My QR classes are active enough that I thought I could sneak in some inquiry based learning (the session provider called it something else but it is IBL).
The lesson was on fractals and fractal dimension. I drew three steps in the iteration used to create a side of Koch’s snowflake and Sierpinski’s triangle. I mentioned the length or area of each image. I wrote what’s next on the board. Created groups of approximately 3 and gave everyone a spot at the board. Chaos ensued. Part of the problem was that the numbers proved a distraction. Students found erroneous patterns in the numbers and tried to draw pictures to match. After a while most groups figured out how to draw the next step in Sierpinski’s triangle.
I talked about how why the figure might have non-integer dimensions and showed how to calculate fractal dimension. I then gave two iterations of new fractals one based on line segments and one on Sierpinski’s carpet. Somewhat less chaos ensued. The next step of the carpet was done by most groups quickly. The graph made of line segments was eventually iterated right by about half the groups. The groups started out with different (and so many wrong) calculations of fractal dimensions. Then they started wandering and talking with other groups and the incorrect fractal dimensions were corrected. All groups ended with a correct calculation. Afterwards we discussed how to determine the ‘ruler-size-factor’ as a class as some students were still uncomfortable with that.
In other words, it was a mess, but I think most student left knowing what N and R were in the fractal dimension formula.
Later, I decided that my intermediate algebra class was not engaged enough so I stopped what I was doing (I do- we do- you do with quadratic-like equations). We had done examples of the form ax^4 + bx^2 + c = 0. I had them work in trios on the board with other formats like ax + b*sqrt(x) + c =0. As I had done less work like this with this class the first few minutes were largely spent with complaining and two high achievers just trying to do the problem for their group. Eventually everyone got to work. Three groups saw how to use substitution as in the prior example (one made an error factoring, but the process was solid). One group isolated the square root and squared and continued from there. The algebra was ugly but they did it accurately. One group essentially did substitution but did not replace the variable. So, it kind of worked here as we had examples in the theme of the lesson and an example of how sometimes you can reapply old approaches.
So, anyway, from frustration to chaos to learning was the theme of the day. I’m going to substantially revamp the quantitative reasoning class for winter. The intermediate algebra curriculum is so skill based that it might be harder to do there.