Tag Archives: desmos

Desmos + Statistics = Happiness

Sunday – a quiet evening before President’s Day – checking out twitter – not looking for trouble – and then,

Wait..what’s this?  Standard Deviation?  It was my birthday this past Saturday, and the Desmos folks knew exactly what to get me as a present.  Abandon all plans, it’s time to play.  A lesson I picked up from Daren Starnes (of The Practice of Statistics fame) is a favorite of mine when looking at scatterplots.  In the past, Fathom had been the tool of choice, but now it’s time to fly with Desmos.  There are a few nuggets from AP Statistics here, and efforts to build conceptual understanding.

CORRELATION, LSRL’S AND STANDARD DEVIATION

Click the icon to the right to open a Desmos document, which contains a table of data from The Practice of Statistics.  In you are playing along at home, this data set comes from page 194 of TPS5e and shows the body mess and resting metabolic rate of 12 adult female subjects. One of the points is “moveable” – find the ghosted point, give it a drag, and observe the change in the LSRL (least-squares regression line) – explore changes and think about what it means to be an “influential” point.

Next, click the “Means” folder to activate it.  Here, we are given a vertical line and horizontal line, representing the means of the explanatory (x) and response (y) variables. Note the intersection of these lines.  Having AP students buy into the importance of the (x-bar, y-bar) point in regression beyond a memorized fact is tricky in this unit.  Drag the point, play, and hopefully we can develop the idea that this landmark point always lies on the LSRL.

Another “fact” from this unit which can easily wind up in the “just memorize it” bin is this formula which brings together slope, correlation, and standard deviation:

The formula is given on the exam, with b1 acting as the slope, so even memorizing it isn’t required, but we can develop a “feel” for the formula by looking at its components.

Click the “Means plus Std Devs” Folder and two new lines appear. we have moved one standard deviation in each direction for the x and y variables. Note that the intersection of these new lines is no longer on the LSRL. But it’s pretty close…seems like there is something going on here.

Ask students to play with the moveable point, and observe how close the rise comes to the intersection point. Can it ever reach the intersection? Can we ever over-shoot it? In the “Rise Over Run” folder, we can then verify that the slope of the LSRL can be found by taking a “rise” of one standard deviation of y, dividing by a “run” of one standard deviation of x, and multiplying by the correlation coefficient, r.


There’s other great stuff happening in the Desmos universe as well.

1.  This summer brings the 4th edition of Twitter Math Camp, to be held at Harvey Mudd College in California. I’m thrilled to have latched onto a team leading a morning session on Desmos. Consider coming out for the free PD event, and join myself, Michael Fenton, Jed Butler, and Glenn Waddell for what promise to be awesome mornings. To be honest, I feel the Ringo of this crew….

2. Can’t make it to the west coast this summer? Join me at the ISTE conference in Philadelphia, where I will present a learning session: “Rethink Math Class with the Desmos Graphing Calculator“. Bring your own device and join in the fun!

3. Are you new to the world of Desmos? Michael Fenton has organized an outstanding series of challenges, with 3 difficulty levels, to help you learn by doing. Try them out – they promise to get you think about how you and your students approach relationships.

4. Merry GIFSmos everybody!  The team at Desmos has developed GIFSmos to let you build your own animated gifs from Desmos files. EDIT – as Eli noted in the comments, credit for GIFSmos goes to Chris Lusto.  Thanks for being so awesome, Chris!

Class Opener – Day 29 – Geometric Series

Aren’t infinite geometric series cool?  If you just shouted “yes”, then you are potentially as geeky as I am. A “proof without words” from MathFail kicked off today’s discussion:

Proof
I wasn’t quite sure what sort of observations I would receive from my class. But just enough ideas were generated to get us going:

There are an infinite number of triangles down the right side.

All those triangles on the right add up to the half-triangle on the left.

Both are great starts for what I hope my students will learn today. A video I made in my driveway continued the ideas of geometric series and their infinite terms.

A few students wanted to argue that the sequence in the video was arithmetic, but some meaningful debate yielded agreement that geometric made more sense.  Groups then worked through a similar problem involving a Superball being dropped, leading to terms and total distance traveled.

seriesMany groups employed a “brute force” method to find their answers. Using the Desmos calculator (many students chose to use the iPhone app), we found value in developing the equation and using tables and summation symbols to find solutions. This was my first time usign Desmos with this particular lesson, and it was an awesome addition, which added value to the need for writing a clear function to define your situation.

Piecewise Functions and Restrictions on Desmos

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I love checking my blog stats. Above are just some of the many search terms which cause people to end up here on the blog. You search, I listen. Armed with Camtasia (provided by my friend and barbecue savant Jason Valade from TechSmith) here is a tutorial I hope you find helpful as you start your school year. Resolve to make Desmos part of your classroom culture this year, then check out the Desmos File Cabinet of graphs to get you started.  Also, check out classroom strategies for using Desmos to explore function inequalities in the second video below.

DOMAIN RESTRICTIONS AND PIECEWISE FUNCTIONS

 

INVESTIGATING INEQUALITIES USING DESMOS

Professional Development: Who Owns the Responsibility?

Last weekend, I had the oppotunity to speak at the Association of Mathematics Teachers of New Jersey Technology Conference on my experiences with Desmos, where I shared a number of the Desmos activities and ideas I have posted here on the blog.  Conference speaking and classroom direct instruction are totally different experiences.  There a few things I need to think about in preparing for a conference which aren’t part of my daily routine:

  • How many people will there be?  Conference sessions are mini-popularity contests…I may have 5 people in the room, I may have 50.  You just never quite know what to prepare for.  Small groups are great for encouraging discussions.  Larger groups have a difference set of engagement challenges.
  • What’s the overall background of the room? People who will come to see me could be a mix of veteran and new teachers, with different stories to share, and levels of comfort with my topic.  Taking the temperature of a room quickly and finding a baseline comfort zone is key.
  • Do I really know what the heck I am talking about? I’ve offered to present because I feel I have something to share; but is my message unique or helpful?  Could there be someone in the room who knows far more about this topic than I do?

It can be a stressful experience presenting in front of peers, but also highly rewarding and a great way to make professional contacts!

Overall, I feel my session went quite well.  After all of the stragglers found their way inside, there was a full house (or lab) of about 35 folks, with a positive vibe in the room.  I think I met all of my goals, and I believe many left the session with tangible ideas for their classroom, and ideas to share with colleagues.  If I accomplished all that, then I feel successful.

For this speaking opportunity, I borrowed some equipment and recorded my talk, something I had never done before (see video below).  Upon review, there is a common theme running through the talk which has cause some post-talk reflection for me.  A few times I ask the group about their experiences with online tools, with some (to me) surprising results from the room of 30-35:

  • Only 1 or 2 had used Desmos before.
  • Only one had used Geogebra before.
  • Only one had heard of Edmodo before.

How could this be?  Here was a group of enthusiastic educators, concerned enough about their craft that they sought out professional development on a Saturday, and very few knew about these tools.  Is this small sample group indicative of all math teachers?  Should I be as surprised as I seemed in the video (really…watch my expression when I ask about Edmodo)? The good news is that hopefully some exponential growth occurs, and these teachers tell their colleagues, who then share with their colleagues…and so on….

But what of those teachers who do not seek out conferneces?  How do they find new resources?  Or are they even looking?  Do teachers have a professional responsibility to seek ro revise their ideas and practices?  I won’t pretend to have any answers in this blog post; rather I’d be eager to hear some thoughts on these questions from my readers.

And while my session has a clear technology slant, does the variation in learning experiences extend to math pedagogy in general?  Can teachers defend their classroom practices, and seek our resources for revision if needed?  How many teachers have considered how Common Core shifts will effect their classroom structures? Have teachers considered the Standards for Mathematical Practices and how they apply to their classrooms? Where do teachers go to find professional development opportunities which meet their unique needs?

And, most importantly, what are the responsibilities of classroom teachers, curriculum specialists and administrators in facilitating these reflections?  It’s a lot to chew on.


Below is video of my Desmos session.  Seeing myself on camera is at the same time cringe-worthy and thrilling…so much to learn from.  Man, do I gesture with my hands…. a lot!  Feel free to comment, share or heckle!

SAY “YES” TO DESMOS – AMTNJ – APRIL 2014

Visualizing Shared Work Problems

Fred can paint a room in 5 hours, working alone.  His friend, Joe, can paint the same room in 7 hours.  How long will it take for them to paint the room, working together?

It’s a shared-work party, people!  Get your party hats on and let’s look at a visual method for exploring these often mundane problems.  This past summer at Twitter Math Camp, I participated in an algebra 2 group where part of our time was spend considering methods to re-think the traditionl approach to rational functions and their applications.  Thanks to John Berray for the great conversations, which led to some changes in how I appoached shared work problems this year.

My approach this year started similarly to previous years: guiding a dicussion with the class, with the goal of developing models for the amount of work done by each painter.  I find that quesitons like “How much of the job will fred have complete after 1 hour? 2 hours…etc” will usually lead to the models we seek.  What I did differently this year was graph the two work functions.  Using the Desmos calculator works nicely, and allowed for a discussion of the problem much richer than if the expressions had been just jotted down on the board.  Many students followed along on their TI calculators.

SW1

From here, we can make connections betweem the functions, their graphs, and make conjectures about the sum of these functions.

SW2

In my class, students certainly completed similar problems (including distance / rate / time), with the graphs serving as a check and visual affirmation.  With the graphs, we could also look at adaptations to the theme, such as “what happens if one of the painters shows up 2 hours late?”

SW3

Also, problems where the combined time was given, with the goal of finding a missing individual rate, were explored and discussed.

SW4

Click the icon below to play with this model on your own.  This is a great opportunity to let students observe function behavior and communicate results from a graph.

UPDATE: The Desmos folks flew with this one, and added a whole bunch of bells and whistles.  Click the graph below to experience their shared-work extravaganza.

Use Appropriate Tools Strategically

This semester, my Algebra 2 students will be exposed to a wealth of math tech tools.  Graphing calculators will be a big part of what happens in my classroom; not only because they are great tools for discovery, but also because I feel some responsibility to have students understand the appropriate use of these tools as they head towards AP classes.  Forcing a tool upon students because it will help them on a test is weak, I know…I cry myself to sleep sometimes…though I do rely on the technology to craft discovery moments in my class.

But I also want my students to experience other tools, like the Desmos calculator (which we will use later for the world-famous Conic Sections project), Geogebra and Wolfram|Alpha (reviewed earlier here on the blog).  So, how do I get my students to experience all of these tools, and start to make measured decisions about how and when to use them?  Hey, we have a Standard for Mathemaical Practice for that!

CCSS.Math.Practice.MP5 Use appropriate tools strategically.

Lost in the great stuff on precision, modeling and reasoning is this awesome nugget, with a specific focus on tech tools:

Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

Nice!  Exactly what I am looking for!  So, how do you do that?  How do you get students to start comparing and assessing tools?

Here is the day 1 assignment I gave to my students in Algebra 2, as posted on Edmodo:

Write a product review for the Desmos online calculator. Consider its pros and cons, and whether you feel this is a site you would recommend to others. One side of a piece of paper MAX. Screen shots allowed and encoraged. Can be turned in via hard copy, or electronically.

Thats it.  Go ahead, kick the tires, and tell me what you find.  I didn’t make students aware of my on-going man-crush with Desmos or that I had done a webinar for them.  I had no idea what I was going to get back.  And since I stress written communication in my classes (moreso, I suppose, than many math teachers), this gave me a first writing sample to analyze.

The results were largely encouraging.  While many students focused solely on the graphing of functions, some students demonstrated evidence of digging deeper, looking for characteristics which make Desmos unique.  Some snippets:

Unlike the normal graphing calculator, it graphs your equation as you are typing it and allows you to delete parts of the equation if your graph isn’t what you wanted it to be. Desmos also provides the general equations for many different lines, parabolas, and other more advanced graphs.

The example graph list on the left side of the screen acts as a jump start and learning tool to give confused students a boost in the right direction.

It is quick, simple, and efficient to use and is recommended to all users that seek a tool for graphing. The designs are not distracting but sleek and a simple white to emphasize the purpose of the tool, for math and nothing else.

The Desmos is different, its not complicated at all, it can do so many things that most calculators can’t, and it’s free. The fact that Desmos is free is really what makes it so much better than all of the other because you don’t have to shell $120 out of your pocket for a calculator that has all the same capabilities that Desmos has.

But not all is sunny, as some students noted some “Cons”:

With the internet calculator, there is the obvious issue of no internet, no calculator. Also, I found some buttons were tough to get to such as the “pi” key which required me to press several buttons in order to get that one.

One last thing about the calculator is the fact that it can be downloaded as an app, but only on apple products. For android users, like myself, you would have to use the calculator through the internet which isn’t as easy to use as through an app.  Also, the app is accessible without wireless internet connection, but android users need the wireless connection to use the Desmos calculator.

All told, a good first writing assignment for my students, followed by some discussions of tools and their appropriate use.  As we travel through Algebra 2, many chances to compare tools, and discuss the best tool for the job.  Looking forward to doing another product review, using Wolfram|Alpha.  Stay tuned.

Model Classroom Resources for Siemens STEM 2013

Online Calculator: Desmos

Balloon Activity:

Bob’s Eyeball:

Resources for the Conic Sections Activity:

Blog Post

Webinar

Global Math Department Presentation Slides


UN-CONFERENCE RESOURCES: Free Math Stuff

The Daily Desmos: teacher-written graphing challenges

Geogebra: dynamic construction software.  Download, or run in java.

Geogebratube: teacher-created demonstrations

SITES FOR PROMOTING INQUIRY –

Visual Patterns – Challenge your students to find and summarize patterns.

Graphing Stories – video starters, from linear to exponential

101qs – picture and video openers for promoting inquiry.  Contribute your own!

Mathtwitterblogosphere – network of teachers dedicated to sharing resources and classroom experiences