A Desmosian Gem

I finally had a chance to do the Function Carnival with my classes.  Thank you to Desmos, Christopher Danielson, and Dan Meyer for their work on this project.

As David Cox captured in his blog previously, the real power of this activity is the immediate feedback.

 

When the graph looks like the one below and 8+ rocket men burst out of the cannon, the students see that right away and adjust for it.

Rocketman

 

Dan had mentioned in a blog post a while back that “this stuff is really difficult to do well.”  After seeing students work through this activity today, I can appreciate the difficulty in creating an online math activity that gives both students and teachers detailed feedback in real time.

Some observations:

  • Students don’t realize at first that you can see their work live.  I allowed them to “play” for a minute, but some may need more encouragement.
  • A tool to allow you to communicate digitally with the class would be nice.  Google chat, for example?
  • Some students don’t realize that the bumper car SHOULD crash and make their graph to avoid it.
  • A student or two misunderstood the graph misconception questions and went back and changed their graphs to look like the misconception graphs.
  • It was interesting to see which students wanted their graphs to be perfect versus which ones said there’s was “good enough.”  It would be interesting to have a discussion about which is appropriate in the particular situation.

Bravo, Dan, Christopher and the Desmosians.  Thank you for creating an online math activity that gives me some faith in online math activities for the future.

Olympians, Tweagles, & Friends in my Phone

I started tweeting in 2008, around the Beijing Olympics. It was cool that actual Olympians would respond to my tweets.  When Summer Sanders responded to one of my tweets, I about fainted. Twitter was new, they probably didn’t know any better.  

I followed a few celebrities. I found some of their off-color honesty hilarious and sad at the same time.  In the meantime, my hilarious brother managed to rack up tens of thousands of twitter followers. (@sucittam if you are looking to add some hilariousness to your timeline). Here’s one of his tweets being featured on Ellen:

He opened my eyes to the idea that following actual REAL people is more entertaining and fulfilling. He was absolutely right.

I went through a phase where I followed a bunch of people who tweet as their beagle.  I’m pretty sure I was the first one to use the term Tweagles, although I have no proof of that. 

Then in January 2013, my indifferent view of people on twitter changed forever. My 29-yr old sister in-law, Danielle, suffered a massive brain aneurysm and it wasn’t certain she would recover.  She was in the ICU at the University of Iowa for almost 6 weeks, and while my brother stayed by her side every day, his twitter followers rallied support that went viral. All of these people, most of which he’d never met, wanted to reach out to help. Benefits were organized, gifts were donated, and memorabilia was auctioned all to benefit Danielle whose recover was slow, but steady. 

Rex Huppke (@RexHuppke) wrote a beautiful article illustrating that the people we interact with on twitter are not just cyber-acquaintances.  Danny Zucker makes the best point:

 “We’re willing to accept the concept that cyberbullying is real, and it is. But if you can accept the idea that the negative is real, then you have to accept the idea that the positive is real. If strangers can hurt you, they can be friends as well.”

And just like that I leaped head first into the T of the MBToS. I realized that people like Fawn Nguyen, Andrew Stadel, Kate Nowak, and Christopher Danielson were real teachers just like I was.  They had great blogs, and they were on twitter too. And if I wanted to get a real benefit from all of the resources I had found online, I needed to start posting feedback of how I incorporated them into my classroom.  And then tell the creator of the activity about how it went. Through this I’ve really been able to experience the genuine human behind all of these @ symbols. These are not only great teachers who don’t just shine on their own. They want to freely share what they’ve done so that others can shine just as brightly. 

A Visual Patterns Trifecta

This is my third (and most exciting) post about my new found love for Visual Patterns.  My enthusiasm stems from a growing appreciation of how these patterns can be used in such a wide range of grade-levels, including advanced algebra.  The use in an elementary or lower-level secondary classroom is easy to see.  However, the teacher and student need to dig a bit deeper into the make-up of these patterns in order to generalize them.

For example, here is Pattern #8.  Kudos to Fawn Nguyen on this one.

It’s not immediately apparent what step 4 should be.  But even more so, the quadratic nature of this pattern is not necessarily simple to comprehend.  From yesterday’s pattern #5, the students had a method for finding the number of penguins in the nth step by converting the penguins into a table and creating a system of equations.  I didn’t want to encourage this method, as it is very procedural and tedious.  However, it was a good place for students who liked to work in a more algebraic way to feel successful.

Also, the table allowed them to explore what the difference of differences really told them.  I had a student, let’s call her Kay, ask “I wonder what the constant difference of differences represents in our equation for the nth step.”  She came up with a conjecture by comparing it to our problem from pattern #5.  Kay concluded that the “a” value in the equation ax^2 + bx + c = y is half of the constant difference of differences.  I challenged Kay to continue to examine these values in future problems to see if her conjecture holds true.  I had another student, Em, wonder if that meant that the “a” value in a cubic function is equal to one third of the difference of difference of differences.  This she will investigate as well.  What is very exciting about these questions is that they were non-existent 5 weeks ago.  It wasn’t that the students didn’t WANT to be mathematically curious, they just didn’t know HOW.  It was a huge thrill for me as a teacher to see these kids move from looking at a math problem with a single solution to being able to ask new questions.  A nod to Christopher Danielson for helping me realize that learning is having new questions to ask.

  Back to the problem at hand:  How many penguins are in step n?  A few of the students were able to get the answer without using a table.  These were mostly the students who like to do things in their head.  The ones who want to fully process the problem in their brain, but not write any of it down.  [Side note:  these are usually the ones who are brilliant with numbers but get lower grades in traditional math classes because they don’t want to “show their work.”] Anyway, I wanted to challenge those who used the table method and set up a system of equations to relate their model back to the picture.  Spoiler alert!  The answer is 1/2n^2 + 1/2n + 1, but I wanted my students to be able to relate that back to the picture.  What do the individual pieces of the expression represent in penguins? This way, the students were able to make that connection of a picture or pattern that didn’t seem quadratic to begin with and flesh out its quadratic properties.

When the students figured this out, it was a magical moment.  I had to capture it:

IMG_2993

Another cool experience with this problem:  the same evening that I did this problem, our school hosted parent-teacher conferences.  One of my students came into conferences with her parents and her three little sisters, ranging in age from about 5 to 10.  One of the little sisters sat down and wanted to be part of the conference.  I pulled up the visual pattern and asked her how many penguins would be in the next step.  It was a validation of my initial thoughts of how open and accessible these problems are to all levels of mathematics.  Here was an 8? year-old looking at the same pattern that her 17 year-old sister explored earlier that day.  And it was mathematically applicable to them both.  Beautiful.