Here is the only setup:
 This diagram shows a circle with one equilateral triangle inside and one equilateral triangle outside.
 Find the exact ratio of the areas of the two triangles. Be convincing in your explanation.
Classes and thoughts of Andrew Busch 

We spent a day working on a fun task from the MARS Team. Here is the only setup:
After spending 30 minutes or so working on this problem in groups, we ran out of time. Meh. We planned on starting class the next day talking about our ideas for solution methods. Normally, with two minutes left of class, they line up at the door like cattle. Some of the students weren't done with the problem yet.
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For some reason, every year I trick myself into believing next year I will do better at managing my time and posting to my blog. Next year. Here's a bit of what we've been up to lately: Backs to front review for our unit on circles. I love how loud this gets! Floodlights lesson introducing similar triangles. Big honking whiteboards. Too tall to measure using proportionality and similar figures. Math Outside! Speed dating review on simplifying rational expressions. We call it 'speed therapy' because dating in middle school causes all sorts of commotion. Exponential Dice (thanks to Kate Nowak!) Any reason to get students out of seats and still focused on content is a good reason.
This is the lesson I intended to use for evaluationsyou know what I meanthe lesson you plan to do in the next week and a half that you think will make you look the best. Things didn't go 'terrible' but neither did we actually get to the cool part of the lesson either. In our study of similar figures, I use Math Assessment Project's: Floodlight Shadows. It does an excellent job progressing with a relatively simple group task (#2) involving similar triangles and then switches things up asking what happens to the length of the shadow when the football player moves (#3). The second question asks for a solution of an instance. The third question asks for a general solution method... a proof. All the resources you need are in the link. Okay, but we gotta back up. I have two sections of Geometry this year3rd and 6th periods. My Principal decided to come for my second section. I'm thinking "Sweet! I'll get some practice in." I had all sorts of grand ideas about how things were going to play out. Today I need to show off my mad teaching skills. It didn't happen. Turns out the homework assignment from the night before ate most of my class period today. Now, in my Algebra 1B class this would have been horrid. In my Geometry class it's usually only frustrating to meI have an agenda people!everyone else seems cool with it.
In 3rd period, as students asked questions on problems, other students volunteered to present their solution methods. Hands were raised. Students called on each other. Mics were dropped. Laughter. Shouts. 10 minutes on the task (up through #2). Whatever, I felt brilliant. In 6th period, I gave the students a choice, we could either go over questions at the beginning of class and have a small start on the Floodlights task or we could start the Floodlights task and go over the homework the next day. Secretly, I hoped they would choose the lesson. We put it up to a class vote and they decided to check the homework. Okay, if things go like they did earlier in the day everything will be fine. As students asked questions on problems, ...crickets. Having them volunteer was sooo much harder than usual. Having them ask each other questions was like pulling teeth. It was akin to losing 2 months of classroom culturebuilding overnight. Sigh. 6th period ended their homework discussion with 15 to 20 minutes left of class. I pulled out the big honking white boards and set them to work on the task. One or two groups were ready by the time class ended but not enough for a classroom discussion. Homework was to complete task #2. We started off day 2 with a quick discussion about solution methods for #2 in the task, the length of the football player's shadows if he stood in the middle of the field. Every group had a solution method. Every group used proportions. It was a short discussion. Then we upped difficulty level. What would happen to the football player's shadow when he walked towards one of the floodlights? Would the total length of both shadows stay the same or change? Many groups attempted to extend their previous solution method using proportions. It worked for a couple of them but there is a lot of symbolic manipulation involved. At least half the groups who tried using proportions got stuck.
The actual lesson from mathshell.org has sample solution methods for students to evaluate after they've attempted the task. Last year I used them. This year I wanted to give it a little more time. I had them printed off, just in case. I stopped the groups with 1015 minutes left of class the second day. In both of my classes, I had one group that either didn't complete a solution method or didn't have a promising plan of how to get one. With our last 10 minutes of class, students presented solution methods. One group had a solution so elegant they received applause from the class (6th period). They dropped the mic with wide grins. Homework was to revise a solution method they'd seen in class. I'm not very good at having students revise their work. I figured this was as good a time as any. We rework drafts over and over in English and History. We do experiments over and over again in science. Why not in math? I asked for complete sentences, diagrams, etc. We'll see. Next year I'm going to split the task slide and give #3 it's own slide. Having #2 and #3 on the same page gave away too much about where I wanted to go. Some students had already attempted #3 at home the night after day 1 and controlled the discussion in their groups for too much of the time on day 2. When teaching proportions or similar figures I generally throw in an example or two of measuring the height of objects you wouldn't normally be able to measure using shadows or various other known distances. Since I've been on my activity/experiment kick, I wanted to go measure shadows to find the heights of trees and telephone poles. However, it's January in Michigan. Nobody wants to go outside and measure shadows, which assumes there will be a sun to cast a shadow. There are sometimes weeks that go by without seeing the sun up here. Enter a brilliant adaptation by Jonathan Claydon for the cannotgooutsidefor5monthsbecauseweliveinthemidwest group of people. I have my adaptation of it at the bottom of this post. Students start be measuring each member of their group (see below). This measurement is known as the group member's 'real height'. Then groups drift off to the gym to find the height of objects that are too tall to measure easily. I picked three of them: the gym ceiling, the top of the backboard, and the height of the railing on the weight room. Using a ruler, group members find the 'fake height' of a each group member standing next to the tootall object and the 'fake height' of the object. We found both measurements need to be recorded each time because of differences in how students hold the rulers. After completing our measurements, we went back to our classroom to finish the calculations. Each group member had to find the heights of the objects using their own fake height measurements. Then groups posted a group height for each tootall object. Lots of good conversations happened when calculated heights didn't agree. Most groups just averaged the calculated heights for each tootall object. Next, we figured out how we would go about agreeing on the actual height since I don't have the actual measurements myself. With our ceiling calculations, we decided to remove the group's height which was very different before averaging the heights. The class's backboard heights weren't nearly as close. Here we thought the median might be a better descriptor of the 'real height'.
Here again, the 'real heights' were all over the map. They thought the median worked out well last time and wanted to use it again. When we talked about difficulties we encountered, several students said they had difficulty eyebaling the measurements because they couldn't hold their hands steady. When I do this again next year, I plan to hand out a jumbo paper clip to put on each student ruler to serve as a marker for the top measurement. Thanks to http://infinitesums.squarespace.com/commentary/2011/9/30/proportions.html for the idea!

Andrew Busch
I teach Math and Programming at Summit Middle School in Boulder, CO. Categories
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