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.

Venns

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:

Angelo

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.

Class

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.

Board2

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.

Class Opener – Day 7 – Probability Fail

Today’s opener comes from the site Math-Fail, which I use often for quick visuals and discussion starters.  This particular post is called Probability Fail:

Fail

There was quite a buzz in my room as students discussed whether this question was “real”, or one of my tricks (aside – they seem to be wise to my style by this point – always be skeptical!). But I have no reason to believe the task is anything but authentic, unfortunately. To see what they thought, I had the class answer the question and assessed their thoughts by a show of hands:

How many of you think the answer is A?

A few tentative hands rise.

How many of you think the answer is B?

More hands, yet still uneasiness.

How many of you think this question is just plain stupid?

Many, many hands….

So, what’s the problem with this question? I was thrilled that the first student to volunteer refered to “independence” – that coins don’t talk to or influence each other.  But what about those 8 heads? Is it possible that heads are “hot”, and more likely?  Or maybe tails is “due”.  Fun discussion today, which leads to a true story about a colleague I have shared many times with classes:


HOW TO LOSE $150 IN 4 MINUTES

Our school is about 90 minutes away from Atlantic City, and when our staff was younger we would take semi-regular bus trips to the gambling mecca for a night to let off steam and enjoy some beverages together.

RouletteThis story centers around a dubious strategy for betting on roulette; in particular, using the results board to your advantage.  Here is the strategy my colleague (a highly repected and intelligent social studies teacher) provided:

  • Stand in the center of the roulette table area. Often there are 4 or 5 tables together.
  • Observe the board results. Locate a table which has had a “run” of a particular color; 4 or more of the same color.
  • Place money on the other color to win, as it is “due”
  • If you lose, repeat your bet. When you win, you will have recouped all losses and made a profit

Realizing my colleague’s strategy was full of holes, I was interested to see the theory of “run” put to the test.  We located a table where there had been 5 red results in a row. Time to make a profit!

$10 on black….spin, spin, spin…..RED!

$20 on black…spin, spin, spin…..RED!

$40 on black….spin, spin, spin…..RED!

$80 on black…spin, spin, spin…..RED!

And then we left the casino floor…$150 poorer yet with a valuable probability lesson behind us!

Class Opener – Day 6 – Foul Balls

In today’s opener, the Cleveland Indians provide the hook:

The article from ESPN.com provides the math:

The odds are one in a thousand just to catch one foul ball at any give game, according to ESPN Stats & Information. So what are the odds of one person catching four at a single game?

A cool one in a trillion, or simply a great day in Cleveland.

Pretty long odds. Almost suspiciously long.

Digging deeper, we can find out where ESPN came up with their trillion odds (linked from theblaze.com):

Trillion

After students looked over this arguement, I asked them how many of them had caught foul balls at a baseball game before.  Where did they sit?

I was behind home plate.

I was near third base.

I was in the upper level near first base.

PNC parkFew people from the top sections, the outfield (where clearly a foul ball would not be an issue), and some other goofy sections ever have a chance at a foul ball.  This led to general agreement that some sections are “ideal” for catching foul balls, while others are not so great.  It appears that our Cleveland friend was probably sitting in one of the “hot” sections. The cool site IdealSeat provides heat maps for a number of MLB stadiums, showing you where to sit in order to optimize our chances of catching a baseball. Based on this evidence, is it safe to assume that the probability of catching a ball at a game is 1/1000? We agreed it was probably something lower, based on the lucky man’s seat location.

This is also a great time to talk about the multiplicationrule for independent events, where we agreed that the rule was (for the most part) applied correctly, though with some uneasiness.

I’ve reached the point where anytime probability or odds are quoted on the TV news or in a newspaper or magazine, I immediately am skeptical of the claim. I hope I transfer this desire to dig deeper to my classes.

Class Opener – Day 5 – Our Favorite Numbers

Today’s class opener features the first of many times I hope to expose my classes to the rich site MathMunch.  In this week’s post, the site discusses Alex Bellos’ “Favorite Number” survery, and my classes participated by sharing their favorites on a combined dot plot. Before showing the video below, I asked my classes why they felt certain numbers would be deemed “favorites”, while others may not feel the love.  The class agreed that 7 would probably be the most popular overall, with one astute observation that 7 tends to provide an anchor to interesting lists: 7 days, 7 seas, etc, and seems to have some interesting mathematical properties – like its primeness.

The students had many reasons for having a favorite number:

It’s what I wear for sports.

It’s my birthdate.

My favorite multiplication fact when I was little was 8 x 9 = 72, so 8972 is my favorite number. My dad played it in the lottery once and I won $200

grapesIn the same post, the Math Munch gang expresses appreciation for Bellos’ book The Grapes of Math. So, what’s on your math classroom bookshelf? Here are some favorites I keep around for stories, and for students to check out after assessments:

 

 

bookshelf

Outliers – by Malcolm Gladwell

Journey Through Genius – by William Dunham

The Wisdom of Crowds – by James Surowiecki

In Code – by Sarah Flannery

Count Down – by Steve Olson

Class Opener – Day 4 – the 36 Officers Problem

As students entered today, each group found on their tables the mystery envelope…{cue Law and Order music}.

DiceInside each evenlope were 36 cut out pieces of paper.  Yesterday we had discussed sample spaces, theoretical and experimental probability involving 2 dice, using applets and online resources to think about similarities / differences, so these pieces were not unfamiliar.  Instructions for the pieces was provided on the board:

Arrange your pieces into a 6×6 grid, so that no duplicate red die appears in any row or column, nor any duplicate white die.

TryingGroups immediately gravitatied towards this task, and after some initial misunderstandings over the directions, got down to business looking for patterns. One student who was a Sudoku fan became quite obsessed. I let this problem hang through the period, just having them stop when we completed notes or had class problems to do.  Many students felt they were close, but couldn’t quite solve the puzzle. Half-way through the period I reminded students that it is often helpful to think about a smaller problem before tackling a larger one. What if you just looked at the dice numbered 1 through 4 – can you complete that problem?  Learn from smaller steps – then tackle the big ones.

If you want to try this task on your own, cut out the pieces from the picture above. Then come back here and visit for the solution.

OR if you just want to know how it end….you unadventurous soul…then keep reading…

We’ll wait for you to come back.

THE BIG REVEAL

The problem I gave today is an interpretation of the 36 Officers Problem, a problem often credited to Euler.  Instead of dice, the Officers challenge is to arrange 6 ranks and 6 regiments into rows and columns.  My alma mater, Muhlenberg College, hosts a high school math competition every February – with a free t-shirt to all who attend. For the 36th year of the contest, the shirt features the 36 Officers Problem, and gave me inspiration for today’s activity.

Muhlenberg

As for the solution….here’s the thing….the 36 Officers Problem is impossible.  It’s do-able for all other numbers of rows and columns (except 2), but is impossible for a 6×6.  Some examples of solvable squares appear on this untamed blog, and I’d love to have others contribute their thoughts and or/resources.