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Explore modeling challenges in a high school Differential Equations course. Teach and learn through hands-on experiences focusing on real-world contexts. Suitable for advanced math students.
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Modeling Challenges for a High School Course on Differential Equations Ted Theodosopoulos
The Main Idea • Modeling challenges as pedagogy • Teach and learn by doing • Focus on the context • Experiences rather than content • Class description • Spring semester elective • 8 seniors, 2 juniors and 1 sophomore • All have taken Multivariable Calculus
The Main Idea too… • History • Pure/applied versions of semester course at Swarthmore • Last two months of year-long Calculus 2 at Saint Ann’s • Project-based Multivariable Calculus at Worcester Academy • WHAT IS an Inquiry-Driven Environment? • Start with questions… • Students lead with questions... • End with questions…
The Nuts and Bolts • Topics • 1st order • Linear higher order • Systems • Linear • Nonlinear
The Nuts and Bolts • Techniques • Power series • Transform methods • Phase diagrams • Topics • 1st order • Linear higher order • Systems • Linear • Nonlinear
The Nuts and Bolts • Techniques • Power series • Transform methods • Phase diagrams • Projects • What do I provide • What do I expect • What have I observed • Topics • 1st order • Linear higher order • Systems • Linear • Nonlinear
The Nuts and Bolts • Techniques • Power series • Transform methods • Phase diagrams • Projects • What do I provide • What do I expect • What have I observed • Mathematics is everywhere • Teamwork is essential and indispensable • Topics • 1st order • Linear higher order • Systems • Linear • Nonlinear
Attraction and Repulsion Consider the motion of a particle in one dimension (x-axis) under the simultaneous influence of two forces.
Attraction and Repulsion Consider the motion of a particle in one dimension (x-axis) under the simultaneous influence of two forces. One force repels the particle from the location x = 1, while the second attracts the particle towards the location x= -1.
Attraction and Repulsion Consider the motion of a particle in one dimension (x-axis) under the simultaneous influence of two forces. One force repels the particle from the location x = 1, while the second attracts the particle towards the location x= -1. In both cases the force is inversely proportional to the distance from x = 1 and x = -1 respectively, but the proportionality constants may be different.
Attraction and Repulsion Consider the motion of a particle in one dimension (x-axis) under the simultaneous influence of two forces. One force repels the particle from the location x = 1, while the second attracts the particle towards the location x= -1. In both cases the force is inversely proportional to the distance from x = 1 and x = -1 respectively, but the proportionality constants may be different. Does the motion of the particle change qualitatively as you vary the proportionality constants?
Viscous Ballistic Motion Consider the motion of a projectile that is given an initial velocity and travels under the influence of gravity through a viscous medium, in which friction is proportional to the square of the velocity.
Viscous Ballistic Motion Consider the motion of a projectile that is given an initial velocity and travels under the influence of gravity through a viscous medium, in which friction is proportional to the square of the velocity. Is it possible to fire one such projectile so that it intercepts another one that was fired from the same location a certain amount of time earlier?
Viscous Ballistic Motion Consider the motion of a projectile that is given an initial velocity and travels under the influence of gravity through a viscous medium, in which friction is proportional to the square of the velocity. Is it possible to fire one such projectile so that it intercepts another one that was fired from the same location a certain amount of time earlier? After back and forth…
Viscous Ballistic Motion Consider the motion of a projectile that is given an initial velocity and travels under the influence of gravity through a viscous medium, in which friction is proportional to the square of the velocity. Is it possible to fire one such projectile so that it intercepts another one that was fired from the same location a certain amount of time earlier? After back and forth… FRICTION IS ALWAYS IN LINE WITH VELOCITY
Spinning Spring Consider a hollow metal tube that has a spring inside it.
Spinning Spring Consider a hollow metal tube that has a spring inside it. One end of the spring is anchored at one end of the tube and the other is attached to a mass, still inside the tube.
Spinning Spring Consider a hollow metal tube that has a spring inside it. One end of the spring is anchored at one end of the tube and the other is attached to a mass, still inside the tube. Both the mass and the spring can move freely and without friction inside the tube.
Spinning Spring Consider a hollow metal tube that has a spring inside it. One end of the spring is anchored at one end of the tube and the other is attached to a mass, still inside the tube. Both the mass and the spring can move freely and without friction inside the tube. You hold the tube from the end where the spring is pinned and spin it in a circular motion.
Spinning Spring Consider a hollow metal tube that has a spring inside it. One end of the spring is anchored at one end of the tube and the other is attached to a mass, still inside the tube. Both the mass and the spring can move freely and without friction inside the tube. You hold the tube from the end where the spring is pinned and spin it in a circular motion. What is the motion of the mass at the end of the spring?
