Reflections

For reflection the students measured the object distance of an object from a mirror and the image distance behind the mirror. Mirror tiles that were still present in the laboratory cabinet four years after having been purchased and left behind along with flat decorator marbles for objects provided the equipment for the laboratory. Yard sticks marked in centimeters on one side provided measurements. 

Elise sets up for measurements

As the students started to make measurements I asked about possible slopes. Suggested slopes of zero, one, and two were offered. 

Judah locates the image while Mary measures

Sina holds the mirror

Renee-Ann-Seshrue and Sue Malia Crystal consider a measurement

Recording data

I had suggested making at least five measurements, which led to each group making only five measurements. 


Mikey suggested that the data he gathered with Elise was probably linear, so that was a starting point for the discussion of the data. The class had settled on d₁ for the object distance and i₁ for the image distance, or i₁~md₁+b for a linear model. Judah felt the data he gathered with Mary was probably also linear. CharlynRose, Sue Malia Crystal, and Jeffan thought their data was probably parabolic, so I suggested a full quadratic equation, i₁~ad₁²+bd₁+c, which fit their data perfectly. Their data did appear to be parabolic. as sketched on the board above, the points forming a smooth visual curve. This left the data for Sue Malia Crystal and Renee-Ann-Seshrue. I suggested a general cubic equation, i₁~ad₁³+bd₁²+cd₁+d which fit their data almost without residuals. Desmos recognizes d₁ and d as being different variables, but this choice was, in retrospect, suboptimal. 

While the quadratic and cubic equations would be questioned by those with a knowledge of optics, the intent was to let the data guide the equations, allow the equations to emerge from the data, the continuation of the theme of emergent truth in the course. Two of the four groups converged on a linear hypothesis, two opted for a non-linear interpretation of their data. That is how science is - mathematical models of new physics rarely arrive in their final form. Sometimes they do, but often there is a time during which a conversation is held that leads to the mathematical equations governing a system.

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