Month: May 2015

Whenever I finish a new post I will tweet out a message about it and I often encourage people to drop on by and share some wisdom. I definitely need some tonight.

Had a great conversation with a colleague today about his Algebra II Honors class. They are examining exponential functions and are ready to talk about logs. He came by with what seemed like a straightforward question – but it no longer feels like it is. He sketched the graph off y = 2 ^ x and marked pi on the x-axis. He talked about working his kiddos through the argument that there must be some power of 2 that yields pi as the answer. He talked about a method of exhaustion making better and better guesses to get closer and closer. We talked about how this conversation could be approached as carefully as possible. I talked about the intermediate value theorem but advised that it not be named yet. I talked about temperature during a day and speed on a car’s speedometer. But as he pushed me I realized that all of these arguments rely on comfortably knowing that this function is continuous and that if 2 ^ 1 = 2 and 2 ^ 2 = 4 that there MUST be some value of x between 1 and 2 so that 2 ^ x = pi. We ended the conversation – because I had a committee to run to – with this questions: How do we convince Algebra II students that this function actually does have to have an input x that yields every output y in a region? How do we recognize a function as continuous? What are the markers? This feels like a question that I should have had a better answer for and maybe in October when my brain is smarter I might have. So, I ask you out there – how can we convince Algebra II students that there is some real x so that 2 ^ x = pi?

Last week it felt as if my Geometry class as making great progress in examining radians, looking at areas of regular polygons, dealing with a new vocabulary word (apothem), and generally doing a nice job of making connections – especially with the right triangle trig that we are now leaning heavily on. Well, Friday morning we had a conversation that I am still unpacking. On their HW from Wednesday night one of their tasks was to complete the table below:

When my students asked me to review this problem I started by drawing an equilateral triangle and its apothem. I know that it is more efficient to use our knowledge of the triangle area formula, but I wanted to reinforce our new formula that the area of any regular polygon is half of the product of the apothem and the perimeter. Unfortunately, when I drew the picture they asked me why I drew what I drew. They wanted instead to draw an equilateral triangle and its altitude. I followed this suggestion by drawing the following figure:                                       

They were quick to identify that this is not, in fact, an example of an apothem. Pretty much everyone agreed to this quickly. Instead they instructed me to draw the following figure:

I have been trying to figure out why so many made the same mistake and, after talking to one of my Geometry teaching colleagues, I have a theory. I keep drawing pictures like this one: 

I fear that seeing this drawing repeatedly has somehow convinced some of my students that the altitude of a triangle is simply the apothem. I know that I pointed out the similarities between the terms when I was trying to help them remember this new word. I talked about how altitudes and apothems were each perpendicular to a side (and that they both begin with the letter a). However, I do not know why some students would simply draw an altitude and figure that, for some reason, we are now calling it an apothem. So this was a disappointing way to start the day. I think I feel a little better now that I have a feeling where the triangle mistake came from. Have any of you experienced something similar?

The other disappointment came when they asked me to address the next HW question. They were again asked to fill out a table, Here is the second HW problem:

I suspect that most of you see some similarities between the first two HW questions. My students did not. I understand that when you are learning something new it is hard to step back and see big picture things going on. However, I also know that this type of HW problem is tedious and I hope that my students are thinking of ways to streamline the process. Instead of recognizing that the triangles in the two problems are in a 1 to 2 ratio and using our knowledge of similarity and scale factors, many students simply reset and did the problem from scratch. I try really hard to build in these sort of  connections in my assignments and my tests. I do not expect students to recognize that in October, but I would hope that they do by May. I’d love some advice about how I can better help my students look for these types of connections. How can I help them step back a bit and see these connections?

Back at it 8 AM tomorrow. I’m optimistic we’ll have a good week. Test on Wednesday – I want them to really kill it on this one.