Integer Sequences and the Need for Proof

This post over at Mr. Honner on unstated assumptions reminded me to share a great visual that shows some sequences are not what they seem:

patterns1

Of course, the pattern that jumps out is:

2, 4, 8, 16, 32, 64, …

Geometric Sequence!  Exponential function!  2^(n-1).  HOWEVA,

patterns2

I have yet to bust this out in my classroom, but I envision it being my answer the question “Why do we need proofs?”  Without proof or a counterexample, it is awfully easy to assume that a pattern exists when it really doesn’t.  That’s why we need proof.  Without being able to logically show something is true or false, you can fall into the traps of assumption.

I think I like the idea of setting up students to fail here. Just give them the question and let them try to find the pattern. Of course, their assumption will be that they can find a pattern and I’m sure most would be satisfied that 2^n works with n=1, 2, 3, 4, so it must work with all n.  As Lee Corso would say, “Not so fast my friend.”

Update

patterns5Thanks to commenter Graeme McRae for pointing out this sequence on OEIS (Online Encyclopedia of Integer Sequences).  The sequence can also be defined as the sum of the first five terms of the nth row of Pascal’s triangle:

Update #2

Andy Huynh notes that this sequence should be specified as the maximum number of regions created by putting n points on a circle and connecting them with chords.  As I mentioned in my comment, I think this observation again highlights the need to be able to prove something.  Andy notes that if you evenly space the points when n = 6 (create a regular hexagon), then you will end up with only 30 regions, not 31.  In fact, that’s what I did when I first did this problem which was shared at NCTM by Jen Szydlik.

 patterns4

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Texting in class: Wiffiti and PollEverywhere

Previously: Getting Students to Ask Their Good Questions

How much wood could a woodchuck chuck?

How much wood could a woodchuck chuck?

This week I gave cell phones in class another shot. I had previously tried this during my student teaching to mediocre results and again in my first year of teaching. Similar to the previous times I’ve pulled out Wiffiti (now branded LocaModa), I used it as a way to submit answers electronically. This worked to varying levels of success, depending on my class.

For my first class, I used it as a way for groups to submit answers to a large pool of questions.  However, this turned into a huge disaster. With 7 groups, text messages were popping up at a rate that was too fast for me to keep track of. Unfortunately, Wiffiti does not store these texts in a way that lets a user easily go back and see what was submitted. Therefore, it turned into a lot of me yelling “Hands up!” at regular intervals so I could catch up and the students could tell me what I missed. I was pretty much simply sitting next to the board scribbling results like crazy instead of walking the room like I wanted.  Strike one.

My second section of Algebra II is much smaller, so I had only three groups to keep track of. Also, I turned it into a jeopardy style game: I asked the questions, they submitted their answers. However, once an answer is up, there’s little to stop a group from just looking on the board and seeing it. It was a set-up that didn’t let students struggle with problems and too easily let them just submit the answer they see on the board, rather than work together for a solution.  Strike two.

The third attempt of the day was my most worrisome. Last period. Largest class. 8th and 9th graders. Madness ensued. Even though I resorted to the jeopardy style so I could keep the rate of texts popping up manageable, there was still too many groups to keep up with. My 8s and 9s did not react well to the down time I needed to keep up and it turned into chaos quickly with few kids working, a few kids waiting for an answer to text in, and a bunch of students just sitting around.  I had to stop the activity and resort to individual practice.  Strike three – at least it was swinging.

Answer: 2 million

In 2012, how many people proposed on V-Day?

None of these times did I feel the benefit from the novelty of being allowed to text in class outweighed the clunky-ness of using the phones.  Basically it was just a way to raise your hand and answer a question – nice and fun for a while, but really I’m not gaining anything by using Wiffiti. It’s flashy and cool looking but it lacks the features a teacher would want: recording data and a multitude of options and features. Plus, I have yet to have smooth cell service and quick results with Wiffiti.

Sorry Wiffiti. We once had something nice – you even gave me a nice line on my resume and some great talking points for interviews. Hey, it’s not you, it’s me.

Unlike the last two years, though, I’m going to stick with it. Nearly 100% of my students have phones and texting, so I’m going to start utilizing that on a regular basis (using partners to ensure everyone has the ability to text).  On to PollEverywhere. I had basically forgot about PollEverywhere. Back in grad school it was mentioned, though at the time it was pretty basic. Now, however, I’m ready to give it another shot.

Closers

I tried to make closers a regular appearance since the winter break, but it was never easy nor convenient to collect or grade them. PollEverywhere should make it convenient. I’ll get fast formative assessment and it will be easy for students to submit their answer to a quick question at the end of class.

polle1It will also allow for a way of quick student self-assessment. If 80% of my classmates select the third option, but I chose the first option, I know I have to re-assess the question and my knowledge.  Maybe that is a gateway to sticking after class for 1 minute to ask for a clarification.

Anonymous Questions

This was the new idea that I was hoping Wiffiti could help with. However, PollEverywhere will be much better. If you’re listening to a lecture or watching an explanation of a solution and get lost it is important to ask for clarification. Yet, we know that our students do not want to single themselves out and ask that question for fear of being ostracized. What if they could text in their question and have it appear immediately and anonymously?  I think it will work with PollEverywhere.

polle2The questions show up anonymously and can be submitted via text or online.  Furthermore, I can go back and view a full history of questions that are asked – a feature not available with Wiffiti.

In large classrooms I can see this being useful. The next step is getting two projectors in my classroom so I can have PollEverywhere up as well as SmartBoard slides or PPT slides or the TI Smart View. Don’t know if I’ll get that wish anytime soon though.

Regardless, I see texting becoming a major part of my classrooms now. Despite the failures of the past week, there’s no way I’m going to keep ignoring the fact that nearly 100% of my students have mini-computer in their pockets that can instantaneously share information. That’s power that needs to be taken advantage of.

Drive or Fly?

Due to two snow days, I had to axe a “Leaving on a Jet Plane” project [full credit to Dan Meyer] where students were going to model flight cost as a function of distance from Detroit. Pretty simple wrap up to linear equations. Now, however, I’m pretty excited it got pushed back since our next stop is systems…

I’m hoping the quick visual I put together will provide a bit of spark and interest. Which circle represents the area where it is cheaper to drive than it is to fly?

fly-or-drive-combo

So now, instead of simply plotting flight cost vs. distance, we’ll also plot driving cost vs. distance, helping us find the answer to the question above. A couple plots, a couple lines of best fit, an intersection representing a break even point and YES YOU CAN USE THIS IN REAL LIFE.

UPDATE #1: Here’s the lab that was whipped together.

Fly or Drive Lab – Word Doc

UPDATE #2:  Next year –

The biggest problem I had was with the data collection. This was a class of 13 8th graders and 10 9th graders. It took much longer to collect data (flight cost & gas cost to each city) than needed. I like having that research in there, but it became far too tedious and drew attention away from what we should have been focusing on. Plus, kids that were absent (whether physically or mentally…) while we walked through how to research the costs fell far behind.

Next year I will crowdsource the data collection. Each student picks one city. Each student researches the cost to fly to their single city and the cost to drive to their single city. I would expect this takes about 15-20 minutes. As they get their data, I’m throwing it into a table that is projected up on the board. Everyone uses the same data, which also allows me to more easily check the lab and make sure the data does what I want (the models intersect at a reasonable distance).