Been Too Long…

It has been quite a while since I have posted anything here. The end of the school year is one main reason, the summer days spent with my kiddos is another reason.

I have also been working on editing the Geometry text I wrote last summer. I have to admit that it has not been as rewarding as the initial writing was. I mostly spent time undoing silly typos, trying to clean up some explanations, and formatting so that it is easier to read along with. I received feedback from our students who made it through the maiden voyage of this text and my teammates were great at catching some silly typos and giving me constructive feedback regarding layout. So, I am happy to share this link to my dropbox where you can find a PDF of the 2nd edition of my text as well as a folder containing all of the HW assignments we wrote last year for the text. We did not make it through all of the sections, so there are some HW gaps. I am hoping we can pace ourselves better this year. If we can, I’ll be uploading any new HW docs as we move along. I will also try to keep track of any other documents we create along the way as classroom practice, as explorations, as review notes, etc.

I have to give credit to a series of amazing blog posts recently by Meg Craig (@mathymeg07) where she has been sharing a goldmine of classroom resources. You should hope over to her website – to see what she has been sharing. Your life will be better after you do!

In five days I will be heading out west to twittermathcamp15. I will report back after that.

Looking for Some Wisdom

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?

Trying to Understand a Curious Misunderstanding

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:Polygon Table

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:                                       Incorrect Apothem

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:

Correct Apothem

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: Polygon with Inscribed Triangle

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:

Second Polygon Table

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.

Catching Up with the Past Week

So there are a couple of activities this past week that I want to write about. However, I have been swamped with meetings so I have fallen a bit behind.

In AP Stats we have finished our required curriculum as of 8 days ago. I am a big baseball fan and my favorite team is the New York Mets. They are having a pretty wonderful start to their year (or at least were until the last few days) so last Friday I posed the following question to my kiddos: Given that the most optimistic projection I saw for the Mets’ season had them pegged as an 87 win team, what is the likelihood of their current record (which, if I remember correctly) is 10 – 5? I liked this for a few reasons. First, it concerns baseball and likely would have a positive outlook for my Metropolitans. Second, it was not so focused on the most recent material at hand. My Stats students tend to know recent material well but struggle remembering other procedures that have not been practiced as recently. Third, it generated some nice thinking out loud about what approach to take. Being more of an algebra stream guy myself I immediately placed this in the context of a probability problem and was prepared to go down a Pascal’s triangle/binomial theorem path. Most of my Stats students don’t tend in this direction so their conversation focused instead on comparing proportions – the 87 – 75 projection with the 10 – 5 proportion. They suggested running a two proportion z test and looking at the corresponding p-value. This opened up the avenue for me to sneak in my approach and make a connection pretty visible to them. Turns out that we felt that we had enough evidence to reject the null hypothesis of the Mets being an 87 win team in favor of believing that they will exceed that win total. Their recent 5 – 5 run of games might adjust that but I do not want to know this – so I will not re-run the test right now! After we checked our trusty TI to find the p-value of this test I reminded them of the probability approach and we set up the appropriate term of the binomial expansion. Guess what happened? This calculation matched the p-value of the two proportion z test!!! This is one of those ideas that we discussed but somehow seeing these results side-by-side seemed eye opening to my kiddos. A triumph on a number of levels!

In my morning Geometry class we dipped our toes into an exploration of radians yesterday using the ProRadian Protractor designed by the fantastic Jennifer Silverman (@jensilvermath) and using an activity that she designed. I wrote a follow up HW assignment that my kiddos worked on last night. I also linked to a fabulous web site that allowed my students to explore radian measure and I shared these notes with my colleagues. There is also a lovely GeoGebra applet (also designed by jennifer Silverman) that is linked from the worksheet. I was totally excited to explore this activity with my students and I had a really nice chat with one of my teammates.

I handed out the radian protractors as well as our regular old angle protractors and we had a nice conversation about similarities between the two protractors. We had a lovely discussion about this but, looking back on yesterday , I think I allowed too many clues to seep into the conversation too quickly. Jennifer’s activity is a terrific one and I got in the way by loading too many conversations in at the beginning of the class. By having students come to my screen and try to identify where one radian measure would lie on the circle AND by having the protractors side-by-side I reduced the mystery element that I think should have been part of the classroom activity. I think I took away the opportunity to discover what was happening here. I did have one student give a GREAT explanation of why the quadrilateral radian measure was twice the triangles radian measure. She invoked a proportional idea and referenced our (n – 2)*180 formula. I had a number of students quickly see that the ratios we had been working with before (\frac{x}{360}=\frac{arc}{2\pi r}=\frac{sector}{\pi r^{2}}) could be easily extended to add one more simple fraction of \frac{x}{360}=\frac{\theta }{2\pi }. That definitely felt like a triumph. So, the lesson I learned here – and I hope I remember it for next year – is to be a little more minimalist in front loading this conversation. I think that we can touch on all of these resources and really let the discovery sink in, but I feel I nudged them a little too much this time around. So the plan for next year is to hand out the radian protractor and work through the worksheet. Then hand out the angle protractor and talk about comparing them. Then, the next day after some time to think, show the web app and have them identify where one radian is. Let this unfold a little more slowly.

A Quick Geometry Snippet

So, we were reviewing this morning in Geometry getting ready to finish up our circle unit. I was reminding the kiddos of the angle and arc relationships we have been discussing. I had been writing things like x=\frac{1}{2}\left ( a+b \right ) where x is an angle formed by the intersection of two chords and a and b are the measures of the intercepted arcs. However, today as I drew my diagram for it one of my students suggested I mark is as seen below. Interior Angles

I think I am delighted by this and will write it this way from now on. It feels to me that it is a more natural way to think about this relationship instead of having a coefficient of 2 or a coefficient of 1/2 that might not seem at all intuitive. I then drew the following Exterior Angles

I’d love to hear from some other teachers about whether this seems at all like an improvement over the more standard way of writing these equations.

That’s all for now, just needed to get that off of my chest!

Highlights of a Stressful Week

So, there have been many scattered thoughts on my mind in the past week but there are also three things that happened that are just completely awesome.

  1. My pal John Golden (@mathhombre on twitter) steered a number of his teacher training students over to a post on my blog and twelve of them chimed in with comments. Totally cool! One of them decided to follow my blog and I took the time to respond to each of them. LOVE the idea of new teachers in training dipping their toes into this rich world of teachers blogging and sharing. I am also flattered that John thought my virtual home here was worth a visit.
  2. I woke up Wednesday morning with a message on twitter from a teacher in Louisiana who asked if he could use my Geometry book at his school next year. I am so excited by the idea that this work might be used at another school.
  3. In my Geometry class this week we are talking about angle and arc relationships. One of my students stayed after class one day this week and she had this to say. “You know, I was thinking, when will this be important? I mean, when will I need to find an arc length like that? Then I realized that the work we are doing to find that length is what is important. Pretty cool.” Wow.

Some Fun Geometry Action

On the heels of learning some right triangle trig I am really trying to develop more proportional logic with my students. Just this week we had a really productive conversation about the following problem.

Screen Shot 2015-04-09 at 6.45.44 PM

Being a bit of a bull in a china shop sometimes, I proposed that we should find the height of each triangle, find each chord length and find the height of the trapezoid by finding the difference of the heights. Not elegant, I know. I was trying to make sure that we remembered some right triangle trig. that we remembered our area formula for a trapezoids, and that we try to develop some patience in solving multi-step problems. That was my plan, but as with many school plans, it did not quite unfold that way. One of my students who is a bright and quick problem-solver pointed out that simply finding each triangle area would be enough. I understand that his solution is pretty much the same as mine, but it certainly sounds more efficient. But as soon as I acknowledged that his idea was more efficient than mine another student trumped each of us. She pointed out that I had already asked them to consider the ratio of areas between the two triangles. So, if we know one area, we can automatically know the other area. If we know both areas, we find their difference as suggested by the first student who chimed in. I was so happy that she took my clue from within the problem and that she was clever enough to really save time and energy this way. I made sure to compliment her in class and I bragged about her work to two of my colleagues yesterday. Oddly, this morning when we were reviewing before today’s quiz I reminded her – and the rest of the class – of her clever idea. She had no memory of this conversation. Sigh…

I’m trying to process this and figure out what it might mean for my classroom practice. I understand that I should be more excited by my students’ ideas than they often are. I understand that I will remember context of conversations more easily than they will because I am not dealing with the cognitive load of trying to learn/understand the conversation. I am simply coming at it from such a different place. What I don’t understand is how a student can be so in command of an idea but then not remember the creative process that made her arrive at this clever conclusion. I discussed this in the faculty lounge right after Geometry today and one of her other teachers intimated that this might simply be modesty on her part. I am not sure how much faith I put in that reading.

So, while I am a bit frustrated and confused, I am choosing to focus instead on the positive energy of yesterday’s conversation, on the clever ideas that my students brought to the table, and on the fact that my students did a really nice job on their quiz today.