Graphical Solutions for Forces and Kinematics
On the Sunday of the AAPT summer meeting (July 27th), Casey Rutherford and I are offering a free, unofficial1 workshop on using velocity graphs and force vector addition diagrams to solve kinematics and force problems. I was really lucky to get to also work with Mike Pustie earlier this month to give a similar workshop for the STEM Teachers NYC (née PhysicsTeachersNYC) group.
Our hope is to show how our students use these diagrams as sense-making tools to solve problems, to give you time to try using them yourself, and to show how and why we’ve found them useful for physics students. These strategies can be more accessible for students who have less experience or confidence with math while also allowing for more subtlety, depth, and original thinking for students who have more math confidence and/or experience, so we think that teachers in a wide range of schools/classes/etc might find something useful in this workshop.
You can find more details about the July workshop including our description, the location, and the sign-up form on our workshop website.
And now, to make this a legit blog post, here’s a preview of the content:
Solving Kinematics Problems with Velocity-Time Graphs
In my observations, the kinematics equations can be a big stumbling block for intro physics students for a couple of reasons. (1) Solving problems with these equations means keeping careful track of algebra and dealing with a lot of symbols at once. (2) The procedure that many students adopt in using equations for kinematics often separates that work from making sense of physical situations. In that case, students may just be hunting for equations with the symbols they want and not doing a lot of (what I would call) real physics thinking.
Enter the velocity graph. (My favorite graph, as my students might tell you. It’s just so useful! The slope and the area both mean things, and (in intro physics) it’s usually a straight line, so the area is made up of rectangles, triangles, or trapezoids—all areas I can find easily.) With this approach, students always start problems in the same sort of way (making for a comfortable-to-students procedural-type feel)—they draw and annotate graphs that represent the situation. They immediately make decisions on what model is useful, direction of motion, how the speed is changing, etc. Their work is all about sense-making.
Here’s a taste of what it looks like when real life students solve problems this way. (Note, the “Yay!” is the student’s—she wrote it while looking over the solutions when she turned in the quiz.)
On the one hand, solving problems this way gives students who struggle with algebra, keeping track of signs, and remembering equations better access to solving quantitative motion problems at any difficulty level. On the other hand, it makes the connection to calculus really clear for students (they come back in subsequent years to exclaim that they are basically generalizing in math what they’ve done in physics—they leave the class with a real appreciation for the meaning of slopes and areas on graphs)—I see their work as being more mathematically sophisticated than it seemed to me when I was teaching students kinematic equations.
Solving Force Problems with Vector Addition Diagrams
When it comes to procedural solutions that can allow students to move through problems without necessarily making physical sense of the solution, Newton’s 2nd Law in component form was a major offender in my classroom. Students were often bogged down in trigonometry (which they might not have even understood because they hadn’t gotten that far in their math classes yet) or sign problems or lining up the correct parts of the equation in just the way I had told them to write it. Every “good solution” to a problem basically looked identical, so there was (sometimes) little of a student’s own creativity or sense-making needed.
I found vector addition diagrams really compelling when I first encountered them, and I started showing them to students as an additional representation. Eventually, I started teaching balanced forces this way (and delaying components until unbalanced forces when students might be more comfortable with trig). And finally, students convinced me that this method was better and told me to stop teaching components. (Never fear—the idea of components comes up naturally for students as they analyze their diagrams, so they don’t really miss out on that concept, and they are ready to break velocity into parts when they study projectile motion.)
I only teach my students to draw vector diagrams to scale (if they are using them quantitatively—they always start with a qualitative sketch anyway). As they learn more math, they start seeing how they can apply trig ideas to their physics work and transition as they are ready. I can always tell when they’ve just learned law of sines (they are always so in love with that idea that they try to use it for everything, even right triangles, as soon as they’ve learned it). There is a lot of ownership and joy from students when they are deciding how to solve each geometric problem.
And again, here’s some delicious student work on force problems.
I think you can see from even just these three examples that this method allows for a diversity of approaches, that students are thinking about relative sizes of forces, and that students do end up inventing their own ideas about components in the context of the problem. Love it!
Here’s an old, relevant post: No More Components
If you will be in Minnesota for AAPT this summer and would like to join us for the workshop, please fill out the form on our website to let us know and to reserve a spot. Casey and I are excited to share this work with you! (Click the sign up button to go to the workshop website.)
1 Casey and I started planning this workshop many months ago (January at least, but maybe earlier). Unfortunately, we didn’t understand how to get an official workshop and later found out we would have had to request that at least a year(!) in advance and not the several months in advance when the contributed talks are due as we’d thought—we asked if we could be added to the schedule once we realized that, but we were told it was too late for this year. We didn’t want to wait over a year to share what we’d been doing with students (and a 10 minute contributed talk wouldn’t have accomplished the same sort of sharing), so we decided to go the unofficial route for this year. So worry not—we didn’t set out to be especially subversive, and we plan to participate in many other, more official, AAPT events at the conference.