Tag Archives: desmos

4 Engaging Ideas From Twitter Math Camp

This past week, over 100 math teachers descended upon the Drexel University campus for Twitter Math Camp 2013.  It was a fantastic opportunity to meet people I had communicated with via Twitter for some time, make new friends, and share math ideas.  It’s a real rush to hang out with colleagues who share similar ideals on math instruction, and a commitment to improve our practices.  Check out the hashtag #tmc13 on Twitter to look back on some of the action and reactions, and find new math folks to follow.

While there’s so much to share from TMC13, I know there are many math friends who couldn’t attend who are looking forward to hearing about the goings-on, so in this post I share 4 ideas from this year’s Twitter Math Camp I am eager to try in my classroom right away.

EliELI’S BALLOONS – Followers of the blog know that I am a big fan of the Desmos online graphing calculator.  The highlight of the week for me, and I suspect for many, was having Desmos founder Eli Luberoff model a lesson using his creation.  Eli’s enthusiasm for sharing Desmos, and his sincere desire to work with teachers to improve the interface, are infectuous.  There were many “oooh” and “aah” moments from the assembled group, and a loud cheer for the “nthroot” command…yes, it’s a pretty geeky group!  (thanks to @jreulbach for tweeeting out the great picture of Eli showing off Desmos’ position when you Google “graphing calculator”)

Eli’s lesson idea has a simple and engaging premise:

  • Hand out balloons
  • Blow up the balloon.  For each breath, have a partner record the girth of the balloon
  • Consider the data set


That’s it.  No worksheet.  No convoluted instructions.  Eli walked us through an exploration of the data set using Desmos, using the table to record the data, and considered various function models: is a square root model?  Is it logarithmic?  The group eventually settled upon a cube root as the proper model – and how often in class do we encounter data best modeled by a cube root?!  Since the explanatory (air entering the balloon) is volume, and the response variable (girth) is linear, the cubic model makes perfect sense. Fun stuff.  But wait…there’s more!  Eli then analyzed the fit of curve by looking at the squares of the residuals.  Click the graph below to check out my best-shot recreation of Eli’s presentation, and play around with the fit of the curve by toying with the “a” slider.

More great new additions to Desmos are coming.  Thanks to Eli for letting us preview some of them!  Was a pleasure meeting you and hearing about your fascinating story.

GLENN’S PROBLEM POSING – Glenn Waddell is a colleague I feel I have a lot in common with, in that we have both experienced the frustrations of trying to “spread the word” to colleagues of the great new ideas, and strong need, for inquiry-based mathematics.  In this session, Glenn presented a framework for problem posing in mathemtics which can be employed equally-well with real-life problems (see the “meatball” example in Glenn’s Powerpoint, which was adapted from a Dan Meyer “math makeover” problem) or with a garden-variety drill problem.

The framework asks that teachers lead students in a discussion that goes beyond just the problem in front of us.  Think about the many attributes of a problem, list them, consider changes to them and their consequences, and generalize results.  Glenn suggests the book “The Art of Problem Posing” by Brown and Walter as a resource for getting started, which employs the problem posing framework.


Glenn led the group through an exploration of a quadratic equation, where we started by listing its many attributes.

Glenn Problem

Now we consider changes to attributes:

  • What would happen if there were a “less than” sign, rather than equals?
  • What would happen if the last sign were minus?
  • What is it were an x-cubed, rather than x-squared?

There’s no limit to the depth or number of adaptations, and that’s why I like this method of problem posing for all levels of courses.

Download Glen’s presentation on the TMC wiki, and explore the wiki to get the flavor of many of the sessions.



If you have never visited Mathalicious, go now….take a look at some of the free preview lessons, and you will become lost in the great ideas for hours.  THEN, make sure you sign up and get access to all of the engaging lessons.  Here is a company that is doing it right: lessons come with a video or visual hook, data which naturally lead to a discussion of tghe underlying mathematics, and just the right amount of structure to encourage students to contribute their thoughts and ideas.  At TMC, Mathalicious founder Karim Kai Ani led the group through two lessons.  A brief summary is given here, but I encourage you to check out the site and subscribe….you’ll be glad you did.

The “Romance Cone” – What is the appropriate age difference between two romantic partners?  Is there a general rule?  A fun lesson, “Datelines” on Mathalicious, where students explore a function and its inverse, without using those scary-looking terms.  I have been looking for an opening activity for our Algebra 2 course, which brings back ideas of function, inverses and relationships, and looking forward to trying this as a my first-day hook.  Also a great activity for Algebra 1.

PRISM = PRISN? – I have led my probability students through an exploration of false positives in medical testing for many years, and I like how this activity puts a new twist, and some great new conceptual ideas, on the theme.  “Ripped from the headlines”, this lesson challenges students to consider government snooping, and the flagging of perhaps innocent citizens.  If a citizen is flagged, what is the probability they are dangerous?  How often are we missing potentially dangerous folks in our snooping?  What I really liked here was the inclusion of Venn Diagrams, with sets representing “Flagged” and “Dangerous” people, where the group was asked to describe and compare the diagrams.  Fascinating discussions, and a good segue into Type I and Type II error for AP Stats if you want to take it that far.

This lesson does not appear to be available on the Mathalicious site yet, (update from Mathalicious – will be released in the Fall) but will be using it when it is completed!  Later that day, the TMC teachers broke into smaller groups to gain behind-the-scenes access to the Mathalicious writing formula.  Thanks to Kate and Chris for sharing, listening, and giving us all the opportunity to contribute ideas.


Webinar Action with the Desmos Folks.

A few weeks ago, I recorded a webinar for the Desmos folks, featuring the semi-famous Conic Sections project.  We had a few blips with playing video during the webinar (darn you, Google Hangouts!), but Jen from Desmos cleaned up the video, and it is now ready for your viewing pleasure on the Desmos YouTube Channel.

Visit the channel for a number of new, exciting video additions:

Daily Desmos – Justin Lanier and Dan Anderson talk about this math sensation.

Moveable Points – Desmos gives a “how to” guide for this new feature.


Conics Webcast

Thursday evening at 8PM EST, June 20, I am looking forward to sharing my Desmos conics project in a webcast.


Go to Desmos’ place on YouTube: http://www.youtube.com/desmosinc where you will be able to view the webinar and answer questions.  This webinar was rescheduled after some snafus with Google Hangout, which we hope (fingers crossed) will be all ready to go tomorrow.  Scroll down to my last post for more info.  Have your computer ready to watch, and maybe your iPad handy, as we will walk through a sample drawing.

Hope to see you there.  What should I wear when broadcasting from my dining room?  Suggestions encouraged.

More Great Conics Project News!

UPDATE – a newer post concerning this project, with rubric can be found with this post.

I can tell it’s the end of the school year, and math teachers are looking for fun math projects to do with their classes, as the search terms which get people to my blog contain lots of references to “conics projects, “math art projects” and the like.  The searches have led to many hits to my conic sections art project blog post from last June.

At my home high school, this is the second year we have done our long-standing conic sections art project using the Desmos calculaor, and this year’s submissions have raised the bar considerably.  The most improvement has come from working with students to restirct domains, which has made more complex drawings easier to manage.  Here are a few to share, but look for an announcement from Desmos, with whom I will present a webinar on June 6 and give you some ideas for getting started with your class.

First off, a tiger, which took over 100 individual equations to create.  Stunning!


Next up, an ambitious student who took 87 different picture “slides” to create this animated gif.  I wish I was half this creative when I was 15!

Falling Man

Check out the recording from the Global Math Department for more information, and be on the lookout for webinar information on the 6th!  Meanwhile, let me check some of our other Algebra 2 classes for some promising projects!

Teachers Sharing Desmos Ideas

This coming Tuesday, April 9, the fantastic online graphing calculator Desmos will be featured in a webinar held through the Global Math Department.  This is part of a weekly math conference series hosted by bigmarker.com.  Some weeks, there is a set theme, while other weeks teachers present their favorite lesson.  It’s exciting to hear some of the math teachers I have come to respect and admire through twitter and blogs share their favorite lessons, and you will always find something worth adapting for your classroom.  You can check out an archive of past webinars on the conference section of the Global Math Department on bigmarker.com.  I am looking forward to sharing my conic section lesson this week, and the agenda is packed with great ideas for Desmos, including:

  • @samjshah – Using sliders in polar equations to study conic sections
  • @Mr_Stadel – an exploration of gemetric shapes
  • @MrOrr_geek – Creating pictures using function transformations

Pop in and say hello, or come back later and enjoy the webinar archive.


Earlier this year, Desmos unleashed its table feature, and it is a seamless addition to an already simple tool.  You have choices for how to implement a table in a Desmos document.  Start a new table, and enter a rule in the “y” position.  Or take an existing function, and “edit” it to become a table.  Or, name your function as f(x) and Desmos will recognize it in a new table.  Here, a quadratic function was converted to a table, and a new column added to compute values of the derivative.

Desmos Capture

Think about the conversations you can with your class about this.  How do the values of the rule “2x+4” relate to the graph of the quadraic function?  When does 2x+4 take on positive / negative value?  When is it zero?

Play with the Desmos graph by clicking on the link, and enjoy the table feature.


Sometimes it’s the simplest idea that produces the biggest wow moments, and the Daily Desmos site earns my kudos for not only its simple, powerful concept, but also its potential for differentiation.  Each day, 2 new graphs generated on Desmos are given.  It is up to you, or your students to determine how the graph was made.  How was this graph made? Daily Desmos

Many of us teach high schoolers how to graph trig functions, and our students certainly know linear functions.  So, how to combine them?

The site also challenges users to contribute their own graphs and provides guidelines for basic and advanced graphs.  What a fantastic tool for differentiation:  allow you quick finishers to pursue a Desmos graph, and show off their ideas to the world.  Print out the graphs, post them around your room, and let math go beyond the mundane and routine.  When you have your first conversation about polar coordinates and functions with your class, when you weren;t planning to have it, you’ll know you are doing something right for your kids!  Thanks to Michael Fenton for starting the Daily Desmos.  Keep up the great work!

You Asked For Piecewise Functions, I Give You Piecewise Functions!

NEW: After popular demand from this post, I have created a tutorial on domain restrictions and piecewise functions.  Enjoy!

UPDATE: Many of my Desmos files are avilable on this page: Desmos File Cabinet Enjoy!

Let is never be said that mathcoachblog doesn’t listen to the needs of its followers!  One of the neat things about having a blog is checking out the routes people take to get to the blog. What search caused them to arrive here?  What countries are my visitors from?  What search phrases cause them to reach the blog?

Every day, without fail, there is a theme which appears in the search terms of blog visitors.  Here is a sampling of terms from just the last week:

  • Online piecewise graphing calculator
  • Graph a piecewise function online calculator
  • Piecewise function calculator online
  • Graphing piecewise functions calculator online
  • Piecewise functions online grapher
  • Online graphing calculator piecewise functions
  • How to do a piecewise function on Desmos

OK, folks I get it.  We want to graph piecewise functions.  So, let’s light this candle.


The Desmos knowledge base provides instructions for graphing a piecewise function, and a neat video tutorial.  But I’ll provide a few examples here, and some teaching tips.  Let’s say we want to graph this piecewise function:

In the Desmos calculator, colons are used to separate domain restrictions from their functions.  And commas are used to have multiple function rules in one command.  So, the piecewise function above would be entered as:

Piecewise Entry

The function then appears quite nicely:


Sliders can be used to have students explore the continuity of a piecewise function.  Consider this problem:

For what value(s) of x is the piecewise function below continuous?

In Desmos, start by defining a slider for the parameter “a”.  For mine, I chose to limit the domain to between -10 and 10, and have step counts of .5.  Then, a can be used in the piecewise function.  Click the icon below to play with the document online.  The sharing features are another aspect of Desmos which have improved greatly in the past year.


OK, smart guy, yes…yes, I have kinda avoided the equals parts of the domain restrictions.  Something neat (odd, goofy) happens when an equals is used in the domain restrictions.  Let’s graph this function:

Click this link to find out what happened when I tried to enter this function on Desmos.  The Desmos folks tend to monitor these things, so let’s see if they have a suggestion here.

Down the road, I want to take a deeper look at the new table feature, and will report out.  But my early impression is that it is a addition which works seamlessly with the existing, awesome, calculator.

Also, while I’m in a sharing mood, here is a quick file I created to use in an absolute value inequality unit.  Click below to check it out.  Would enjoy your input!

And finally, I started this post by sharing some of the search terms which cause people to find my blog.  Most of the time, I can explain those terms, and why people would end up here.  But this….well….this, I got nothing…..

search terms