I provide my students with a simple geometric problem, similar to earlier ones, featuring a multi-part force versus distance function. This problem reminds students about the big idea that we're working on: that work can be defined as the area under a force versus distance graph.
After students have had a chance to work through the problem, I have a student show a solution on the board. We discuss the ideas associated with the problem and I probe for understanding by asking questions as we proceed. Finally, just before ending this section of class, I pose an extension question. I ask students to put their hands in the air if they IMMEDIATELY know what will happen if I only change the sign of the multiplier in the third segment of the function. I wait and count how many hands go up. After seeing 6-10 hands, I call on students to explain the impact of the sign change. In this manner, I can check for true understanding as the "what-if" nature of the probe affords me insight into student thinking.
We pause our study of electrostatics to have the next student presentation from the "Hot Rocks" investigation of the previous unit. Students generated interesting secondary questions while exploring calorimetry experiments that featured heated rocks and cool water baths and pursued those questions by designing their own procedures. I use this kind of culminating activity - as opposed to the more traditional lab report - because the challenge of facing a live audience increases the authenticity of the learning; I ask our students to be more like scientists and, by submitting to this "peer review" we can approximate that.
These presentations provide opportunities for students to report conclusions to their peers, based on the distinct evidence they collected on their chosen question. As a result, the entire project allows each team to engage in many of the NGSS Science and Engineering Practices including Asking Questions, Planning and Carrying Out Investigations, Analyzing and Interpreting Data, Constructing Explanations, Engaging in Argument from Evidence, and Obtaining, Evaluating, and Communicating Information.
In addition to a presentation score, students receive an "audience score." Above and beyond the obvious expected level of attention, I challenge each student to ask at least four questions over the course of the four presentations they'll see in the next week or so. At first, the questions seem a bit contrived but the spirit of honest inquiry inevitably takes over and the questions become more authentic over time. For example, an early question might be "How much water did you use in your trials?" which would count as a question but leads to little, if any, depth of understanding. Later, however, the questions become more thought-provoking like "What did you expect to find by altering the experiment the way you did?" I use a blank roster sheet to record questions and comments along with a scoring sheet for the presenters that focuses on several of our school's learning expectations.
I reserve time for one presentation per day. This pace allows us to continue our focus on work without allowing the previous unit's ideas to become forgotten. Below is an overview of the presentation from today's class:
Before asking students to continue with their consideration of the ideas of work and electric forces, I present a quick summary of what we've learned previously. Specifically, I review the shape of Coulomb's Law (force as a function of distance), the dilemma of finding the area under a curve for which no obvious geometric shapes fit, and our agreement that a simple, first approximation can be accomplished with a single trapezoid.
My intention here is to frame the next segment of class where students are provided some time to practice the single trapezoid method. I want to ensure that all students use the same method for a first approximation of area. In addition, it's important that students really see how poor an approximation this method provides. The two practice problems should provide that experience. The hope is that that recognition will motivate their thinking about better approximations since, in the final segment of the lesson, I ask students to consider improvements and share them at the board.
Students spend some time practicing the estimation of work and area by the trapezoid method we've developed. I provide two Coulomb's Law practice problems with scaled graphs for them to work with. I circulate and "teach between the desks," addressing specific concerns of individual and small groups of students. One important aspect of these practice problems is the capacity of students to visualize how large of an error we are making with our single trapezoid method. By seeing this, it prompts further thoughts about enhancing our method later in class. This is a good example of assessment FOR learning as I can see how well the ideas are being internalized while students get the necessary practice.
Having done the practice problems, students are ready to consider ideas that will help to minimize the estimation error. Indeed, many have been objecting to the single trapezoid method and are happy to be set free to create better approximations.
I show the sample force graph from the previous lesson and draw in a single trapezoid to replicate the kinds of problems we've been doing. Having had earlier conversations about this and having just circulated around the room soliciting ideas about the estimate, students are primed to think about minimizing the error. I invite students with ideas to come to the board to draw in their vision of minimizing error. Here is one student's example of minimizing error. His basic line of reasoning is that the top line of the trapezoid obviously captures too much area. His desire is to have the top of the trapezoid cut through the function, thereby cutting down on the excess while simultaneously creating some "lost area." Ideally the excess and the loss balance perfectly; determining how to create that ideal balance, however, was not a question he was ready to address.
Eventually, we stop to consider the most promising strategy: typically, the use of a multiple trapezoids method to "shave off" the over-estimation that is inherent in the trapezoid approach to this function. While the vast majority of my students have not had calculus, it is very satisfying to see these students replicate the kind of thinking that underpins integration.
If time allows, students return to the first question on the practice sheet and attempt to minimize the error with their new approaches. Students are welcome to try their personal approaches; the investment they've made can be honored by allowing them to pursue their own solution. Whether that time is available or not, we come to the end of this class with a refined approach to our problem of calculating work with electric forces. In coming classes, we can benefit from this thinking by replacing our earlier, naive methods with more sophisticated ones.