What resources (Desmos, Geogebra, math3d, etc.) have you built to help your students better understand or visualize topics in mathematics?
6 Answers
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Fundamental Theorem of Calculus Explanation
Error bound on integral for increasing function
"Secant Circles" and "Tangent Circles"
- Some students of mine worked out what would happen if you find the unique circle passing through three nearby points on the graph of a function, and then take a limit as those points get close to each other. Similar idea to the tangent line development, but with circles. They ended up needing Taylor series to get the exact equations for the "tangent circle"!
- This visualizes the one form by looking at level sets of the covector in a small rectangle around each point. You integrate the 1-form along a curve by just counting how many level sets you cross (taking into account the spacing $h$ between level sets).
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2$\begingroup$ Hah I like how we both have very distinctive styles with Desmos demos. These are great! $\endgroup$ Commented Aug 4, 2022 at 19:15
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$\begingroup$ In my opinion, tangent circles make more sense in the more general context of plane curves (say, parametrized curves) rather than the particular case of the graphs of functions. The (reciprocal of the) radius of said circle is perhaps the most common definition of the curvature of a curve. $\endgroup$– ArthurCommented Aug 5, 2022 at 6:38
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2$\begingroup$ @Arthur I agree, but these were Calc 2 students who were "discovering" all of this on their own. $\endgroup$ Commented Aug 5, 2022 at 11:46
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Here are some physics-related ones... many not quite as polished as they could be.
update:
webvpython / glowscript (also supported on trinket.io)
- EMwave - EMwave on trinket.io - an interactive 3D animation that shows how the Faraday and Ampere-Maxwell laws propagate the electric and magnetic fields in a plane electromagnetic wave.
desmos
- add Covectors (equipotenials) (draft version)
- covectors map a vector (draft version)
- 161-robphy-Electric-VectorField-Superposition-v2 (draft)
- Adding Equipotentials - point charges
- Circular Wave Interference on a Screen (play the time variable)
- spacetime diagrammer for relativity v.8e-2021 (time upward)
https://www.geogebra.org/u/robphy
- Vector Fields, Divergence and Curl, and Flux and Circulation Integrals
- Spherical Coordinates
- Refraction-Snell Law
- CircularLightClocks-VisualizingProperTimeInSpecialRelativity
- Relativity-LightClock-MichelsonMorley-2018
- Relativity on Rotated Graph Paper (robphy) - MAA2016
- Virtual Torque Lab
- Vernier Caliper
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2$\begingroup$ I love the spherical coordinates demo you have. I had a kind of crappy version that was not remotely as clean. Having the volume element represented is fantastic. $\endgroup$ Commented Aug 6, 2022 at 1:18
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2$\begingroup$ The special relativity Desmos demo is also great. $\endgroup$ Commented Aug 6, 2022 at 1:19
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Desmos
- Clickable Venn diagram with three sets and universe set
- Unit circle
- Secant lines motivating instantaneous rate of change
- Tangent lines
- Rolle's Theorem
- Mean Value Theorem (for derivatives)
- Slopes/concavity for building questions from
- Inverse Function Theorem
- Riemann sums with arbitrary (random) sample points
- Riemann sums with left/midpoint/right sample points
- Upper and lower Darboux sums
- Method of exhaustion for area of a circle (part of a scaffolded assignment I give the students)
- Riemann sums for areas between curves
- Areas between curves
- Trapezoidal rule
- Simpson's rule
- Arc length
- Fourier series (warning: slow)
Geogebra
- Volumes with equilateral triangle cross sections
- Volumes with isosceles right triangle cross sections
- Volumes with square cross sections
- Volumes with semicircular cross sections
- Volumes with circular cross sections
- Volumes with washers
- Volumes with washers
- Volumes with cylindrical shells
- Volumes formed by rotating around horizontal line
- Volumes formed by rotating around vertical line
- Cavalieri's principle
Colab
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Automatic-Algebra.org
I built this website for the purpose of timed self-quizzes on topics that should (mostly) be prerequisite, "automatic" knowledge prior to a number of basic college courses. Available quizzes include:
- Times tables
- Negative numbers
- Order of operations
- Sets of numbers
- Special products
- Finding intercepts
- Special lines
- Graphing lines
- Decimal rounding
- Decimal relations
- Decimals to percent
- Sampling distributions
- Chained relations
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I usually just use html/SVG combination and do it on a case by case basis. Examples include
https://www.math.kent.edu/~nazarov/switches/switchgame.html (Linear system in $Z_2$)
https://www.math.kent.edu/~nazarov/Hillcipher/hill.html (Linear cipher)
https://www.math.kent.edu/~nazarov/Buffon/buffon.html (Buffon's needle)
https://www.math.kent.edu/~nazarov/T-intersection/tint.html ("real life" linear equations problem (ill-posed without prohibiting left turn))
https://www.math.kent.edu/~nazarov/turmach (Turing machine emulation)
There is more stuff in that directory (mainly games made for Math. Club). It doesn't relate to the question directly, but you are welcome to take a look at it too just to see what can be done in two or three evenings with basic HTML skills (except Jeopardy: that one took a couple of weeks and was a work of several people). Some of these are not immediately obvious how to operate, but if you get interested in any of them, just let me know.
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I've put together a number of online interactives in Javascript for College Algebra, Precalculus, Statistics, and Linear Algebra, available here: https://trkern.github.io/