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Class Openers

Class Opener – Day 14 – Many, Many Meals

Starting one of my favorite units of the year: permutations, combinations and the binomial theorem. I stumbled upon this article proclaiming the arrival of 140 Million Burger Combinations, heading to New York City – and when I see combinations in the news, it’s time to investigate.

burgersThe article comes from 2010 and heralds the opening of an exciting new direction in burger construction (which has since closed). The website for 4food is sill active for now, and has a cool applet where you can build your own burger. There are many decisions to be made, and some exotic choices (a scoop of roasted brussels?).  I offered students the opportunity to create their own burger, with stations set up on my laptop and on my ipad. Much pro-con debate over the appropriateness of burger pickles ensued!

The choices 4food offers (or, offered, as they are closed…) were summarized by my students:

  • 4 choices of bun
  • 4 choices of “add-on”
  • 10 condiment choices
  • 5 cheese choices
  • 3 “slice” options
  • 12 “scoops”
  • 6 patty options

But multiplying these numbers does not get us near 140 million…so what gives? My classes will explore this problem deeper in the coming days, but for now some seeds have been planted. Soon, we will consider the possibility that you could select multiple condiments, cheeses, and scoops, and work to derive the final count.

This scenario brings to mind a counting principle challenge I have provided classes in the past:

The Tastee Donut Shop charges eighty-nine cents for its Mix N Match selection, which allows you to select any three doughnuts from among the following varieties: plain, maple, frosted, strawberry, blueberry, vanilla, chocolate, glazed, and jelly.  How many different Mix N Match selections are possible?

Here is a printable version of this problem you can share with classes.

I enjoy this problem because students need to think beyond a one-step counting problem. This challenge is more sophisticated than many worksheet problems in that we need to consider a number of possibilities – could a customer buy 3 of the same donut? 3 different donuts? 2 and 1? In the end, the solution comes down to the sum of 3 distinct possibilities, each more challenging:

  • Buy 3 of the same donut (easy): 9 ways
  • Buy 3 different donuts (medium): compute 9 choose 3
  • Buy 2 of one type, a 1 of another (hard): we need to pick two flavors. But picking 2 glazed and 1 jelly is distinct from 2 jelly, 1 glazed. Order matters. Compute 9 pick (permuation) 2.

 

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Class Openers

Class Opener – Day 13 – Serenity Now!

Is Mr. Lochel asleep?

Today is the first major test for my freshman classes, and for all of them it is their first big test as high schoolers. And after a Back-to-School Night the evening before where I discussed some study strategies for the 9th graders, it’s a pretty stressful day for the young ones.

As students entered I heard the usual cacophony of frenzied papers being shuffled and concept cramming. It’s just too much noise and too much distraction before a test. Time to change the culture some with a video:

5 minutes of restful waves and ocean breezes to clear the mind. But it took a few minutes to take hold. At the start, as I sat in the front, eyes closed, silently contemplating the day, most students continued their frenzied studying. But eventually a few joined in, resting their heads, shushing each other, and taking advantage of a few moments away from math.

I’m hoping that today’s test will mark improvement for a number of my struggling students. I find that students coming from middle school often suffer from a similar mindset when it comes to taking math assessments: every problem must be done rigidly, teachers grade with an eye for missign nuance (arrows at the ends of lines, that sort of thing), papers are returned and go into a folder, and we move on.

This cycle isn’t good enough if we want students to reflect on their progress and grow.  The usual test study formula, where students shuffle through notes and seek more practice problems, isn’t sufficient. And while it is difficult to cause students to completely change study habits, I provided some tips for students as they progressed through the unit:

  • Every time you encounter a sticky classroom or homework problem, place a star next to it. In the days leading up to a test, redo these problems. Have the concepts had time to marinate? Are you now able to complete these problems with less difficulty?
  • Reflecting upon past classroom quizzes is essential. This year, my students are required to re-do all missed quiz items (excluding minor errors) as homework and attach them to their original quiz. I’m happy that a handful of students visited to discuss their corrections, while many more re-visited their previous math sins.
  • Deep breaths and long pauses matter. Undo obsession over that one test question, the one you have been working on for 20 minutes, is probably not healthy. Think about the warm ocean breezes, move on to items you CAN do well, and remain upbeat.
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Class Openers

Class Opener – Day 12 – How Big is Big?

We’re coming to the end of our first unit of the year on basic probability, and headed towards the fun world of counting principles, including permutations, combinations and the binomial theorem.  To review ideas regaring factorial and size, students were faced with the following question on the board:

2014-09-17_0005

Many students ignored the exclamation point right off the bat, giving replies like “it’s a little bigger than 51”, or “pretty big”, until a student realized that I clearly meant factorial here.  This genrated classroom discussion about what factorial meant, and some side discussion about how big a number this could be, including some calculator experimentation. We’re off to a good start!

But just HOW big is this number?  To get students thinking, I asked them to consider what a quantity that big could represent, being as creative (within reason) as they like. Some of the responses were awesome fun.  Did you know Kanye had THAT much swag?

2014-09-17_0011 2014-09-17_0010

To finish this opener, I played one of my favorite clips: from the British panel show QI, Steven Fry uses a simple deck of cards to do something never before done by man! I’ve dicsussed this clip on the blog in the past, so visit there for more info regarding this card shuffling experiment.  Enjoy.