Acceleration day one
Summer 2019 I read Steven Strogatz's Infinite Powers: How Calculus Reveals the Secrets of the Universe. 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 retool 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 began class on Monday by sketching a time versus distance graph onto the board with a speed of 1.69 meters per second found from the RipStik run of the previous Monday.
I then sketched a time versus velocity graph for a constant velocity of 1.69 m/s on the board. I noted that on this graph the velocity is on the y-axis. If the velocity stayed the same, then the graph is a horizontal line. I noted the importance of labelling the axes properly.
Then I drew a diagonal line through the origin on the velocity versus time graph. I asked the class, "For a RipStik that has a velocity that increases linearly, where twice the time is twice the speed, what will the time versus distance graph look like? No one ventured an answer. So I asked the students to sketch what they thought the time versus distance graph would look like for the RipStik going faster and faster. There was a broad variety of answers.
The above notebook records the graphs I drew on the board.
Here a student proposed a linear solution. Linear solutions were a common choice along with simply leaving the graph blank. This problem, posed this way, demands thinking graphically, reasoning graphically, and at a fairly high level. This is general education program student learning outcome 3.2 Present and interpret numeric information in graphic forms made manifest.
Two to three students produced a half parabolic shape. Bear in mind that the class had just learned in laboratory two that slope is speed, thus increasing speed must be increasing slope. That said, these solutions were very insightful. At this point in the class I do not give any indication of which drawings might be correct.
This was another proposed solution. I then noted that science is not that which is taken on faith. Science is that which can be tested, a truth that emerges from the data or, put another way, an emergent truth. I used to focus on the philosophy that science is that which can be proven false, and I suspect that philosophically falsifiability is the correct approach to truth in science. Certainly statistical hypothesis testing runs on what can be shown to be false - either the null hypothesis can be shown to be false at some level of confidence or there is a failure to show that the null hypothesis can be shown to be false. Yet at the end of the day students happily climb into a piece of metal that leaves the surface of the earth using wings running on Bernoulli's theorem. Falsifiability may be philosophically sound, but the reality is that there is an emergent truth that arises from data.
I then put on my graph on the board a smiley face and said that I thought this is what the graph would look like for time versus distance for a RipStik going faster and faster.
I then took the class outside to make the RipStik start from a velocity of zero meters per second and then go faster and faster. I had a number of timers timing my runs, and I went with one who felt sure their times were accurate. I recorded the data to a poster sheet from the chronograph.
As one student noted with a touch of surprise, before I had added the line, "That's a curve." Time was running out in the class by this point - the first few minutes had been devoted to covering the quiz from the previous Friday. I entered a linear relationship v₁~mt₁ and the line could be seen to not work.
I then asked what I might do to get the line to fit the points better. That yielded no answer. I then asked if there was a name for the shape of the points. One student suggested a parabola. I asked how I might get that parabola. Use a squared came the answer. So I squared the time.
Note that I have deliberately omitted the lead coefficient of ½ as I have no way to explain the presence of that coefficient at this point. I will tackle that subtlety on Wednesday along with the deceleration and acceleration curve.
In a post hoc analysis my speed did increase relatively linearly.
That said, the triangular areas under the line are not coinciding with the distance covered, and the area under the above should accumulate to be the distance covered. I could use the lecture path of using triangles to get at the distance as the area under the line, as I did last term, but this proved complex and I had the sense that this did not lead to students to better understand the factor of ½ that comes into the acceleration equation.
The average acceleration is roughly 0.38, which is well under the 0.49 value that derives from multiplying 0.2454 by two. So there is not good clean explanation available down that path for the ½ either.
I do want to spend more time on the full parabola and what that graph will mean, the explanation used last term took too long and did not lead to any sort of clarity for the students. Ultimately I want the students to understand the graph and its relation to reality more than the geometric approach to integration and calculus.
The graphs provide a visual, qualitative approach, a hand waving argument that linearly increasing velocity produces something like a parabola on a time versus distance graph. The numbers do not easily support a numeric, geometric argument. Some terms the numbers do line up, the acceleration from v = at (on the order of 0.36 m/s² by the graph above, perhaps 0.38 m/s² based on the average acceleration) produces the same acceleration as the d=½at² equation (0.49 m/s² this term). When that happens, the path to arguing for the factor of ½ is open.
This term perhaps the approach should be approach that one can sometimes see places from a mountaintop that one cannot get to, may not ever get to. That ∫ a dt is v = at + v₀ and ∫ (at + v₀)dt is d = ½at² + v₀t + d₀ and that this is the form y = ax² + bx + c, a quadratic which produces parabolas on graphs. And then go create one with the RipStik.
I began class on Monday by sketching a time versus distance graph onto the board with a speed of 1.69 meters per second found from the RipStik run of the previous Monday.
I then sketched a time versus velocity graph for a constant velocity of 1.69 m/s on the board. I noted that on this graph the velocity is on the y-axis. If the velocity stayed the same, then the graph is a horizontal line. I noted the importance of labelling the axes properly.
Then I drew a diagonal line through the origin on the velocity versus time graph. I asked the class, "For a RipStik that has a velocity that increases linearly, where twice the time is twice the speed, what will the time versus distance graph look like? No one ventured an answer. So I asked the students to sketch what they thought the time versus distance graph would look like for the RipStik going faster and faster. There was a broad variety of answers.
The above notebook records the graphs I drew on the board.
Here a student proposed a linear solution. Linear solutions were a common choice along with simply leaving the graph blank. This problem, posed this way, demands thinking graphically, reasoning graphically, and at a fairly high level. This is general education program student learning outcome 3.2 Present and interpret numeric information in graphic forms made manifest.
Two to three students produced a half parabolic shape. Bear in mind that the class had just learned in laboratory two that slope is speed, thus increasing speed must be increasing slope. That said, these solutions were very insightful. At this point in the class I do not give any indication of which drawings might be correct.
This was another proposed solution. I then noted that science is not that which is taken on faith. Science is that which can be tested, a truth that emerges from the data or, put another way, an emergent truth. I used to focus on the philosophy that science is that which can be proven false, and I suspect that philosophically falsifiability is the correct approach to truth in science. Certainly statistical hypothesis testing runs on what can be shown to be false - either the null hypothesis can be shown to be false at some level of confidence or there is a failure to show that the null hypothesis can be shown to be false. Yet at the end of the day students happily climb into a piece of metal that leaves the surface of the earth using wings running on Bernoulli's theorem. Falsifiability may be philosophically sound, but the reality is that there is an emergent truth that arises from data.
I then put on my graph on the board a smiley face and said that I thought this is what the graph would look like for time versus distance for a RipStik going faster and faster.
As seen above, timing marks were at 1, 2, 3, 6, 9 and 12 meters
I then asked what I might do to get the line to fit the points better. That yielded no answer. I then asked if there was a name for the shape of the points. One student suggested a parabola. I asked how I might get that parabola. Use a squared came the answer. So I squared the time.
Note that I have deliberately omitted the lead coefficient of ½ as I have no way to explain the presence of that coefficient at this point. I will tackle that subtlety on Wednesday along with the deceleration and acceleration curve.
In a post hoc analysis my speed did increase relatively linearly.
The average acceleration is roughly 0.38, which is well under the 0.49 value that derives from multiplying 0.2454 by two. So there is not good clean explanation available down that path for the ½ either.
I do want to spend more time on the full parabola and what that graph will mean, the explanation used last term took too long and did not lead to any sort of clarity for the students. Ultimately I want the students to understand the graph and its relation to reality more than the geometric approach to integration and calculus.
The graphs provide a visual, qualitative approach, a hand waving argument that linearly increasing velocity produces something like a parabola on a time versus distance graph. The numbers do not easily support a numeric, geometric argument. Some terms the numbers do line up, the acceleration from v = at (on the order of 0.36 m/s² by the graph above, perhaps 0.38 m/s² based on the average acceleration) produces the same acceleration as the d=½at² equation (0.49 m/s² this term). When that happens, the path to arguing for the factor of ½ is open.
This term perhaps the approach should be approach that one can sometimes see places from a mountaintop that one cannot get to, may not ever get to. That ∫ a dt is v = at + v₀ and ∫ (at + v₀)dt is d = ½at² + v₀t + d₀ and that this is the form y = ax² + bx + c, a quadratic which produces parabolas on graphs. And then go create one with the RipStik.
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