This is the kind of strategy that I think is really important for my future physics teachers. I oscillate over the years about whether and how much to do this with students.

I added a few more correct examples, just playing with different formats, and show-casing different strategies.

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I’m sure you already know this, but for the graph example, you can use the formula for the area of a trapezoid to find the displacement – which is the same as using the average velocity because the average velocity equals the average height of the trapezoid. Your could even show the average velocity on the graph along with its area.

Yes, and this specifically is something I make sure all my preservice teachers know well. How all the equations relate to different ways of conceiving of that area. I also make use this method them “prove” that xf = xi + vf t – 1/2 a t^2 is a valid equation.

Awesome! I always struggle with how much of this to do with students. I always show all possible methods least once. I struggle with if it’s something to always do and/or always ask students to solve using multiple methods (e.g., finding displacement using area method AND average velocity method).

Yes, definitely a struggle. For me, it’s partly an issue of cognitive load. How much and how many to off load, and over what period of time. How we package them for students and how we ask them to connect them probably has a big impact, but I also don’t know an optimal path.