Students often do not (for lack of time, motivation, skills, reward) connect their mathematical work to their conceptual understanding. This page will house a collection of curricular pieces and references that guide students to connect meaning and mathematics in physics courses.
Most of these pieces were written for the upper level undergraduate classical mechanics course on the level of Marion and Thorton or Cassiday and Fowles. We refer the interested instructor to Intermediate Mechanics Tutorials for other useful curricular pieces for the same class. Some of the tutorials below assume students have done specific IMT tutorials already; this will be in the "notes" column. All of these have been class-tested with 2-3 groups of 15-20 students.
|Level||Topic and format||Notes||Files|
|Introductory||Electric field due to a bar of charge; guided inquiry tutorial and homework||Helps students become develop a view of integration as a sum of physical quantitities.|
|Sophomore-junior||Scaling Equations; guided inquiry tutorial, overview sheet, and AAPT talk||To show how scaling can help see the story that an equation tells.|
|Sophomore-junior||Phase space for a damped harmonic oscillator; guided inquiry tutorial. Physics topics: simple harmonic motion, but also charged particle in magnetic field and non-inertial reference frame are solved by coupled linear differential equations.||To show how to use the eigenvalues and eigenvectors of a coupled, linear differential equation can be used to understand the organization of phase space of a damped harmonic oscillator. This piece assumes students understand phase space rules at the level of theIMT tutorial on "phase space - simple harmonic motion."|
|Sophomore-junior||Identifying separable differential equations; worksheet. Physics topics: velocity dependent forces, rockets,||This sheet is practice (not guided inquiry) on recognizing separable diff eqs. The second page gives students a chance to see why second order diff eqs are never separable.|
|Sophomore-junior||Solving first order differential equations numerically; guided inquiry tutorial; physics topics: oscillators with non-linear damping||This sheet requires access to some programming language. It builds on students' understanding of the derivative as a measure of change.|
|Sophomore-junior||Finding roots of transcendental equations; guided inquiry tutorial||This sheet guides students to see how a linear approximation to a function creates a simple root-finding method.|
|Sophomore-junior||Connecting conditions on integrals to physics; worksheet. Physics topics: velocity dependent forces, central force problems||This is a very simple worksheet to give students a chance to see that conditions on integrals are connected to different physical (or unphysical) situations.|
|Sophomore-junior||Inertia tensor; guided inquiry tutorial.||This helps students understand what it means for inertia to be a tensor and not a scalar. As a supplementary topic, the matrix sheet just reminds students how to tell if vectors are parallel.|
|Sophomore-junior||Solutions to linear, any order, constant coefficient, homogeneous differential equations. Physics topics: simple harmonic motion.||The purpose is to help students understand why x(t)=Re^(nt) is a solution.|
|Sophomore-junior||Modified Atwood with a massive string using Lagrangians.||This is a guided problem solving worksheet that asks students to a) get a foothold, b) choose a method, c) solve the math and d) check and learn.|
|Sophomore-junior||Equations of motion for a ball tossed upward from the surface of the earth, including coriolis force.||This is a guided problem solving worksheet that asks students to a) get a foothold, b) choose a method, c) solve the math and d) check and learn.|
|Sophomore-junior||Taylor series; worksheet (this is not really guided inquiry, but hits some key confusions)||Reminds students of how to find Taylor series and convergence issues; introduces the idea that the expansion variable must be unitless; does an example; gives a practice problem|