Illustrating Volumes of Solids with Known Cross Sections

My posts are few and far between these days.  I hope that changes soon, but for now, it’s just a post here and there when I can.  And today I can!  Let me share some fun Desmos-ing with you!


In this post:

  • A new Desmos feature: how to use labels!  
  • Attending to the clarity and usability of graphs.  
  • Fun cross-sectional solids graphs to play with!!  What could be better?

It’s no secret with my colleagues that I love Desmos.  For real. SO. MUCH.  And I love it when people ask me to help them look for Desmos graphs or activities for particular topics.

Which brings us to today. One of the calculus teachers I work with was asking me about whether I knew of any good Desmos graphs of rotational solids.  I sent him some awesome graphs made by Geoff Patterson.  Graphs that use such advanced math that my jaw drops with awe.  Seriously.  If you haven’t given his blog a look, do yourself a favor and check it out.  Pronto. Here’s a link:


This picture is of one of Geoff Patterson’s creations. Image can be found at

But what I didn’t know Continue reading


Circular Thinking

My freshman geometry class has been asking me all year when we would be able to use Desmos (they know how much I love it.)  Now is there time.  We are doing a whole unit on coordinate geometry, and so we’ve been Desmos-ing it up.

Here’s the activity we did in class: Circles Activity.

One of the slides in the activity allowed the students to make a simple drawing out of circles.  I suggested a Mickey Mouse silhouette or the Olympic Rings.

While most students decided to make the Mickey Mouse Silhouette, a couple went a bit further, one even asking how to make an ellipse.

Circle Activity Screenshot.

Face 1

Of course, working with circles got me to thinking about them.  In particular, I got to thinking about how to make use of lists to get Desmos to graph some pretty cool sets of circles.  Here’s one that I came up with:

gifsmos (3)

…and that led to a quick exploration of cycloids and epicycloids…

Cycloid and Epicycloids:

gifsmos (5)

…followed by a little project to trace the path made by the center of a circle that rolls over a sine curve.  Here’s what I came up with:

gifsmos (7).gif

I’m pretty sure that my way of making this happen is not the most efficient way.  Do you know a better way to do this?  Let me know in the comments!

Thoughts for where to go from here: at some point, I’d like to trace a point on a the edge of a circle as it rolls over a sine curve.

Tracing Parametrics, Chains of Events, and Constructing a Pentagon

Last week I attended a PD session by Suzanne Gaskell, an art teacher at the high school where I teach.  She presented on a work of art she has created and calls the Kavad of a Sacred Geometer.

This work of art is a wooden box that tells the story of ancient geometry and its relation to various sacred traditions.  After her presentation, she instructed us on how to perform some interesting constructions, including one I had never done before, the regular pentagon.

It inspired me to create an illustration of the construction for a regular polygon using Desmos.  Here’s what I came up with:

Pentagon Construction


First skill: how to trace a parametric curve.  Prior to creating this, I didn’t know how to make a parametric curve trace itself out.  Here’s how to do it: Continue reading

Using Restrictions in Innovative Ways

Let’s talk restrictions.  So helpful, so amazing.  I’ve seen people use restrictions in some very creative ways lately, so here’s the lowdown on cool ways to use restrictions.  Make sure to scroll to the end to see an Angry Birds activity that makes extensive use of restrictions.

First, the classic use: restricting x and/or y.

You can restrict just the x values, or you can restrict just the y-values, or you can stack restrictions.  Desmos even knows the difference between AND and OR:

Here, I’ve shown a graph with x > 0 OR y > 0
In Desmos, type the restrictions in the same bracket, separated by a comma: {x>0,y>0}

BTW, this is super helpful if you want to do something like this:

Restriction by List 2

Here, I’ve shown a graph with x > 0 AND y > 0
In Desmos, type the restrictions in separate brackets: {x>0} {y>0}

The team at Desmos has also made it possible to place restrictions on both x and y at the same time in an implicitly defined inequality.  Here is one example:

Link to Desmos: see how this is done!

Second: restricting parameters.  This allows certain aspects of the graph appear or disappear depending on a certain parameter.  It also can be used to make whole images appear or disappear.  Here are some of my favorite examples lately:

Here I built upon a graph that Desmos had featured, and set it up to turn on tangents by sliding a point on the graph.  This technique is awesome for building teacher activities where you want to make a graph into an interactive exhibit (i.e. students can’t access the expression list, just the  graph itself.)  Check out the graph here.

For another example, here is a tweet by Stefan Fritz showing a fantastic Angry Birds activity he created that gives students feedback as they work through the clever use of restrictions on parameters.  Here’s the link to the teacher activity for anyone who wants to use it!  Such a fun activity!


Sinusoidal Fun

Link to some totally rad sinusoidal picture graphs I’ve been working at lately.  The first 3 pictures have movement!

Turns out sinusoidal curves can be amazing for creating really cool animations.

All 4

These pictures were part of a nerdy Valentine to my husband.  (Valentine? Yup!  These pictures are Desmosified versions of some of our favorite things!  And, he’s nearly as nerdy as me, so it’s cool.)  I already showed them to him because I was too pumped up on Desmos-ing to wait until the 14th to share them with him.

I hope you enjoy!

Peace out Desmos world!

Ideas for Desmos PD Day

On Feb. 16, I will be running a PD session for my department on how to use Desmos effectively in the classroom.

I would say more than half the department is coming with little to no experience with, and limited experience with the calculator.  I am planning on assuming no prior knowledge.  We will have 3 hours together.  Everything is still a work in progress, but I’m hoping to get suggestions and feedback!

So far, here’s the plan: Continue reading

Rotating a Function

You can pretty easily use parametric equations to rotate a function through any angle of rotation.

  1.  Define a function, f(x)
  2. Either choose an angle measure, a, or leave a as a slider and type in this parametric equation: (t·cos f(t)·sin a, t·sin a+f(t)·cos a)
  3. You’ll need to specify the values of t.  I generally use -20 to 20, because that will cover what is visible in a normal zoom.  If you want to zoom out quite a bit, though, you may want larger values as your bounds on t.
  4. If you want to animate the rotation, specify the bounds on the slider a.

Check out this example!  Click on the Continue reading