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…
View original post 372 more words
In geometry we had some fun with kite like shapes for the last couple of days. On Wednesday we asked:
See it in Action (click image to open)
I asked the students to play with the applet. I prompted them to ask for more information. Some noticed the sample questions below the applet and asked those to start the discussion. The key question had to do with decomposing the shape into other shapes that we had more familiar tools to work with. The students were quick to see the right triangles. Once we were able to identify that, the next question was, “How does that help me?” Getting students to connect and then apply relationships they know into a seemingly new context is a constant challenge, but they are getting better. I reminded them to recognize what are we trying to find. Once they narrowed in on the task of finding a length, and the length was a part of a right triangle, some started to see it, “Pythag Thyrem.” (this seems to be a tongue twister for the general high school math population). “Okay, how do you mean?” was my reply. Some didn’t see the given values for the hypotenuse that was also a radius.
Circles seem to be a pain for many geometry teachers, and I feel that it’s because so many people approach circles as a never ending list of formulas. We need to find ways to simplify the overall question and give students an opportunity to fill in the structure(s) needed to respond to the question. I know there are some awesome activities out there dealing with volume and area with round objects. Here I’m trying to put together a series of interactive questions the see the overlaps and relationships with circles, segments and angles.
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.