Got some great feedback on my last post on Soccer and Math from Howard Phillips, @MathButler and @sholvah2010, and I’m thinking particularly about transformations.
In thinking about what students need to understand transformations in 8th grade (my grade), I’m thinking about two big buckets.
Bucket number one is being able to conceptualize what it means to perform a transformation. Knowing that reflections, rotations and translations produce congruent figures, and dilations produce similar figures. That reflections produce a mirror image the same distance from the line of reflection, that rotations produce an image equidistant from the point of rotation. That translations have no fixed points, rotations have one, and reflections have a fixed line. Not that every student needs to be able to name all of these, but students should be able to describe the properties of transformations.
Bucket number two is being able to perform transformations. This is where Mr. Butler’s comment got me thinking. Here, concepts are important, but so is practice, both seeing and doing transformations. And it’s hard to get lots of authentic practice through pencil and paper. The applets I linked to above are great, and Robert Kaplinsky created this great lesson about Ms. Pac-Man. There are other resources on the internet — a quick Google search reveals plenty of resources of varying quality.
Anyway, what I think I was most lacking in my transformations unit this year was giving students lots of chances to both see and perform transformations. This is where I want to incorporate more technology next year, and Mr. Butler’s resources are a great example of ways to get students seeing and performing transformations — gaining fluency in what the basic transformations are and what they do — without the transaction cost of lots of pencil-and-paper practice — which is where my class was last year.
In the comments, Howard Phillips mentions wax paper. I’m looking forward to another go-around there. I used wax paper this year for rotations, and I think it helped, but I did a day of rotations before using wax paper and I think it’s best used as an introduction before moving to more abstract ways of performing rotations.
Final thought: transformations are hard. The most important thing, in my opinion, is that students get lots of chances to visualize what happens to an object as it goes through a transformation. There are lots of tricks students can lean on — algebraic motion rules, wax paper, counting squares on a coordinate plane. These are crutches for the big takeaway that students can transfer to other areas of mathematics and life — being able to visualize what an object looks like after a function or operation is applied to it. I think I’d rather have a student struggling and unable to perform some transformations than blindly applying motion rules or memorizing a rule they don’t understand.