Introduction
If you ask a student of physics what a vector is, you can expect the standard answer: “a quantity with a magnitude and direction.” They may even have an innate notion of why we care about direction so much in physics, but a deeper understanding of what vectors are and what they represent is something that takes experience and insight to develop.
Since the mathematical foundation of introductory physics is based on the concept of vectors, doing justice to physics requires using vectors and using vectors requires that students learn some “new math.” Here lies the challenge: How can I teach the foundational math in a way that is more intuitive and accessible so that my students’ conceptual understanding is not limited by a computational bottleneck?
My students arrive having never really been held truly accountable for considering units. They do not have much experience making measurements. Quantities they have studied are typically only related to countable amounts of objects, areas and volumes. Physics cares about numbers in lots of exciting ways that students have never had to care about them before. Positions change as time elapses, and the way you describe a space dictates how you observe it and how you quantify the observations made.
The purpose of this unit is to provide a language and understanding of number lines as coordinate systems we can construct and place at will in the physical world. Units are intrinsically tied to the coordinate systems we create and measurements are made against the coordinate systems in multiples of the units. More than anything, I want my students to realize that arithmetic operations change things, and understanding physics involves understanding of stretches, sides and spins applied to objects in our world.
Demographics
I work at a small magnet high school with an incredibly diverse student population. Despite being a magnet with academic entrance requirements, it is not atypical for students to make it to 11th grade lacking expected math skills. With the exception of having one section of AP Physics 1, our school does not track students for my classes, so I have a full range of abilities in each classroom. In addition, the population of students I teach is indicative of the demographics of Philadelphia as a whole. I have a large number of recent immigrants and English language learners, both from Southeast Asia and Latin America. The wide range of backgrounds and skill levels I have in my room on any given day is always in my mind as I plan.
This is designed to be the second unit for my 11th grade General Physics and AP Physics 1 classes. My school is set up so that physics is the required science course for all juniors. Over two thirds of my 150 students will be concurrently enrolled in Algebra 2. The remaining one third will be in a Precalculus class. Of the students in Algebra 2, a small handful will be simultaneously repeating Geometry. Of the students in Precalculus, twenty or so might also be in AP Statistics. Because of this huge variety, I am targeting students who have passed Algebra 1 and Geometry and allowing room to differentiate my instruction toward students who come in above or below this mark. It is also important to note that this unit does not depend on any specific physics content knowledge, with the exception of some connections to introductory kinematics. The unit, with little modification, could also fit well into a Geometry, Algebra 2 or Precalculus class offered at the high school level that is concerned with grounding mathematics in the physical world, measurement or transformations.
The final demographic consideration is that I teach in the Philadelphia School District, which is the eighth largest school district in the country, serving the entire city of Philadelphia. The financial state of the district is such that schools function with operating budgets so slim that basic needs go unfilled on a daily basis. Realizing this as my own reality and the potential reality for other teachers, the activities and strategies are designed to be free or extremely cheap.
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