### Reflection and refraction

Laboratory eleven centers on two linear regressions and continues the course emphasis on mathematical models underlying physical systems.

The first system is the object versus image distance for a plane mirror. This is a simple enough system that classes usually hypothesize the distances to be equal, which implies a slope of unity. This often becomes the hypothesis against which to test the theory. Each term this hypothesis picks up the name of the student who proposes the hypothesis. This term Gayashalane Augustine proposed an equal distance theory.

Trisden set up to explore the Augustine law of reflection

Lexus and Maxon determine and record measurements

Osbert holds the mirror

Gayshalane stands back and observes, Francina on the right

Ashley locates the image of the object

In the second half of the laboratory the students use apparent depth to determine image and object distances for submerged pennies. The slope of this relationship should be the index of refraction for water, about 1.333.

Vandecia, Diane, Misko working with the mirror

This term the arrival of a Chromebook has me working almost exclusively in Google Docs. The recent addition of the LibreOffice.org statistics functions in Google Docs has brought the full array form of the LINEST function to Google Sheets. With that information 95% confidence intervals can be calculated for the slope and intercept.

The 95% confidence interval for the slope of the mirror data included a possible slope of one, leaving the Augustine slope theory standing.

Nagsia and Michelle record data

The 95% confidence interval for the slope of the apparent depth data ran from 1.26 to 1.56, well bracketing 1.333 for the group above. There was at least one group who underestimated the apparent depth, finding deeper depths than expected, which reduced their slope (image distance is on the x-axis) sufficiently that a 95% confidence interval did not capture the index of refraction for water.

Beverly Ann observes as Josey, Yuta, and Anster make measurements

In class I do not discuss the 95% confidence interval for the slope and intercept. For those students who have either had statistics or who are in statistics, I do, if I get the chance, show them the confidence interval calculations for the slope. For the students who have not had statistics, the confidence interval concept is a bridge too far. These are students who still struggle with the physical meaning of the slope.

Nagsia, Glenn, Michelle

That said, this sort of laboratory provides a vision of where one could take a mathematics class that was untethered from the traditional subject area constraints of mathematics. College algebra focuses on polynomials. Statistics focuses on statistical measures. While statistics includes linear regressions, the topic is but a single chapter. Imagine instead an algestats or statigebra course where a linear regression is generated and then chased all the way to the 95% confidence interval and p-value for the slope against an expected value. What stops such a course? Finding the program that such a course would belong in. Explaining such a chimera to a curriculum committee. Lack of articulation and transfer options for a course that exists nowhere else. The need to build support materials from scratch to support students in such a hybrid. There exist some quantitative foundations courses - transition courses from mathematics to algebra for developmental students, but rare must be the actual college level combination course.

Kanga the kangaroo