We found several good tasks to use throughout our unit on polygons.
I had heard of the NRICH site before but had not used it. A search on NRICH for polygon angles turned up some out-of-the-ordinary tasks.
I was surprised at how quickly my students thought to draw a concave polygon to explore this problem. I went straight to a convex polygon first when I thought about it. I think this has to do with look for and make use of structure. We’ve asked “what do you see that’s not pictured?” We’ve learned geometry by drawing auxiliary lines. I’m convinced that my students think differently than I do because of how they’ve learned geometry.
I can’t remember where I got the next problem. But I like that it asks for both a calculation and a proof of why the triangle must be isosceles. When I saw the student results (27 out of 30 correct), I had to think quickly about whether it was worth the time to go through the proof. I decided that it was. My students had been asking for more practice with proofs, and so I figured this was an opportunity to let them compare their work with the rest of the class. A student led us through her reasoning.
I can’t remember where I got the regular pentagon task, either, and my attempts at googling have failed. I have two favorite responses.
For the trapezoid requested in part (d), a few students listed a parallelogram. I love that my students have embraced our inclusive definition of trapezoids, and I love that they feel confident enough in their understanding to include a parallelogram as the answer to a question about a trapezoid.
For the similar triangles and justification requested in parts (e) and (f), a few students justified the similarity of triangles using transformations. This student suggested that ∆DJC~∆EJB because of a reflection and a dilation. (Of course the response could use more attention to precision. She didn’t say about which line to reflect one of the triangles, but the line is drawn in the diagram. And you wouldn’t believe that I have actually had a conversation with my students about the word being dilate instead of dialate. Dialect prevails in the South.)
We have incorporated Shapedoku into this unit as well after reading Reinforcing Geometric Properties with Shapedoku Puzzles in the October 2013 Mathematics Teacher.
Finally, we tried something different for conversation about angles in quadrilaterals. My students do need some practice with using the properties of quadrilaterals to calculate angles measures … but the regular types of problems can be so boring. So I gave the regular types of problems with a twist.
Given one angle measure, for which figures can you determine all remaining angle measures?
This gave students practice with properties and angle measures, and it provoked conversation that giving problems like this one at a time wouldn’t have.
(I neglected to save the Quick Poll I sent to collect their work, but it went well. I had students work alone for 2-3 minutes before talking with their team.)
The next class, I used the same diagrams but gave part of one of the interior angles formed by a diagonal. The question was the same: Given one angle measure, for which figures can you determine all remaining angle measures?
I watched as students worked.
And eventually had them talk with each other.
I can show student responses separated (as above) or grouped together (as below) to help decide which students to ask for explanations.
Again, we had good conversation that wouldn’t have happened had I asked the problems one at a time.
Incorporating new tasks each year keeps the class interesting for us as teachers as much as it does for our students. When is the last time you looked for a new task at NRICH or Illustrative Mathematics or the Mathematics Assessment Project? Or when is the last time you used a task you read about on someone’s blog or found through someone’s Tweet?
And so the journey continues … with thanks to all of you for your suggestions along the way.
And so the journey continues … with thanks to all of you for your suggestions along the way.
Easing the Hurry Syndrome 