I begin this lesson with a conceptual warmup problem focused on forces and the definition of "vector addition." This exercise requires little to no computation and allows students the luxury of thinking about the real meaning of a "'tip-to-tail" construction that is the hallmark of vector addition. We have already introduced the idea of "tip-to-tail" addition of vectors and I want today's work to build on that. The warmup problem gives students a chance to develop a deeper understanding of vector addition. In addition, our work today helps to establish the concept of a "net force," a phrase we've been using though in limited ways. If successful today, student understanding of net forces will be enhanced and an important seed will be planted for a later unit in this course.
Though I expect that students will recognize the 3-4-5 right triangle combination in part a of the problem, I project a solution to the first part of the warmup on the board after 3-4 minutes have passed. For those who recognize it already, this provides no additional insight. For those who don't recognize it, my solution provides a clue as to how to proceed with the latter problems. In this way, virtually all students can get deeper into the problem and recognize the importance of direction in vector addition.
I choose this problem because I want to ensure that students have a broader sense of vector addition and don't walk away thinking that it only applies to right triangles. Though most of their solutions DO result in right triangles, the combinations of the "ones" in the problem (often linked together to add or reversed to subtract) opens up the possibilities for adding vectors in all kinds of directions. The resulting conversation, after the warmup, is often rich with questions.
Here's a surprising result from one student, as he addressed the final problem:
Note: the blue arrows are all valued at 1 "unit." This student's 'detour' represents a break from a strict triangle and, yet, achieves the goal of creating the appropriate final vector suing the rules of vector addition. He drew it with the red vector (the result) horizontal (as opposed to "north of east" as the problem prompt requires) only due to personal comfort with drawing on the board.
Having seen this response, I quickly created a "net force" (shown below as a red vector) and asked to see a show hands to the following prompt: "If the red vector is the result of the sum of two vectors, how many different pairs of vectors could create it?" I wait until I can see about 6 hands, then ask students to respond. We establish that there are an infinite number of combinations and we look at three combinations (shown in green, black, and blue in the diagram).
In the previous lesson, we only had a a chance to try one right-angle-charge problem. Today, I allow some significant time for practice. The practice problems are differentiated: the first three provide direct practice with right triangle geometry, while the remainder require the use of Coulomb's Law to calculate the forces and a close inspection of the signs and arrangement of the charges to establish the directions of the forces.
During this time, I circulate to provide precise, timely feedback to students as I see the kinds of issues they're struggling with. Students are encouraged to seek peer-tutoring as well: hearing an idea from a peer is sometimes the best way to make sense of what the teacher has already said! Because of time issues and because of my desire to provide individualized feedback, I do not take time at the board to review answers to these problems. Rather, I make it a point to check in with each student to see what they can do and what they are struggling with. Students who get the ideas will receive affirmation from me and spend their time practicing the problems. Those who are struggling will get some targeted feedback which may take some time away from the practice. In the end, perhaps they only do two problems. But having done two right is better than having done six wrong!
At the end of the practice time, I give my students a one-question quiz. Once they complete the quiz, I hand them the next assignment (right angle charge homework) which they may begin if they are waiting for others to finish the quiz.
I hand out small pieces of scrap paper on which students will do their work. This signals the somewhat low-stakes of this event. Students can access their notes while doing this quiz but must work individually. Students put their names on the paper and the lowest score out of 20 points that they would like to have count. If a student does not make their desired score or better, I will not count the quiz (see the associated reflection section for more thoughts).
As it is just a single question, it is often the case that the entire class is done within 15 minutes. I quickly assess the papers, looking for key ideas and processes and being pretty generous about mathematical mistakes, and, if time allows after, I can choose to share the correct answer.
In the final few minutes, I introduce some new thoughts about "work." My immediate goal is to have students recall the way in which we've defined work (as an area under a Force vs. Distance graph) and to recognize the somewhat clumsy nature of that approach. I hope students will see the value in a more algebraic or computational approach.
I show the redefining work notes on the board, but reveal just a few lines at a time as I want the conversation to proceed in a certain manner. The goal is to get students to see that, when an object is truly worked on, its overall energy will have increased - some outside agent spent some energy to change the energy of the object under question. Furthermore, as we have formulas for both kinetic and potential energies, evaluating those energies before and after being "worked on" can provide a precise measure of work. Though there are other ideas - the specific formulas for various kinds of energy - to add to this discussion, we end the day with these notes, setting up our next class as a time for expanding upon and applying these new ideas.