Andrew Busch

Original problem by Robert Kaplinsky. I use the problem directly after we learn about special right triangles in Geometry (45-45-90 and 30-60-90). Lot's of great overlap between Algebra and Geometry here. Lot's of great opportunities for students to present different solution methods.

Here's the slide deck and handout I use with my students.

We spent a day working on a fun task from the MARS Team.

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.

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. 

This is the lesson I intended to use for evaluations--you know what I mean--the 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 year--3rd 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 me--I 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 culture-building 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.

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 10-15 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. 

This is a sweet little one day task I use for helping students get their minds around rational functions (from EMATHS). The concept is simple enough--find the total miles per gallon for the Chevy Volt.

Students use the table at the top to help make connections between the miles traveled and the amount of gas used. Most students are able to come up with a function modeling the situation without much scaffolding from me.

I've used it with both Algebra 2 and advanced Algebra 1 students. Both groups did great.