(More Of) The Most Beautiful Math I Have Ever Seen

Check out the first post on what my students came up with while doing Sam Shah’s Families of Functions project.  Here’s some more awesomeness:

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The Most Beautiful Math I Have Ever Seen

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There’s something that is just amazing about this graph.  I love it.  It’s perfectly, slightly asymmetrical.  It is deep and detailed and simply beautiful.  And my student made it which makes even more awesome.

For those of you unaware, this is a product of Sam Shah’s Families of Curves project that I shamelessly stole for my algebra II students.  This piece was created with the function:

famfun_1aThe parameter, k, varied on the interval [-7, 7] with steps of 0.25.  The window dimensions are approximately [-3.5, 3.5] in the x direction and [-120, 120] in the y direction.

I’m printing this out as a poster to hang in my room and will be giving the artist a print of their own.  I can’t wait to do this project again next year to add to what will become quite the mathematical art collection!

Tomorrow I’ll post a few of the other amazing pieces my students came up with!

Barbie Bungee and the POWER OF MATH

I’m riding my bike to school a lot in an effort to both save money on gasoline and enjoy the fact that I live just 3.5 miles from school.  The result of this, though, is that I have to be very careful about what I cram into my bag to bring to school.  Long story short: We did Barbie Bungee and not only did I leave my video camera at home, but I also happened to forget my phone.  Sigh.

However, thanks to the fact that we live in 2013 and a ridiculous number of my students have high definition cameras in their pockets, I was able to have a student email me this video, which I edited to slow down.  The video isn’t great, nor upright, but I think it hammers home a significant point:

Like, seriously, how cool is math?  A group modeled the number of rubber bands needed to maximize Barbie’s thrills (but not kills). They were given a height, used their model to predict how long their bungee cord needed to be and look at how amazingly close they got.  Go math.

Quick Note: Holy smokes this is post #99 for this blog.  I guess I need to figure out something special for the next one!

Paper Plate Unit Circles, Done Efficiently

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I like the idea of doing paper plate unit circles and today was my third go at it.  However, in the past it had always been more of an exercise in using a protractor or compass than actually focusing on the unit circle and trig.  So today, I decided I would take the time and do the grunt work for the kids so they could focus on the math.

Step 0: Materials

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Brads, scissors, a Paper-Mate flair fine point marker, the best Google Image search of a unit circle that is blank and has the lines drawn in, paper plates, a spare manilla folder

Step 1: Draw the Circles

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Step 2: Mark off Anglesunitpaper3

This is the true genius of this method, if I may say so.  I simply cut down the unit circle I printed off so that it was slightly smaller than the circle on the paper plates and drew in the tick marks.  Perfect 30, 45, and 60 degree angles in a matter of seconds.

Step 3: Draw the Axes

The axes were drawn simply with a straight edge – I cut an easy-to-wield piece of the manilla folder to do so.  Piece of cake.

Step 4: Cut Some Terminal Rays

Just sliced some appropriate length pieces out of strips of the manilla folder. Then, I took two snipes at an end to create a point.

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Step 5: Teach the Unit Circle, not Protractors/Hand-writing/Neatness

So. Much. Better. Than in the past.  We were able to construct (mathematically speaking here) the unit circle: review degrees to radians, use special right triangles to find the first quadrants, use symmetry and reason to fill in the rest.  Additionally, I still had a good 15 minutes to spare to do some exact value trig problems at the end of class (i.e. tan(30) ).  What has taken twice or three times as long in the past was done in about 25 minutes.

The process of creating 35 “ready-to-go” unit circles, took me 45 minutes. In the past, it would have taken each student about 25 minutes, on average, to do the same for their one paper plate using a protractor.  Much better use of time and we really got to get into the nuance of the unit circle and why it is so simple, elegant, and important.  Students finished it off by using the brad to connect their arrow and then filling in the angle measure (degrees and radians, of course) and the coordinates of the points on the circle.

unitpaper5Voila, here’s your unit circle.  Love it, cherish it, keep it forever and ever and post it in your dorm room freshmen year of college. You will thank me.

Exponent Properties, As Explained By Star Wars Characters

Knocks it out of the park, my student does:

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Seriously, I am suspending my “over 100% is not a possible grade” rule of thumb.  This was all hand-illustrated and some of the color/sharpness did not survive the scanning/jpg-ifying process.  Plus, the cover page has been omitted to keep the student’s name off the Intertubes. I was totally blown away.

I like my timing of this project: Instead of having a test on the exponent properties, we’re doing a project since half of my class is missing this week to a yearly field trip.  However, despite how amazed I am by this student, I wonder: Can she use all these exponent rules?  On its face, this is an awesome alternative assessment, but am I really assessing my students aptitude with exponent properties?  (Clearly, there are mistakes in there, though more of the mismatching examples variety).  Right now I don’t really care – this is awesome, different, and there’s something to be said for holding students accountable to project specifications (see below) and deadlines in addition to cranking out math problems.  But I will let that thought/worry creep into our future assessments, just to make sure.

The project outline and rubric for those who are interested (note: heavily borrowed from an Internet source that I’m struggling to find at the moment. Update hopefully forthcoming):

Exponent Review Project

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:

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Of course, the pattern that jumps out is:

2, 4, 8, 16, 32, 64, …

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

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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.

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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).