Once similarity intuition has been built with circles, we can start getting into more specific relationships with angles and segments. This post will look at using visual information from central angles and inscribed angles.
Students sometimes lack intuition for the measure of something. Andrew Stadel has developed this idea into a thorough curriculum on estimation. In my classes we started reasoning through similar exercises. Once we had a decent understanding of circle parts and whole, we moved on to other types of angles.
At this point most students have the common sense that a circle has 360 degrees, and a triangle is half that at 180 degrees. Built with this intuition in mind, we look at a triangle created by inscribed angles.
The next day we get to see the formula that collapses 3 ideas down to 1.
Dynamic Angles in…
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Today Geometry started with a question:
I was purposely vague, and no student went so far as to talk arc length (as opposed to arc measure in degrees). I told them to give their best guess but back it up with process and reasoning. This is what they came up with:
Students we able to trace the process of splitting into portions or multiplying into larger parts. They also tried using the context of percents without being prompted to do so. The first class didn’t automatically use units in degrees so I prompted, “If I told you the measure of this arc was 22.5 degrees, how do you think I came to that.” Students were able to articulate my process by relating it to their own. The second class had a few students actually use the process of putting the measures into degrees of a circle. Thanks to a recent tweet by NCTM illuminations:
Attn all teachers! Thx for all the hard work you do day in & day out! Let Dynamic Paper make ur life a little easier! http://t.co/Q5OlDu8hgp
— NCTM Illuminations (@NCTMIllum) May 5, 2014
I was able to quickly put together a worksheet of “spinners” as a follow up the this discussion.We practiced these in the same fashion. I made sure the students included the calculation of the arc in degrees.
I could have made this worksheet much longer, including measures in terms of percent, number, or even with angles. I wanted to keep it simple for now. Tomorrow we go further with Algebraic expressions, and start the discussion chords, area of slice of pizza and the length of string cheese. Now I want some pizza.
There is a little bit of mystery and magic to these relationships, if you don’t believe me just ask Mr. Vaudrey. Students trust that a triangle is simple, yet if you asked them to communicate anything beyond the magical balance of 3 angles, and 3 sides, most wouldn’t know what else is true. Sometimes students see triangles as snowflakes, each one of them unique. Little do they know how much all these triangles are alike.
- Get quarter sheets of graph paper
- Draw your own unique triangle
- Color in the angles in each corner
- Cut Out the triangle
- Tear off each corner
- Piece together the puzzle, and what do you see?
Describe in your own words what’s happening.
This is HUGE. Students need time to digest this transformation. If it feels like the engine is stalling, change gears:
Start practicing lo-tech with some paper examples.
Constantly bring back the triangle with the transforming corners. Have the students take some
Then go back again to the applet. Get the class to a point where students are articulating what is happening in the triangle. Have them say it in multiple ways. These angle relationships have patterns and consistencies, but often get lost in the multiple perspectives (what about this corner, or that one, or the inside, or the out, or what does a parallel line have anything to do with it). If students can transform a triangle and its angles, then adding in the relationships with parallel lines is only a half step away.
Don’t forget to check out the other gems over at Transformulas.