desmos.com
Dynamic and Dynamite Desmos Demos

A helpful resource to get you more familiar with the great things Desmos can do!

An attempt to assemble as many of Desmos' useful features on one page.


Some useful links from the folks at Desmos.com

Desmos home page

Desmos User’s Guide pdf (a great resource for beginners!)

Desmos Quick Start Guide pdf (If you're too impatient for the User Guide)

Learn Desmos (a collection of instructional videos categorized by level from 

                          the folks at Desmos)

Youtube video tutorials (more instructional videos made by the folks at Desmos)

Math examples (Desmos staff picks)

Creative art (Desmos staff picks)

Classroom Activities (searchable activities that are "Designed by Teachers. Built

                                     with love by Desmos.")

Desmos Activity Builder ("A DIY tool for creating Desmos-based activities")

Daily Desmos (a daily challenge for you and your students to reproduce given graphs)                                 (not actually by the folks at Desmos but still fun & often challenging)


Saving your graphs

First you will need to set up an account; you can either create your own new Desmos account or sign in using a Google account. You can then save graphs by clicking on the green floppy disk button in the upper left. Have any high school students ever seen a floppy disk?


Restricting x and/or y values


Click an image below to open that Desmos graph in a new tab. 

Piecewise functions

Here is a way for Desmos to draw a piecewise function. The syntax is different from all other functions in Desmos. I imagine they will improve this at some point.

Click the image below to open that Desmos graph in a new tab. 

https://www.desmos.com/calculator/mswqtujl9r

Sliders for dynamic transformations & animations


If you type a variable other than x or y, Desmos will automatically ask if you’d like to create a slider for that variable.

Click here for a video tutorial from Desmos about sliders animations. 

Click an image below to open that Desmos graph in a new tab. 

https://www.desmos.com/calculator/k9lutkm8t1
 
https://www.desmos.com/calculator/pnvqceta2a

https://www.desmos.com/calculator/virpjtckyg

"Whitney Chromatic" by Bryan P Johnson
         https://www.desmos.com/calculator/xxnpucqbeh

         (Click here for a beautiful non-Desmos version with sound!)


Lists (to create multiple related graphs simultaneously)


The newest Desmos feature...

To use lists, type a variable equal to a list of numbers (separated by commas) inside brackets in one entry cell, and then type an equation for a graphing using that variable. All of the various graphs will be graphed simultaneously. 

To create a list longer list, set the parameter variable of your choice equal to

[ <value1> , <value2>... <final_value>] 

The first two values will determine the increment. The final value will determine an upper limit. Separate your second value from your final value by three periods (no commas). 


Click an image to open that Desmos graph in a new tab. 

https://www.desmos.com/calculator/pz518jxcss


Here's a list where the values can be changed (and animated) with a slider:

https://www.desmos.com/calculator/3ulzcag72z


Here's a 40 element list (where the values form an arithmetic sequence):

https://www.desmos.com/calculator/f16foocy7o
(To learn more about polar graphing, click here.)


Here's a pretty amazing use of lists that was shared by the folks at Desmos:

https://www.desmos.com/calculator/vcjn07wybe


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Function notation

Desmos understands function notation too! This can be very useful for teaching transformations and compositions of functions.

For example, you can define f(x), then graph or 2f(x - 3). If you’ve also defined g(x), then you can graph f(g(x)).

Click an image to open that Desmos graph in a new tab. 

https://www.desmos.com/calculator/foghsbxiv4

https://www.desmos.com/calculator/gkayfjtjvc



Inequalities


Click here for a video tutorial from Desmos about inequalities.

Click an image to open that Desmos graph in a new tab. 




Apparently repeating an inequality can make it "rise to the top" of the layers of overlapping graphs, as the document linked below illustrates...



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Inverse relations/functions

I've found Desmos very useful for teaching the properties of inverses to my students since graphing inverses is as easy as switching x and y - the symmetric properties of the two graphs are much easier for students to see, especially if you add sliders. 

Click an image to open that Desmos graph in a new tab. 
 
https://www.desmos.com/calculator/ut2qa32i7q


Movable points

Slider variables can be attached to a point that you can move around.

Click here for a video tutorial from Desmos about movable points. 

Click an image to open that Desmos graph in a new tab. 

https://www.desmos.com/calculator/1kvnubcsz8


https://www.desmos.com/calculator/2xwqfm8fni

Adding your own images to your graph


Click here for quick instructions on how to add images to your graphs.

You can even adjust the image's center and dimensions with sliders.  

Click the image below to open that Desmos graph in a new tab. 

https://www.desmos.com/calculator/exfp8szpji


Statistics - Data & Regression


Click here for a video tutorial from Desmos on Tables. 

Regressions drawn to fit your data are a new feature that many teachers have been asking for for some time. It will also plot residuals if you want to see them. 

Click the image below to launch tutorial from Desmos on the Regression tools in a new tab.        

 https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CB4QFjAA&url=https%3A%2F%2Fwww.desmos.com%2Fcalculator%3Ftour%3Dregressions&ei=lmqkVJqWBoGfgwTDz4HQCg&usg=AFQjCNEART5saAR8CBpu2aMqVNVzI8XsKQ&sig2=0O0DOMxlljBpITKzIZoTGA

Click here a video summarizing Regression tools from Desmos.

The regression graphs aren't limited to just linear functions; any type function can be "fitted" to your data. 


Transformation Project with Examples

Click here to open (right-click to download) our warm-up assignment.

Click here to open (right-click to download) the Transformation Project assignment.

NOTE: Tom Reardon (Austintown Fitch HS and Youngstown State University) originally created this project with input from Ellen Browne (Pomfret School). We thank them for their assistance and willingness to share.

Here are some examples from our Algebra 2 classes’ Transformation Project. Each image below is hyperlinked, so click on an image to open that student’s project. Each drawing in the top two rows includes some animation!

https://www.desmos.com/calculator/1vn0xjjtng         https://www.desmos.com/calculator/3obymiwjec     


https://www.desmos.com/calculator/hgniwhasfo        https://www.desmos.com/calculator/544qx94sez 


https://www.desmos.com/calculator/wgc9adygki         https://www.desmos.com/calculator/rpwjei7r6d

        

   
              https://www.desmos.com/calculator/qp56tfcv88  https://www.desmos.com/calculator/rl8p0kzist

This was not made by one of our students, but it is pretty amazing (shared by the folks at Desmos):
https://www.desmos.com/calculator/q4ctl6mhrz
"Snowglobe" by Shanmin


Distance function/locus graphs



Desmos has a Distance function which you can use if you name at least two points. Naming a point with coordinates (x,y) lets you create locus graphs.


Click an image to open that Desmos graph in a new tab.










p = eccentricity (the variable "e" was already taken)
0<p<1 is an ellipse; p = 1 is a parabola; p > 1 is a hyperbola



The sum of three distances is a constant... fun to think of real-world applications:



The product of three distances is a constant? Are there any real-world applications? I don't know but it's fun to drag the points around:

https://www.desmos.com/calculator/a8tvckfdnu


Parametric functions


To draw a parametric function, create a point where the coordinates are functions if t
You can lick on the upper and lower bounds of the domain to change them. 

Click an image to open that Desmos graph in a new tab.  

https://www.desmos.com/calculator/tb8avdktax


https://www.desmos.com/calculator/fe8entyfzk


https://www.desmos.com/calculator/qepemyowpv

https://www.desmos.com/calculator/rqwgxu7sym

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Polar functions


To create a polar graph, type r equal to the function you want. To type θ, you can either type the word theta or find it in the alphabet keyboard within Demos. 

If you want to use the function in other parts, name the function r(θ) instead of just r. 


Click an image to open that Desmos graph in a new tab. 

https://www.desmos.com/calculator/dswnc6cypl


You can set a slider variable equal to the upper bound.

https://www.desmos.com/calculator/f5glcops68

https://www.desmos.com/calculator/y3kcytvz4i


https://www.desmos.com/calculator/rcmt7rstov

Here’s the problem that, after a bunch of calculus, yields the polar equations found above: Starting from the corners of a square of side 10, each bug chases the one clockwise from it. If they all start at the same time and run at the same speed, how far has each bug run by the time they all collide at the center of the square?

Calculus - Derivatives


d/dx can be found under the "calc" tab on in app keyboard. You can also type 'd/dx' (and then press the right arrow key). 

Click an image to open that Desmos graph in a new tab. 

https://www.desmos.com/calculator/wh0bf3dvjm


If a function is defined, you can also graph its derivative using apostrophes. For example, define f(x) and then simply type f'(x) or f''(x) for a second derivative (two apostrophes, not a quotation mark). In fact, you can keep adding apostrophes to graph more and more derivatives.
 
https://www.desmos.com/calculator/kdftjtmbaa


Calculus - Tangent lines


Here's a way to draw a movable tangent line to a function. 

Click the image below to open that Desmos graph in a new tab. 

https://www.desmos.com/calculator/8cysfnncdf 


Calculus - Integrals


An integral can be found under the "calc" tab on in app keyboard. You can also type 'int'.

The bounds of the integral can be constants, functions, or even variables attached to sliders or lists.

Click an image to open that Desmos graph in a new tab.

https://www.desmos.com/calculator/1qsalfuczx

https://www.desmos.com/calculator/rtjx3jki7n

https://www.desmos.com/calculator/32geeecuu3


Calculus - Functions defined with sigma notation

Desmos can quickly and easily graph Taylor Series and Maclaurin Series written in their sigma notations. 

You can get the sigma by typing “sum” or it can be found in the functions menu under the “calc” tab. Let the upper limit be k.

You’ll then be asked if you want to create a slider for k. Click on an endpoint of the slider, set the endpoints to be 0 and some positive integer. Make the step equal to 1.


Here is the Maclaurin Series for sinx:

Click the image below to open the Desmos graph in a new tab. 

https://www.desmos.com/calculator/inrdcmnbad


Here is the Maclaurin Series for e^x:

Click the image below to open the Desmos graph in a new tab. 

https://www.desmos.com/calculator/owtqulxffw


Here is a Fourier Series example:

https://www.desmos.com/calculator/sgujtkkoip

Desmos can also graph functions defined as products:


Thanks


I tried to cover as many of the features I know Desmos has. I hope you found this all useful and that you and your students will enjoy using Desmos. Thanks to my students and colleagues for helping me discover Desmos. 

Please feel free share with me any interesting things that you or your students create with Desmos. I'd love to learn more! 

This page was originally created for a CWIC titled "Dynamic and Dynamite Desmos Demos" at the 2014 Anja S. Greer Conference on Mathematics, Science & TechnologyPhillips Exeter Academy. Thanks to them for letting me make the presentation. If possible, I urge any math teacher to attend this amazing conference.

Thanks, 

Chip



(617) 800-2161





Disclaimer: I have no affiliation with Desmos.com. I’m simply impressed with their graphing software and wanted to share how useful it can be. 

Update: About a year after I created this site, the folks at Desmos thanked me by mailing a package full of Desmos schwag - some stickers, a t-shirt, and Desmos sunglasses!