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.


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.

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:


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.

Class Opener – Day 11 – the Sequence that Pays!

Here’s the picture which greeted students as they enetered today:


Those are all odd numbers.

Why are they purple?

What’s with the dollar signs?

They are prime numbers.

Are they all prime? How can we tell? Many students remembered the “trick” for determining if a number is a multiple of 3, but how do we check 8191?  Maybe 13 or 17 goes into it, or some other funky prime.

But once we establish they are prime, what’s special about THESE primes. I let this sequence marinate for a bit as I continued instruction, and eventually offered a hint.

Add 1 to each number in the sequence.

This led to a clear observation in one class, and one I didn’t expect in another:

They are powers of 2

They are numbers from the 2048 game

So true! But what is special about these powers of 2, and why did I exclude others?

MersenneThe big reveal here is that the numbers shown at the start are Mersenne Primes: primes of the form shown on the right, where P is also prime.  And the Great Internet Mersenne Prime Search provides an opportunity for the amateur mathematician to participate in a quest to find bigger and bigger Mersenne Primes, and perhaps score a cash prize for your effort! Visit the GIMPS site for more information about the search, and the current status of the project, where the largest prime found boast over 17 million digits. While many in the class wondered why anyone would care about such primes (we will discuss codebreaking later our course when dealing with Matrices), others seemed intrigued by the pattern. And as we begin to discuss counting priniciples and large numbers tomorrow, this was a neat way to foreshadow.  Finally, I want to live someplace which values math so much that they put math on their postage, that would be cool….

Postage 1

Postage 2

Class Opener – Day 10 – the Venn Menu

After a probability quiz on Friday, students were given a problem to tackle and finish for homework. The problem, Come Fly With Me (shared below), features many overlapping events which students need to process. Ideally, the problem is best summarized using a Venn diagram, though certainly other methods can be used to reason it out.

While I find that 9th graders have generally been exposed to Venn diagrams, they also have little conceptual understanindg of how these diagrams are used to process overlapping events. To generate discussion, this photo appeared on the board as students entered:

Venn Menu

If I buy a sandwich with bacon and sausage, where should I place my name? Should I place my name in the bacon only space, as I am getting bacon? How about sausage? And how do we feel about the placement of that mushroom circle?

Now it’s time to go over the “Come Fly With Me” problem, given below, and find out if the class absorbed anything from our brief Venn discussion.

So, did our opening discussion help students use Venn Diagrams more effectively? Results are mixed, as some groups altered their assignment based on the discussion, while others kept the numbers as they were. But hopefully a few students were reminded of the power of these organizational tools.


Class Opener – Day 9 – Pregnancy is Like Drawing Marbles

Isn’t it the best feeling in the world when a former student checks in to let you know about their expereinces, their adventures, and how much your class influenced their life?  Today’s opener is a Facebook post from my former AP Stats student Aneglo as he starts medical school. He shared this probability nugget from a course regarding risk calculation:


His observation regarding probability in the medical-world setting, and class advice for me,  is priceless:

Just tell them that pregnancy is basically like reaching into a bag of different colored marbles.

Message received and relayed. Thanks Angelo!

Class Opener – Day 8 – 9/11 Memoial

Can it really be 13 years since the Twin Towers were attacked?  I clearly recall my old classroom, where a teacher across the hall told me I needed to turn on CNN.  Now the students in front of me today have little recollection of that day: they were 1 or 2 years old.  But they are all familiar with the day’s events, and I hope today’s opener brings some math context to this awful day.

I shared my post on Meaningful Adjacencies, the method used to arrange the names on the 9/11 Memorial, both online and during a “My Favorites” session at Twitter Math Camp this summer. And I am thrilled that many contacted me this week looking for resources and information.

In the activity, students are asked to list their 5 favorite TV shows on a card, then appraoch the board. Their task is to connect with classmates, and placee their card so that people with similar interests are as close as possible to each other. After they have completed the board, I show a video which demonstrates parallels between this activity and the arrangement of names on the WTC Memorial.


One of my classes organized themselves into 3 “pods”, thinking that their interests were isolated. But by finding “Family Guy” as a shared interest, we can challenge small pods to come together and embrace similarities.


September 11 will never be an easy day, but having this activity to share with kids and think about the important task of memorializing those fallen makes it special.