A Graphing and Learning Journey

After a little archaeologic research, I can tell you that my earliest saved Desmos graph is from March 26, 2014.  That’s just over three years ago.

I remember thinking then what a game changer Desmos was. I was stoked about how this could potentially transform teaching.  The graphs!  They were dynamic! And you could graph even non-functions. And the graphs were pretty!  I was truly in love and I thought about how finally (FINALLY!) my students could visualize the beauty and wonder of math in a way they had never been able to before.

This was going to change things for my students. I was 100% sure of it.

And I wasn’t wrong. But that’s not what I’m here to tell you about. 

What I never imagined was the profound impact using Desmos would have on my own mathematical understanding.

Through making more and more involved and intricate graphs, I have discovered myself learning how to do things I never ever imagined I would be able to.  And how did I learn it? Through a little bit of research combined with a whole lot of thinking, graphing, and playing.

Bear with me here for a quick walk down memory lane:

2014: A year of using Desmos to do teacher-y things I would normally have just used a TI calculator to do.  (plus a little more than TI does…like an awesome movable tangent line! Go Desmos!)

2015: A year of getting a little more creative with my graphs. A fish? With chomping teeth? Wowza. That’s some pretty sweet stuff. And check out the shading on that beach umbrella. Is it any wonder Desmos is my latest favorite thing?

2016: The year everything changed.


This pentagon construction? This is the graph that changed everything. This graph taught me more about using and writing parametric equations than I ever learned from a mathematics course.

Pentagon Construction (2)

And since then, I have continued to learn about parametric equations, about writing and composing functions, and about using sliders & lists to get some beautiful graphs:

cross sections - proportional rectangles


Heart Chain Reflect over Circle

Light Art


Non-Function AsymptotesParametric Butterfly

If you had told me in March of 2014, when I had saved my first graph, that in 3 years’ time I would be writing the equations for graphs like these, I would never have believed you.  And yet, it happened.

Desmos is not only a powerful learning tool for our students, it is a powerful learning tool for us, the teachers.

If you’re a math teacher, I challenge you to begin getting inspired to create graphs. There is so much to learn, all of which can enrich your content knowledge and help you to become a better teacher for your students! If you’re not a math teacher, but just like graphing for fun, keep at it!!!!!!!

Where can you get inspired?

  • Start following people on Twitter who are doing interesting mathy graphy things.
  • Check out Dan Anderson’s site: http://dailydesmos.com/.
  • Try some graphing challenges from learn.desmos.com.
  • Or? Look for inspiration in your daily life. Are you watching raindrops hit puddles? Maybe consider making ever-expanding concentric circles! Are you watching a bicyclist pedaling down the road? Try graphing the path the pebble takes!

For me, the next challenge is to make a tornado (at the request of a student.)  Keep an eye out for it on Twitter!

Happy graphing, people! And happy learning.

*link to the title graph.

Light Art


Changing the Domain of Polar Graphs

In today’s post: a work-around for specifying the domain of a polar functions. (hint: it involves a quick switch to parametric equations.)

What’s your favorite type of equation? I have always loved polar equations, but in the last year or so, parametric equations have taken polar’s place as #1.  This post may help to explain one of the reasons behind my change of heart.

Graphing polar functions is super fun. The curves polar functions generate are fun and unexpected. And when you graph them on Desmos, they’re simply stunning. But as of right now, you cannot specify the domain of a polar function on Desmos.  As far as I can tell, it will only draw polar graphs for values of θ from 0 to 12π radians:Polar Spiral

This suffices for many basic graphs, but even in the case of this spiral, it means that we aren’t seeing the whole picture.  For example, what happens when θ is negative? What about for larger values of θ? Continue reading

My Favorite Thing

It’s coming up on one year since I launched this blog, and in honor of the #MTBoSblogsplosion, I’m actually writing my 2nd post just this week.  It’s fitting that the prompt is “My Favorite Thing,” because that’s literally what this entire blog is all about.

I mean, don’t get me wrong, I have more than one favorite thing. I like cooking, baking, dancing, singing, mathing, drinking tea, breathing in the springtime air as the ground is just thawing.  I have lots of favorite things.

But ask my students what my favorite thing is and they will all, without hesitation, tell you it’s Desmos.  They’ve all come to expect that on Fridays I wear Desmos shirts.  And they pretty much think I spend all my free time playing on Desmos.  They also all know how to sing the Desmos song I wrote.  Well, it’s more of a “jingle.”  You can sing it too!  Here’s how you do it.  You sing in the highest note you can, with the most enthusiasm possible “Des-mooooos!”*

Adults also know Desmos is my favorite, but they’re skeptics and require some winning over (which usually just consists of them actually trying it). This fall I was fortunate enough to be a part of Desmos’s first ever Teacher Fellowship Cohort, where I had the opportunity to go out to Desmos HQ in San Francisco. People warned me: make sure it’s not like a time-share, where they’re just trying to sell you something. There are also people who think I’m just promoting Desmos so much because I must be getting something out of it.

Honey, please.

Desmos didn’t come find me. I sought them out. Continue reading

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: http://www.geoffofx.com/.


This picture is of one of Geoff Patterson’s creations. Image can be found at http://www.geoffofx.com/uploads/1/5/8/9/15895202/732203_orig.png

But what I didn’t know Continue reading

A Walking Machine

Hi everyone!  Due to an incredibly busy end to the school year and start to the summer, I’ve been MIA, with no new blog posts in a few months.  But I’m back.  This post is inspired by a gif my brother-in-law shared with me from Beautiful Engineering’s Facebook page.  Here it is:


In the comments, someone mentioned that it looked like one leg of a walking machine.

As soon as I saw it, I thought it looked like something I could create in Desmos.  So I did.  Here’s what I ended up with:


My best friends for this project were the Law of Cosines and parametric equations.

In order to trace the curve, I had to write nearly everything in function notation.  If you want to play around with any of the parameters, you can open the folders and try to make sense of it, but if there’s one thing I’m still learning, it’s how to make a Desmos graph that I can look back at later and understand.  For this graph in particular, what makes my work hard to follow is the fact that I wanted to make the yellow segments have variable lengths.  So each of their lengths is L1, L2, L3, etc. There was no good way to denote which length went with which segment, so I did what I could.  Either way, it was a lot of fun to put this together.

By changing some of the lengths of the segments involved, here are some of the other versions I ended up with.  Click the image to go to the graph.

The Birthday Problem

It’s been a couple of weeks since I last posted.  Things have been BUSY!!  I have been Desmos-ing quite a bit, just haven’t had a chance to write about it.  Here’s a little post about one illustration I enjoyed creating on Desmos lately.

In Pre-Calculus, the students at our school are just beginning to learn various counting principles and ways to calculate probabilities.

One of my favorite probability questions is “The Birthday Problem.”  If you’re not familiar with it, here it is:

What is the probability that at least two people in a room with n people in it share the same birthday (not counting year)?

The answer is a bit counter-intuitive (as are many questions in probability).  In fact, in a room with just 23 people in it, the probability of two people having the same birthday is already better than 50%.

Here’s a fun little Desmos graph to illustrate the probability for different values of n (based on a 365 day year).  The black points represent the falling probability that each person in the room has a unique birthday.  The chances that at least two people share a birthday is shown by the green points.

Birthday Problem.JPG

If you’ve never explored this problem before, despite its surprising result, it isn’t terribly challenging to do the calculations.

Wikipedia does a decent job of explaining the problem in more detail,  or you can also go to Wolfram Mathworld for a bit more of a math-intense explanation.  🙂  Loving Desmos!

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.