I’ve done a lot of professing my new found love for Visual Patterns lately, and today will be no exception. If I haven’t convinced you of the flexibility and differentiation available in these seemingly simple patterns, let me have one more stab at it.
Today, my College Algebra class looked at pattern # 28.
I took out the unifix cubes for those who wanted to actually have the three dimensional shape in front of them. This was helpful for some, however, I realized the limitations of the cubes…the fact that they only will “fix” to one other cube (hence the name UNIfix). This may not be mind blowing information to many of you, but I just put those two things together in my brain today. Because of the one fixture, they were hard to take apart in usable “chunks” without the whole figure falling apart.
Anyway, back to the pattern. I wanted to workout ahead of time all of the possibilities that students would come up with so that I could more effectively use the 5 Practices of Orchestrating a Mathematical Discussion and anticipate their responses. I’ll tell you what, I played around with those expressions so many times, and thought for sure I had came up with at least the majority of responses I would encounter. They were all quadratic. Then, out of left field, the students threw me for a loop. The majority of students came up with n^3 – (n-1)^3!
Now, you might be thinking, duh! It’s 3 dimensional AND a portion of a cube. However, my algebraically trained brain started with quadratic expressions and stuck with them since I saw from the difference of differences table that this pattern was in fact, quadratic. Yes, the n^3 terms get cancelled out when the expression is simplified and the simplified expression becomes quadratic but this opened up a whole new avenue of discussion with my class. We were now able to talk about the misconceptions of expanding something like (n – 1)^3, because if they found the expression for the nth step another way, they could use that as a check for simplifying their answer.
What was eye opening for some of the students that chose an algebraic method (such as using a table of differences and then setting up a system of equations) was that the “c” value in ax^2 + bx + c was hard to conceptualize. It was very difficult for students to grasp that the first term and the non-existent “zero term” had the same number of cubes.
Finding the surface area formula for step n was even more awesome, because it was in this portion of the pattern that I was able to see real growth in my students’ willingness to attempt a more conceptual method. There are certain students whose default method is to set up a system of equations using the table of values for the pattern steps. These students are noticing more that they encounter errors much more often than those who have a conceptual understanding of how the pattern is built. I found this time around, less students relied on the algebraic method (about 7/35) whereas last week, probably 15/35 of them were starting algebraically. As we are covering more and more concepts in this course, the students are realizing that they do not remember specifics about formulas and procedures from their previous algebra courses. They remember “learning” the topics, but they usually can’t quite nail down the specifics of each method. I really feel that we are making some good headway toward solidifying their conceptual understanding of the algebra as I see more and more students break away from the procedural methods toward a more conceptual one.
We talked about this pattern for an entire 60 minute class period. You know it’s a good day when kids look at the clock and say, “whoa! Class is over already?”