Category Archives: Discrete Mathematics

Boston Marathon Times

This morning thousands and thousands of people did something I can not even imagine doing . . . they ran the Boston Marathon (It was actually 35,671 entrants to be exact)!

This year the winning men’s time was 2:08:37 (Meb Keflezighi from California) . . . that’s an average speed of about 1 mile every 4.88 minutes.  (As a comparison I re-started Couch-to-5K last night . . . and I ran about 1 mile every 11 minutes).

Anyway, the whole Boston Marathon thing got me thinking . . . I wonder how Meb’s time compares to other people who have won the Boston Marathon?

The first Boston Marathon was run in 1917.  John J. McDermott (NY) won that race with a time of 2:55:10.  He was still averaging about 1 mile every almost 7 minutes.  So, is Meb just exceptionally fast?  Was John just exceptionally slow?

4.21.14

The graph above represents all of the Boston Marathon times–from John to Meb and all of the marathoners in between.  What do the data seem to tell you?  Was John exceptionally slow?  What about Meb?

Averagetime

This shows the average time for 10 year time spans of Boston Marathon winners.  What seems to be happening to marathon times-over time?

If you had to model Boston Marathon winning times, based on the number of years since the first marathon what type of model would you use?  Exponential Growth/Decay?  Linear Increase/Decrease?  Quadratic model?  Why?  Do you think there might be anything noteworthy about the graph as people continue running the marathon?  Will anyone ever run the marathon in under 2 hours?  1 hour? (if someone ran a marathon in under an hour they would be averaging 1 mile approximately every 2.25 minutes)

I’d love to know what you think!  In the meantime . . . I’ll be trying to get under the 10 minute mile mark with my Couch-to-5k app!

Happy Valentine’s Day 143

1 4 5 11

Do you?

It just makes me think of Valentine’s Day.  As a little kid I remember getting them from my parents, in their little cardboard box.  In high school, I used to tell my sweetheart of the month that all I wanted for Valentine’s Day was one of those little boxes . . . forget the flowers and stuffed bears!

What’s that you say?  You don’t know what 1 4 5 11 means?  It’s code.  1 4 5 11 is code for I love Necco Sweethearts.  You know those chalky little hearts with the Valentine sayings on them?  I didn’t make up this code, Necco did.  Have you seen these hearts?

Screen Shot 2014-02-13 at 1.38.46 PM

(Find the top 14 conversation hearts here, including 143).

Do you know what 143 stands for?  I (1) love (4) you (3).  Get?  If not don’t feel bad.  This person Facebooked Necco to find out why in the world she ate a heart with 143 on it!

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Anyway, I think its cute and all but as far as codes go . . . its really not that great.  I mean 143 could stand for lots of things couldn’t it?

But here’s the thing, Necco’s touching on something that mathematicians have used for a long time.  That is, numbers as code for something else.  Try this one:

91215225251521.

OK, OK.  So its not that hard, right?

Its either:

IABAEBBEBEAEBA or ILOVEYOU.  I’m not sure that I would want to send some sort of top secret code through cyberspace if my coding technique was A=1, B=2, C=3, etc.  But what about this code?

4560776126257605

Do you want a hint?  OK.  It looks like that number might be divisible by 5.  Oh!  It is divisible by 5.  I wonder what you get when you divide the number by 5?

Hmm.  We might be on to something.  It seems to me that a great way to code something might be to do the whole A=1, B=2, C=3 thing and then to multiply it by another number.  If the person I’m sending the code to knows the number to divide my code by, the code is pretty darn easy for the receiver to crack and its fairly difficult for a spy to intercept and figure out what it says, don’t you think?

So, find a sweetheart and send them something in code for tomorrow.  My sweetheart is getting this message.  Can you crack it?

22125135191513519235520851182019

Math Dice Game

My mom bought me this little game called “Math Dice” for Christmas this year.  Have you heard of it?  I hadn’t, and truth be told, I’m not sure my mom realized it was a game she was buying it for me.

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When I opened the package she said “I thought you could do something creative with those dice and your math blog.”  In the days after Christmas, I scooped the unopened box of dice into our “junk drawer” (sorry mom) and rediscovered them this weekend while cleaning.  On the back the of the box were the directions to the “Math Dice Game.”

photo 2 copy

This morning I had a little extra time, so I thought I’d give the game try!

Step One: Roll the 12-sided target dice

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Step Two: Roll the three 6-sided scoring dice.  Combine the three scoring dice in anyway to match or come closest to the Target Number.

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Ummm, really?  1, 1, 2?  The closest I could get to 80 was 24.  This is what I did:

(1+1+2)! = 4*3*2*1.  Can you get closer?

My second roll of the Target Dice was 36

My Scoring Dice roll was 6, 4, 2

Super easy: (6*(4+2))=36.  Did you get 36 another way?

My last roll was 20

And my scoring dice were 6, 3, 1

I couldn’t get 20.  I could get 18 and 21, but 20 right on the money was a little tricky.  Can you do it?

Much, much more to come about this fun Math Game, with my new Math Dice . . . I’m working on a table of possible Target Number combinations as we speak!

Math for High Ability Learners–at the University of Iowa

Just a quick note to let you know that I’m teaching Math for High Ability Learners, a 3 week virtual workshop offered at the University of Iowa beginning January 21.  The workshop is worth 1-hour of graduate or undergraduate credit from the University of Iowa.

To register for the course click here.

The focus of the course will be designing mathematics lessons to meet the needs of both the typical and the high-ability learners in K-12 mathematics classrooms.

To get a better idea of the coursework here’s the syllabus:

MHAL Spring 14–Syllabus.

I hope you’ll join me!

 

M&M’s Revisited (for the Last Time!)

If you haven’t been here before, then you don’t know that we’ve already talked about M&M’s twice (here and here) and you don’t know that we’ve talked a little bit about the colors of M&M’s in the bags.

Well, today I want to keep talking about the different colors of M&M’s in the bags.  Except today I want to talk about the percent of M&M’s that are red, orange, yellow, green, blue, brown.  Before we continue our M&M discussion, do you have a guess?   That is, what percent of the M&M’s manufactured are red, orange, yellow, green, blue, brown?

Hmmm . . . Let’s pretend that we don’t know (maybe you really don’t!).  I think a pretty educated guess would be that 16.67% of the M&M’s are red, 16.67% of them are orange, 16.67% yellow, etc., etc.  Can you live with that guess?

I’m going to use the data I collected in my last M&M post, except instead of individual bags I’m going to look at my entire sample of M&M’s.

Here’s the percentage breakdown of M&M’s:

Screen Shot 2013-11-26 at 9.52.31 AM

Let’s make a nice table, based on what I would expect to get, given my educated guess of 16.67% of each color and what I actually got:

Screen Shot 2013-11-26 at 9.52.41 AM

So, I wonder if the distribution of colors I got in my sample would be likely, if the colors of M&M’s really were distributed evenly at the manufacturer?

Luckily for us there’s a statistical test we can use to answer that exact question.  And, luckily for us its a pretty straightforward test to understand!  It’s called the Chi-Square Goodness of Fit Test.  The Chi-Square Goodness of Fit test compares the observed values (in our case my M&M colors) to the expected values (if our initial assumption was true).  In our case we would subtract the expected value from the observed value and square the difference.  Then, we would divide by the expected value.  We’d do this for each color of M&M and add up the results.  Don’t worry, I’ll do it (actually, I did it with the help of this website). . .

Based on the Chi-Square Goodness of Fit Test it’s fairly reasonable to assume that I could have gotten this distribution of M&M colors given the fact that M&M Mars makes 16.67% of each color of M&M’s.

Screen Shot 2013-11-26 at 10.10.29 AM

So, here’s my next question?  Do they?

(So here’s the thing, about 5 years ago the M&M Mars website used to answer this exact question, but in 2008 they stopped.  This person wrote to M&M’s and posted the response)

Use the distribution for Milk Chocolate M&Ms detailed by M&M Mars and run another Chi Square Goodness of Fit Test with my data (or your own, if you collected any).  How does this compare to the 16.67% guess?

 

Reblogged from The Atlantic: The Myth of I’m Bad at Math

Today I’m reblogging a post I saw in The Atlantic this morning.  I’m reblogging it because I think the authors, Miles Kimball and Noah Smith, do a great job of articulating a sentiment that I think many, many mathematics teachers share.

(And because they talk about Terence Tao, and any true Math Warriors fan knows his importance in the world of mathematics!)

Enjoy.