Back on the MTBoS Bandwagon!

The first quarter of the school year is over and grades are done, which means (among other things) that I finally have time to blog again! Time to catch up with some of the great MTBoS missions of the past few weeks. I'm responding in this post to MTBoS Mission #3. It's been a while.

I've played with the Desmos online graphing calculator before and liked it, so I used this mission to explore the Daily Desmos blog. Part of the reason I didn't explore Desmos much in the past was that it didn't seem to do anything GeoGebra couldn't. (Full disclosure: I am a huge GeoGebra fan.) But Daily Desmos gave me some pretty good evidence that the simple interface and streamlined visuals of the Desmos graphing calculator make it a potentially better independent exploration tool than GeoGebra.

Every few days the folks at Daily Desmos post a pair of related graphs, one Basic and one Advanced, with the deceptively simple instructions to use Desmos to recreate the graphs. The easy problems can often be solved with Algebra 1 level knowledge of linear and quadratic functions, while the difficult problems can be time-consuming even for undergraduate math majors. This, of course, means serious bragging rights if a student in your high school classroom manages to solve one of the Advanced problems.

The Daily Desmos problem that hooked me and subsequently ate up an hour of my Sunday afternoon is here, and it's based on a (much, much) simpler problem here. The Basic version of the problem requires graphing ten different linear equations which can be found by simply finding the slopes and y-intercepts of the lines in the graph.

Way more fun than a set of ten practice problems, but essentially the same thing.

The Advanced version requires finding the envelope curve for the family of lines in the Basic version. The given graph is below.

Looks like a fun Sunday afternoon, right?

When I saw this graph, I thought to myself, "Piece of cake. Just generalize the equations in the Basic version!" I did that, and then I had to parametrize the equation, then combine the resulting parametric equations back into a function, then square the function and make it implicit to allow for both positive and negative square roots. [SPOILER ALERT] After some serious algebraic gymnastics, I ended up with the equation and graph shown here. It turns out it's just a parabola with an oblique axis of symmetry!

So what did I take away from this that could help my teaching? Well, for students naturally inclined toward math, the Advanced problems on the Daily Desmos site are engrossing. And for those not naturally inclined, the simplicity of the Daily Desmos approach is brilliant. It doesn't give these students anything they don't need. It just gives them a picture and a simple question: "Can you make yours look like this?" For some of these problems there may be more than one solution, leading to interesting and heated debates among students. Really, I've seen this happen. Even if there's only one final solution, there may be many routes to get there. Daily Desmos problems require ingenuity and develop problem-solving abilities. But unlike some problems I toss at students, they know for certain that they've reached the end. If their graph looks exactly like the one they were aiming at, they're done! That's very appealing for them.

For now, I'd like to use Daily Desmos problems for extra credit or for independent work on those weird half-day schedules, but the approach could lead to some seriously cool unit design.

All a-Twitter

This post is in response to Exploring the MTBoS Mission #2.

I've had a Twitter account for about a year and never tweeted once until last week, so I am definitely a newcomer to the Twitterverse. When I started my Twitter account my intent was to use it as a professional development tool. I never followed through on that because of the overwhelming sense of chaos I felt as I looked at my feed. There's just so much STUFF piling in all the time. It seemed impossible to keep up with it all. And to be honest, I was (and probably still am) put off by some of the linguistic doomsday prophets who warn of the dangers of reducing discourse to 140-character spurts.

After using Twitter a little more concertedly this week, I'm still overwhelmed by the inexorable march of tweets streaming across my feed. However, I've seen some of the advantages of the 140-character limit--for one thing, it's a lot harder for people to dominate conversations the way they can on Facebook. You'll never see anyone write TLDR in response to a tweet. I've also seen what Justin Lanier (writing on the MTBoS blog linked above) calls the "funny combo of synchronous and asynchronous communication" that happens on Twitter.

Probably the best part of Twitter (and weirdest, for me) is the ability to follow and even tweet at people whose work you admire. There are blogs I've read for a while written by gurus in my field, and I can send a tweet their way just like that! Nothing stops me! I don't have to track down an email address and carefully craft an introductory letter hoping that I can disguise my inner fanboy enough to sound like a competent professional, all so I can maybe hear back from someone I look up to. I can just tweet them! And on their side of things, they don't have to worry about 1,000-word emails from fans. One hundred and forty characters, max.

In summary, I'll continue using Twitter, but I don't know if it will be a go-to tool for me.