off of my chest!

7 thoughts on “A Quick Geometry Snippet”

  1. I like that method. Due to various decisions, I probalby will not do these relationships at all this year, but it works.

    One notice: since a and b are arc measures (not lengths) here, they should probably also include the degree symbol, especially since the x has a degree symbol.

    This type of relationships seems to make the reason for my question last week about why we measure arcs in degrees a little more clear; it means we don’t need to ALSO draw central angles here just to formulate the rule. Even though the a and b come from that. Definitely more ugly if you do draw those angles in.

  2. David

    Important point about he degree symbol. I get pretty lazy with this and it may interfere with my Ss comfortably knowing what we are measuring and how.
    Thanks for dropping by!

  3. SUPER-LIKE!!!!! This was a brilliant insight. I love the “x + x = a + b” label and now will use it always. Thank you for flagging this student’s amazing idea!

    – Elizabeth (@cheesemonkeysf)

  4. I like this representation and I will show it to my students when we return from April vacation. I always like to encourage multiple representations for relationships because I’ve noticed through the years that different “formulas” work for different kids.

    I’m a fan of the traditional representations because I like relating them to the idea of “average.” The measure of each angle formed by the intersection of two chords of circle is the average of the arcs intercepted by the angle and its vertical angle. It’s an interesting discussion to try to extend this thinking to inscribed angles and to angles formed by secant a and tangents intersecting outside the circle. I find that this is one way to encourage my students to remember that factor of 1/2.

    Thanks for the post and the discussion!

    Tim (@CoachMcQuade)

    1. Tim
      You raise an important point about multiple representations. When I get excited by an insight like this one I am tempted to make this the point of emphasis. I need to constantly remind myself that there are different pathways for my students to really develop their own understandings. I’ll definitely use this in the future, but not to the exclusion of the averaging idea which is so important in a number of situations.

  5. I like this representation and I will show it to my students when we return from April vacation. I always like to encourage multiple representations for relationships because I’ve noticed through the years that different “formulas” work for different kids.

    I’m a fan of the traditional representations because I like relating them to the idea of “average.” The measure of each angle formed by the intersection of two chords of circle is the average of the arcs intercepted by the angle and its vertical angle. It’s an interesting discussion to try to extend this thinking to inscribed angles and to angles formed by secant a and tangents intersecting outside the circle. I find that this is one way to encourage my students to remember that factor of 1/2.

    Thanks for the post and the discussion!

    Tim (@CoachMcQuade)

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