Friday, August 29, 2014

Problem Solving with NPR

A friend of mine introduced me to the free app "Stitcher" this summer (thanks, Solana!), which essentially is a way to organize podcasts that you enjoy listening to.  I wasn't much of a "podcaster", but with the help of the app, I've learned a TON about a lot of really cool stuff!  One of my favorite podcasts is the NPR Series: Sunday Puzzle.  Considering I'm a math teacher, that should be no surprise to you!

The way the Sunday Puzzle works is they start the podcast with a recap of last week's puzzle and they reveal the answer.  Then, they contact a random winner (someone who answered the puzzle correctly online) and have them on the show (via phone) to go through a series of smaller puzzles/riddles.  These puzzles are not necessarily related to the Sunday puzzle.  Finally, they introduce the new Sunday puzzle and give you until the following Thursday to submit your answer.

On August 17, the title of the show was, "Is There An Echo In Here?" and the Sunday Puzzle was one from Sam Lloyd.  Here it is...

"You have a target with six rings bearing the numbers 16, 17, 23, 24, 39, and 40.  How can you score exactly 100 points by shooting at the target?"

I thought this would be a great puzzle for my students to try.  After introducing it, they had some clarifying questions, which I thought were great!

"How many times do you shoot at the target?"  I told them that it did not specify, so they could decide.

"Can you hit a number more than once?"  I told them, again, that it did not specify, so they could decided.

I gave them about 3 minutes to work independently, trying it on their own, and it was completely silent.  Every student was engaged.

After this independent work time, I wanted to tap into the minds of my students, so I just asked for their initial strategies.

Lots of students said that they were just randomly adding numbers to see what happened.  Others said that they added all 6 numbers to get 159 and then tried to see if they could combine any of the numbers to get 59 that could be taken away.  Students found that they could get close to 100, either just over or just under.  Students also explored finding a factor of 100 and then repeating that process.  Lots of good stuff here!

Students who arrived at a solution found that they could get to 50 by adding 16 + 17 + 17 and then they doubled it.  Another approach that was shared was very sophisticated and inspiring.  A student said that she added 16 six times and found that she got 96, which was 4 short of 100.  She knew that 4 was not an option for a ring, but thought there might be a way to gain 4 by adding in a 17.  She tested it by adding 16 five times and then 17 once to get 97.  Her strategy was working...she was getting closer to 100.  Ultimately, she found that four 17s and two 16s was the combination she needed.  I loved it!

As I described earlier, the 5 minute podcast starts with the answer from the previous week's puzzle, so I shared the solution with my students (Ben Parks, August 24, "A Puzzle Hokey Pokey, That's What It's All About").  They were delighted to know that they had arrived at the (only) correct solution.

I decided to play the rest of the podcast for them, which they loved!  The riddles involved 2-word phrases (first word has 5 letters, the second word 4 letters) that were generated by dropping the last letter of the first word, then reading the remaining 4 letters backwards to get the second word of the phrase.  (Try this one...Where Peruvian pack animals shop: Llama Mall) :)

Finally, the moment came to hear the new puzzle for the week.  Students were excited to get in on the action.  Here's the latest Sunday Puzzle from Jason Zuffranieri:

"Name a world leader of the 1960's, 2 words, change the last letter of the 2nd word, then switch the order of the words, that is putting the 2nd word in front, and the result will name a hit song of the 1990's.  Who's the leader and what's the song?"

I didn't know what to expect, but I sent my students home with the challenge of figuring this out.  I offered extra credit for getting a correct solution, but they had to somehow defend their process, so I would know that Google didn't deserve the points. :)

I returned to class today (Friday) after introducing this on Wednesday and I was pleasantly surprised that two of my students in my 3rd period class had figured it out!  One young man said that he had his cell phone open to Google looking up world leaders, while his mom's phone was open to iTunes looking up hit songs of the 90's.  He started piecing things together and arrived at his answer.

The other student said that she started with song titles from the 90's, looking for ones that included names (Mr. Jones, Mrs. Robinson, Jane, etc.).  Then she looked at a list of leaders and found a last name that could easily be changed to a common word and pieced it together.

Here's what they came up with...

SO proud!  I can't wait to incorporate this into my classroom on a more regular basis and even encourage my students to submit their answers online!  It was obvious that these two students were not motivated by the extra credit, they were just enthusiastic puzzle solvers.  What more could I ask for from a math student?

Sunday, August 24, 2014

"What Do You Mean, There's No Right Answer?!?"

     Wanting to start the school year off with a bang, I decided to introduce my students to my classroom culture through a mathematical modeling problem that I did not have an answer to.  The Laptop Battery Task from Illustrative Mathematics ( is one that I've used with teachers during professional development, as well as with students, and I like that it makes everyone uncomfortable.  Teachers and students alike have something to gain from engaging in this problem, and other problems of its kind.  Students are uncomfortable because this is possibly their first encounter with a math problem where their teacher doesn't know the answer, and teachers are uncomfortable with presenting a problem that is so open ended.  It's scary stuff!!!
     On the first day of school, I thought I would take a risk and see how my students would handle The Laptop Battery Task.  I went over several things on the first day, laying groundwork for classroom culture, and with 15 minutes left in the 90 minute period, I put this task in front of them.  I asked them to work independently and brainstorm a solution for when Jerry might have a fully charged battery. Students exited the class without discussing their work and their homework was to continue working on the task.

     The next day, as students entered, I asked them to take out their work, being thoughtful to not use words like "answer" because I wanted to emphasize that this was a work in progress and their work could be revised.  Students began bouncing ideas off of one another, comparing strategies and initial solutions.  It became immediately clear that no two students arrived at the same answer, or if they did, they got there in different ways.
     As I circulated the room, I recorded times that students predicted Jerry's battery to be fully charged, as well as what methods students used to arrive at their solution.  I then wrote the times on the whiteboard and students saw that there was a large range of solutions from 10:28 to 11:23.  As far as a teacher move, at this point I wanted to make sure that students saw the two main approaches that students took in solving this problem (graphing and average rate of change), but that we also looked at student work from an individual on the 10:28 end, one on the 11:23 end, and one in the middle somewhere.
     I selected students whose work was to be shown to the rest of the class, but it was the task of their partner to articulate their reasoning and process.  I took this as an opportunity to build collaboration and a culture of being responsible for understanding the reasoning, not just a passive audience member.  It seemed to work very well!
     As the first two students presented, the class saw that graphing would have been a useful option/tool to solve this problem (only one or two students in each class chose to graph), but also that even though their own solution did not match exactly, their reasoning was very similar to those of their peers.  The final student presentation was of the solution of 11:23.  The time of 11:23 didn't sit well with the class, but they had a difficult time articulating why.  They knew in their gut that 11:23 was too late (even the students who had this solution knew something was off), but I loved seeing the perseverance and the unwillingness to back down, as opposed to admitting defeat in arriving at an "incorrect" answer.  What came next was simply magical.
     The partner of a student who had a solution of 11:23 came to the document camera and displayed the work for us to interpret.  Students were instantly engaged and curious to know how their peer arrived at 11:23.  The student work showed proportions, which many students had used, so there was a certain timidness for students to question the solution.  Essentially, this is what the student work showed:

     Students were able to make sense of the proportion and found the setup to be quite useful.  They understood the meaning of the 132 minutes, but they still weren't buying the 11:23 final charge time.  Finally, one student raised their hand and asked the question, "But why are they adding the 132 minutes to 9:11?"  Silence.  Nobody answered...they just looked at me, waiting for me to answer.  I took this opportunity to have them turn to each other and brainstorm their answer to that question.
     Ultimately, we came to the understanding that THIS was the issue with the 11:23 charge time.  The 132 minutes of total time made sense to them, but they grappled with where that fit in the context of this problem.
     A student asked if they could come to the board to draw something that they thought might help make sense of this particular issue.  After picking my jaw up off the ground, I said of course and handed over the whiteboard marker.  This is what she drew:

     Without me saying anything, students looked at the image and began digesting its meaning.  Students were arguing, the discussion was getting interesting.  Finally, I asked if someone could articulate their connection to the diagram and how it helped them understand the problem?  The original student who arrived at a solution of 11:23 raised her hand and gave a thoughtful response to how she had added the 132 minutes to 9:11, which would have meant that the battery charge was at 0% at that time, which was not true.  That was the crux of her issue, and she was able to work through it without me saying a thing.  Other students articulated their thoughts of where the other 66 minutes should go or even proposing another proportion we could set up from here.  They were making sense of the argument of a peer to help build their own understanding.
     Once the excitement had died down a bit, I brought closure to the discussion by highlighting some of the elements of this lesson.  I even asked students to share with me some of their observations of the process as a whole, and they said things like, "You didn't lecture on this topic before giving us this problem", or "You placed a lot of emphasis on how we got an answer, not on the answer itself".  Bingo.
     Of course, it would have been too good to be true if it ended there.  Before excusing them for the day, I had a student raise their hand and ask, "So, what's the answer?"  With all eyes on me, a quick shrug of my shoulders communicated to them that that was not my priority, and I was sticking to it (and, oh, by the way, I have no idea what time it will be fully charged).  Some students left irritated, but overall, I think the students understood that this was not going to be a typical math class.  
     Mission accomplished.

Sunday, August 10, 2014

Memes on Day One

     Much like many of my friends and family, I am getting ready to start another year of teaching.  As always, summer has gone by way too fast, my brain is filled with ideas of how I am going to make this year the best one yet, and I'm excited about the potential that this year holds.  I wanted to do something a little bit different this year for the first day of school.
     Like many teachers, the first day of my class involves reviewing my syllabus, discussing expectations, setting boundaries, and other things that students will most likely not remember in a few hours.  I thought I would mix things up this time around.  One thing that always guarantees a laugh from me is a good meme.  My personal favorites are the "Most Interesting Man In the World" memes that are in the form "I don't always...but when I do..."  They get me every time.
     I've compiled a few memes in a powerpoint to share with my students on the first day (I did not include any of the most interesting man).  My hope is that this presentation will accomplish two things; 1) they will learn and remember what my expectations are for the class, and 2) they will begin to understand my teaching style; a balance of humor, hard work, and a belief that they can be successful, regardless of their previous experiences in a math class.
     Stay thirsty my friends