Today I had a good day teaching. Firstly I had the class who, on Monday, were questioning assessment, and we had a much better lesson. The boy who stated he did not care about learning behaviors did his homework thoroughly, nevertheless!
Anyway the 80 minute period today was with a different 10th grade class. We have been looking at rational expressions, equations and functions. It has been very algebraic. In past years, when I have taught Algebra 2 and Trigonometry, this topic has felt like a mundane, typical part of the course, with no relevance to students’ lives. However as we have moved from probability, to an in depth investigation into the workings of functions, this unit actually feels like a quiet relief for students.
Today we did change tack though: I could let students spend 3 periods factoring and cancelling then drawing asymptotes without making any connections. We moved on to inverse variation. To introduce the idea, there is quite a good discovery streaming video which I showed. Then students collected data, graphed the points on geogebra, and worked with the basic function y = a/x and a slider for varying a, till they found a good fit for the points.
They collected data from two possible means: measuring height of a given amount of water in cylindrical containers of varying widths, or measuring the “apparent height” of an object from varying distances. The water and containers idea was found on a website called learner.org, and @r_w_wright,@daveinstpaul on twitter gave me the idea for the heights.
It went very well. I think the students enjoyed exploring the practical aspects of inverse variation. I do think I need to refine the activity somewhat though, as we ran out of time I realised there were some details to clarify. We did not get to talk about the domain and range for the applications, which is important. Secondly, many students wanted to model the data with an equation of the form y = a/(x – h) + k because we had spent a lot of time on transformations, and they now associate sliders with a, h and k! I did not get time to ask them to think about the position of the vertical asymptote. Maybe a characteristic such as this should be considered even before the data is gathered.
Below is one trio of students’ work. They measured the “apparent height” of one student at varying distances.