Man, my Geometry students are on a roll right now. Today we went through our same new HW procedure again. I was quiet for the first 5 – 8 minutes of our 40 minute class while my students shared their HW with each other. They were asking each other good questions and catching each other’s mistakes. They are still a little shaky at times on their line equation writing skills and their line intersection skills, but the mistakes they are making are much more of the arithmetic and detail type rather than broad conceptual mistakes about what to do.
Today we were concentrating on medians and they seem convinced that the medians should always intersect inside the circle. Last night’s HW (which you can find here along with all our other HW assignments) was on Section 6.2 and they were finding medians and their point of intersection. I also asked them to find the perimeter of an original triangle and the triangle made by connecting the midpoints of the original. This allowed us to do a little noticing and wondering in class together. Everyone seemed pretty convinced that the perimeter of the interior triangle should always be half the perimeter of the ‘parent’ triangle. We displayed this on GeoGebra and I asked them to pick a certain type of triangle that they wanted to explore. One student suggested that a right triangle would be fun so we moved out vertices to the origin, the x-axis, and the y-axis. The interior triangle was still half the size and now the noticing began. They noticed that the smaller triangle not only was also a right triangle but that its acute angles seemed to be the same as the acute angles of the original. They noticed that the four right triangles formed inside were probably all congruent. They noticed that the centroid of the original triangle was also the centroid of the smaller triangle. Then Tara asked about yesterday’s peek at angle bisectors and whether they would meet where the medians met. I asked if there might be a special triangle they could think of where this is true and Miranda guessed that our favorite right triangle, the 5 – 12 – 13 triangle might be special enough. Sadly, it was not, but I was happy to hear a quick guess at this familiar old friend. Then Julia suggested that an equilateral triangle might fit the bill. I worried about how to manipulate our given GeoGebra sketch to match up and she cleverly told me to start a new screen with a regular polygon. Class concluded by seeing that GeoGebra was confirming that Julia’s guess was correct. What a great 40 minutes! I also made a point of telling them that they were on a roll and I hope it carries over to our GeoGebra lab day tomorrow. This is called Chapter Six GeoGebra activity in the dropbox file I had the link to above.
I should have dwelled a little more on my second stats class yesterday. I was really pleased with the three different formulas that those four groups generated. I was especially intrigued by the group that decided that the minimum number in their sample plus the maximum number in their sample should be a good estimator for the true max in the population. I discussed this idea with my morning stats class and we had a pretty vigorous debate over how appropriate this was. Playing with our TI and drawing random samples of 5 from a group of 342 (kind of like the German Tank problem!) convinced them that this technique actually turns out to be pretty accurate.
It’s easy to actively blog when it’s fun to relate what’s happening in class. I hope I can keep up a reasonable pace, if not daily, for the year.