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.

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:

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.

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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:

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?

Never fear! You CAN view a polar graph for any values of θ that you want, you just have to convert it to parametric first.

For my example of how to convert polar to parametric, I’ll use this pretty little curve.

If you’re unfamiliar with parametric equations, this is truly your lucky day. These little guys are the bread and butter of so much of the amazing stuff that can be graphed on Desmos. The idea is just that you have separate equations to define what x is doing and what y is doing.

In order to define x and y independently of each other, we will use the polar-to-rectangular conversion as follows:

Let’s do a quick substitution:

Desmos requires parametric equations to be written in terms of the parameter *t* and not *θ, *like you see below. However, if you put them into Desmos like this, you won’t get the graph you expected at all.

In Desmos, parametric equations are written as an ordered pair, kind of like a point:

Notice that it allows you to specify a domain. This is fantastic if you want to graph something where the curve won’t be complete after 12π radians.

Let’s take another look at that spiral, using parametric equations instead of polar.

Here it is for only positive values of θ, or *t*, after the polar-to-parametric conversion*:*

Now check it out with positive ** and **negative values of

Pretty sweet!

Or let’s look at this polar function with a period of 17π. Here’s what Desmos will graph:

But if we convert this to parametric, we can see the entire curve!

One other benefit of switching polar to parametric is that in parametric form, one can do certain animations that wouldn’t have been possible in polar.

For example, check out this polar butterfly, made up of two mirror-image polar curves:

I wanted to make the wings flap, but it was going to be pretty hard in polar form, so I converted to parametric where I could single out the x-coordinates to make the wings beat.

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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. They need no advertisement because their product *IS* its own advertisement. Just try using it and I can’t imagine you won’t get hooked too. It is, in their own words, *beautiful, free math.*

It’s dynamic. Fun. Free. Beautiful. Engaging. Enriching. Powerful. Playful. Intuitive.

Desmos had me at hello, but its teacher and student sites are the icing on the cake. I have used those companion sites to create engaging lessons and materials for students to explore, practice, review, play with, and create math. Students are literally high-fiving each other in class.

And me? I have grown so much as a mathematician in the last year just by using Desmos and networking with other people who are also using Desmos.

Also. In case you were unaware, the people behind the scenes at Desmos are awesome. I have been fortunate enough to meet them, and they really are the bees’ knees. Every last one of them. The team of engineers, designers, teachers, and others who work every day to make Desmos what it is are some of the most down-to-earth, amazing individuals. They care *deeply* about students and teachers alike. Their guiding principles say it all:

- Do no harm
- Trust teachers
- Design for real classrooms
- Design for delight
- Works every time

I consider it a privilege to witness this small company going from small (but awesome) online graphing calculator startup to global, paradigm-shifting, ed-tech superstars.

**I am at a new school this year, but teachers at my old school have reported back that students are still singing my Desmos song any time Desmos gets mentioned.*

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**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/.

But what I *didn’t* know was that this colleague *also* wanted an awesome Desmos graph illustrating solids made with known cross sections. Apparently, he had seen stuff on Geogebra, but nothing was fitting the bill.

So I took note of his wish list. It included these must-haves:

- editable functions
- editable bounds
- varying numbers of visible cross-sections, from very few to nearly filled-in
- rectangular cross sections with constant height
- rectangular cross sections with proportional height
- semi-circle cross sections
- equilateral triangle cross sections
- isosceles right triangle cross sections

I told him it would certainly be doable in just an evening, but that if he didn’t need it right away, I may not tackle it immediately. He left school for the day, and assured me that he didn’t need it soon. As in, he just wanted it for next year. So I put it off, procrastinator that I am.

Just kidding! As if I could wait to get started on such a fun-sounding graphing challenge! I love tackling this sort of problem, so I worked on it, despite having other tasks on my to-do list.

So what have I been learning recently that I applied in making these graphs? Let me share.

Labels! This is a new feature as of this week, and it was awesome to be able to label f(x) and g(x) on the graph.

**Pro-tip #1: To get a label, make a point, then click “show label” and write whatever you want the label to show. **

**Pro-tip #2: Want just the words and not a point? Click on the circle to the left of the point to hide it! The label will still show up.**

I made labels for f(x), g(x), the lower bound of the region, and the upper bound.

Also, with regards to clarity and usability, I’ve begun making brief annotations in my expressions list to label what’s being done where, and what certain variables I define mean. I haven’t always done this, and have regretted it later when I come back to a graph and want to change something, but can’t remember which variable represented what, and therefore have to think through all the math again.

I’m going to urge you to begin doing this, too, if you haven’t already! It’s easy, and makes such a difference!

**Pro-tip #3: ****Yes, it’s annoying to have to use a mouse and click to “add note.” So don’t! Did you know that you can just type ” and Desmos will add a note instead of an expression? Ah. Keyboard shortcuts. Gotta love ’em.**

And here are the resulting graphs!

Rectangular Cross-Sections of Constant Height:

Rectangular Cross Sections with Proportional Height:

Equilateral/Isosceles Triangular Cross Sections:

I’m still working on the right triangle cross-sections. I’ll post when it’s done.

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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.

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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

npeople 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.

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!

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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.

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:

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

…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:

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.

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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:

**TWO GREAT SKILLS I LEARNED**

**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: (link to graph)

- Set up a parametric curve and define the domain.
- Make a slider for a parameter (I used
*p*) that slides between the exact values of the domain for*t*. **Inside**the parametric equations, place the restrictions on*t*as I’ve shown here:

**Second skill: how to set up a chain of events**. (Hint: use only a single slider and make each stage appear for a different range of the values.) Try making something like the graph below.

(how I did it.)

Both of these skills are certainly things I’ll use in the classroom, and in particular, the skill of tracing parametric equations when we get to that topic in Pre-Calculus this spring.

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You can now hide student names by clicking on the little dude with the hat in the corner! Desmos will (temporarily) change the names of the students to the names of famous mathematicians. Now you can show the class what they’re all working on while keeping the students’ identities hidden. Love this new feature!

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**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:

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!

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