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.
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.
Two-dimensional area starts and ends in pretty much the same place, with base and height. Kids in elementary school calculate space by counting grids. Calculus classrooms do the same thing (on a more complex level of course), but through the short cut of integration. Somewhere in the middle, with geometry and the like, it gets complicated and students lose the conceptual understanding.
Here’s what we did instead.
Start with a few applets:
Then we document our thoughts. Some people call this notes.:
Students watch in amazement as if this were a magical experience. Audible comments of “wow” and “that’s cool” are common.
So then we conclude that there really is just one way to calculate 2 dimensional straight line areas:
base times height (and sometimes half)