Not ODEs – The Variation of Parameters Method

VersatilityWhen one attempts to define versatile, something like “able to be adapted to many functions or activities” may come to mind. As a classroom teacher (and now working K12) this idea comes even more naturally as I know have the opportunity to see how successful and adaptable an idea or lesson may be with learners of various levels and ages.

Recently, I had such an opportunity to try marry a couple of ideas I have brewing in the back of my mind with four classes, two 6th grade classes and two 8th grade classes. The original idea stems from two separate sources a random search and from Mark Driscoll’s amazing book Fostering Algebraic Thinking in which the heart of the lesson follows. The other idea is my love of working a 3 Act lesson into anything that I can teach. So I will discuss both lessons and what I learned in what follows, but I’ll first set the stage for the lessons.

This year, I have had the pleasure of working a lot in the sixth grade, a grade I felt comfortable with my background (as I have described before) doesn’t lend itself naturally to the younger learners. I am working on getting some practice with many folks favorite age group, the infamous eighth graders. So I had this opportunity lineup, within a day of each other, I had the time to teach the 6th graders on an introduction on Equations and Expressions, and an introduction for the 8th graders on slope.

Enter my normal teacher brain, whose thoughts in no particular order went something like this: 1. I don’t have much time to prepare two very different lessons, and make them meaningful, 2. Students need hands on connection, an anchor, that we will be able to refer to throughout the unit,Teacher Brain

3. Students like to build things and I like to facilitate through questions and setup versus talking to myself in a lecture format, 4. Using 3 Act format gets a lot of buy in early, prompts students’ thinking in the direction I need, and makes for a natural direction to move forward, 5. What do I have around me that would allow me to use the same format, but change the questions (or purpose) of the lesson with the same manipulatives?….

Enter the work setup by Driscoll, he posed a Matchstick problem that extends, this would work perfectly, and the dollar store has popsicles by the hundreds for, well….

My manipulatives clearly in hand, my thinking went to the idea that I wanted the sixth graders to build the triangles, as shown in the video, counting the number of triangles formed and relating that to the number of sticks required to make the triangles. There is a natural differentiation here too, advanced learners may recognize that there are more triangles depending on how they build their triangles together. The relationships that start to form here, are well beyond the scope of this lesson, but what a great learning experience for those kids ready to go beyond. As the students build the triangles, they record the data, and then the extension question(s) go like: 1. How many sticks would we need if we were building 100, 1000, or 1,000,000 triangles? 2. How many triangles could we build if we have 100, 1000, or 1,000,000 sticks?

Good ThingsThe math nerd in me love these questions because it extends to the idea and in light of actually building it, this is a natural way for me to introduce the idea of generalizing, enter expressions and equations. In addition, this also shows the doing and undoing process of the relationship between the number of triangles and the number of sticks, an important piece that we sometime overlook.

Using a similar idea in eighth grade, the idea there was to tackle the misconception that these learners had in regards to scaling, the idea that the ratio of two similar triangles is the same though the lengths of the respective sides may be much different. The learning that I wanted the students to see was that as they build the right triangles, make measurements (by counting sticks) and comparing the ratios of “rise” to “run” they would discover that the ratio was the same. I was expecting some errors, like how the students build and counted the right triangles, which I was hoping would be a great “teachable moment.” Students would get a lot of collaboration and would work with a small data set of 9 other ratios calculated from other student groups, reinforcing their understanding.

What was used is shared is here.

What I learned through this experience is the importance of modeling and being very clear with what was required of the learners. After reflecting on these lessons for a while, and thanks to invaluable input from #MTBoS and #mathchat folks, the lessons need some tweaks here and there. Specifically, the amount of scaffolding and teacher direction are not calibrated for optimum performance. Having giving them a test run, I like the core ideas and I think the experience was very valid, but I know some major revamping I will be doing for next year.

What did go well was the conversation and engagement when students were actually building and working with the manipulatives. I am forever amazed that something as simple as small, wooden sticks could get even the most ardent stalwart working through the lesson. The classroom was buzzing, that’s the moment I love to hear, kids actively involved with their learning. I will update this and other thoughts as I move forward, getting back to blogging and catching up has been a hard road to toll.

Happy Mathing!