I built off of an approach inspired by Steven Strogatz's Infinite Powers: How Calculus Reveals the Secrets of the Universe . While reading this during the summer of 2019 I was reminded that the earliest steps towards calculus were taken by those who were trying to solve physical science problems. And their approaches were both algebraic and geometric in their reasoning. I realized I could try retooling my approach to acceleration from a more Galilean approach, to not provide the quadratic regression from the get go on day one, but to edge into this relationship more carefully. To build more slowly from the material of the second week, and to not presume that a parabolic curve is the automatic result of a constant increase in speed. I first tackled this approach in the fall of 2019 and tried to develop the approach further during spring 2020. I think I relied too much on Desmos spring 2020. Then the global pandemic intervened and there was no more "day one of acceleration." Notes do not exist from fall 2021. This spring I think I stumbled into a way to tackle this material.
I used the data from the prior Monday on linear velocity. On Monday the RipStik velocity of 1.42 m/s. The chart on the left side of the board reproduces the previous Monday's RipStik run. I then graphed the velocity versus the time. I asked "Where is the distance in this graph?" One student said, "distance is velocity times the time" which is exactly where the distance is in the second graph. The time is the length of the rectangle, the velocity is the height, so the area is the distance. We are at the very edge of calculus here.
Then I said, "But what if I increase the velocity at a steady rate with respect to time, instead of holding a constant velocity. Start from a speed of zero. How far will I have gone? The answer is still the area "under" the velocity line. A calculation of ½ base* height. But if I now double the time, how many "triangles" are under the new triangle? When we double the time, we double the height. We can fit in three of the original rectangles in the new area, for a total of four rectangles. If I triple the original time, I can fit in an additional five triangles for a total of nine triangles. Each additional time interval, shown above as a two second interval, adds an odd number of triangles and the sum of odd numbers of triangles are the sequence 1, 4, 9, 16, 25,... when I asked what the next number would be many students were still not seeing the pattern. Then someone said, "add eleven... 36!" Still, no one was yet "seeing" that these were squares: that the area was going up as the square of the time. And area is distance. Hence I went explicit with y = x².
I asked "What shape does y = x² make on a graph?" No one answered. I entered y=x² into Desmos on my phone and showed everyone in the class the shape I was getting. I then asked what the name of the shape is. I checked a few students individually, but they did not know. Only one student, with some reluctance, said, "Parabola?" I think the students were being honest - they did not remember the name of the shape. I think they must have seen the name once - almost all have completed MS 100 College Algebra and some completed MS 101 Algebra and Trigonometry.
I also learned that the students had not seen that the perfect squares are the sum of a sequence of odd numbers. That was also new to them. This is almost magical the first time one sees this connection. There is beauty in mathematics, but the typical algebra curriculum is so focused on algorithms for hand solving equations that the beauty is lost. The patterns, the symmetries, the things that intrigue.
I noted that my argument was a not a proof, just a diagram on a board. I also said I was not going to try to prove the argument mathematically, but was rather going to see if I could show that this is what happens when I ride the RipStik at a steadily increasing velocity.
I used a seven meter stretch of sidewalk, suggesting measures at 1,2,3,5, and 7. But one student opted to take measurements every meter, and their data looked promising, so I used that data.
The time versus distance data is the solid purple circles. This was perhaps the third run, and one in which I was slower and more deliberate in the first meter. Going out too fast is problematic for this exercise. The data well argues for a parabola. Below, in open blue circles, are the meter to meter velocities. My velocity increased in a moderately steady manner.
This approach felt right and makes visual sense. This also has a suggestion that ½ could play a role in the acceleration formula from the triangular area formula. If the base and height were both x, then one gets ½x² as the area, as the distance traveled by a RipStik with a unitary acceleration.
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