Birthdays & Probability

We are putting the finishing touches on our probability unit and there’s one more problem that I want to show my kids.  The problem goes something like this:

“What is the probability that in a group of 30 people, at least 2 people have the same birthday?  Assume there are 365 days in a year.”

For this particular problem, I like to lower the level of abstraction to get everyone to buy in.   I say the following:  “We have 29 people in this room (including me).  There are 365 days in a year.  Take a guess at what the probability would be that 2 of us share the same birthday.”  We aren’t talking about the birth year here, just the birthday.  I ask the kids to take a guess and write it down.  I want them to own their thoughts in writing. We talk to people around us to compare answers.  There’s a shy kid in front who nobody turns to so I pair up with him.  I poll the class to get the lowest and the highest answer and write them up on the board.  Today the guesses ranged from .05 – .49.

I then ask the kids to think about how they could possibly calculate the theoretical probability and give them some time to brainstorm either by themselves or with each other.  We entertain their ideas and as I’m walking around eavesdropping on conversations, I’m writing things down on the board that I’m hearing.

“365^29 – Geez that’s big”
“It’s a permutation”
“We need to use the complement”
“Let’s make this easier by using only 5 people”

After letting them wrestle with the problem for a while I realize that they are getting closer, but not quite there yet.  It’s time for me to interject.  I go through the solution using this strategy I read about on the “Ask Dr. Math” section of the math forum.  There’s a wow factor buzzing through the classroom as we determine that in a room of 30 people there’s a .71 chance that at least 2 of them share a birthday and in a room of 90 people there’s a .99 chance.  We go back to their guesses and the kid who guessed 49% gets a prize out of my ‘Box of Wow’ for being the closest.

Now it’s time for the Johnny Carson video. In this short video clip, Johnny is marveled by the probabilities of this particular problem and then polls the audience trying to find a match.  It’s worth the 2 minute watch.  I ask the kids to talk about their reaction to the video.  Did Johnny do something wrong here or we witnessing a statistical anomaly right before our eyes?

There are still some doubter’s in the class so someone suggests we try it out. I jump on the opportunity to quickly discuss the differences of  theoretical and statistical probability.  There’s 3 minutes left in the period so we begin going up and down the rows having the kids state their birthdays.  I start with September 23.  Up one row and down the next, no matches.  As we get to the last two rows there’s both excitement yet disappointment showing on the faces of my kids.  Now there’s 3 kids left.  “March 22” says one of them, the next kid in line smiles wide as she calmly says “Me too.”  The bell rings.


SBG In A World of Percentages

When it comes to standards based grading (SBG) I consider myself ‘All In’.  I’ve been practicing SBG in my classroom for about 4 years and for the first time in my 22 year career I’m happy with my grading system.  Up until this semester I’ve been using a 10 scale rubric because it makes for an easy transition to percentages.  I stole this from someone on twitter and I apologize that I cannot cite my source for this.  If anyone knows where it came from, please give this person credit in the comments section.


This semester, looking to take things to the next level, I tried a the classic 4 scale rubric.

4- Exceeds Mastery
3- Demonstrates Mastery
2- Working Towards Mastery
1- Showing Little Mastery

In the beginning of the semester I was quite excited however I quickly learned that this rubric has several drawbacks which made for a difficult semester to say the least.  First off, there isn’t an easy transition to percentages that worked for my grade book (we use powerschool).  I used the Marzano model to convert to percentages however I had to do this manually for each student.

MarzanoIt proved to be a major pain so I only did this at the time progress reports were due.  My students grew frustrated with me because they wanted to know their percentage after every summative assessment and I wasn’t able to provide it to them.  The other major hangup with this system was concerning academic eligibility.  The school requires me to report a grade for each of my students every week. Normally, they pull the percentages from the grade book however since my percentages were not up to date, I had to provide a manual letter grade.  Many, many, many of man hours later, I’m deciding to go back to my 10 scale rubric for the next semester and future semesters after this.

Today Was A Good Day

I’ve been given 2 missions from Exploring The MTBoS regarding blogging and twitter usage and to be honest, I’ve been trying to find the motivation to go after it.  The last thing I want is for one of my blog posts to be forced.  I want my blog to be a written record of “things that work well”.  Ya know when you go home and say to yourself, “Dang, that was a good day today.  What made that work so well?”  So here’s something that went right in my class this week. It involved a new twist to an old lesson that I’ve been doing for years.  The concept is graphing sine and cosine functions.  We’ve gone through all of the transformations and today we are just tying up loose ends.  The old version of this lesson goes something like this:  As students walk in there is a hand drawn graph up on the screen and for a warm up the Ss are to write an equation for the graph in their notebook.  After a minute or two I ask them to compare answers with the people around them in the hopes that there could be some interesting conversations.  I go over the answer(s), we talk about how there could be more than one answer, and we move on to another graph and another equation.  Repeat a few times and Voila, call it a day.


Today, I took used as a tool and the results were crazy cool!  Ss came in, grabbed a chrome book, and clicked a link on my website which took them to this graph.  Desmos Graph small

The challenge, I say, is to use to recreate the graph.  I think it’s important for me to say here that we have been using chrome books, and pretty frequently in class so there’s not much of a learning curve here.  Kids log in and off they go.  They start out with some random guessing and they are getting instant feedback from the graph which then leads to more guessing and more feedback.  It becomes addictive and they start noticing patterns on how the graph changes when the input bigger and smaller numbers.  Most are using guess and check and some of them decide that this strategy can only get them so far.  One girl gets so close but can’t quite bring it home.  She is losing interest and calls me over.  I tell her that she is one digit away from striking gold and ask her to figure out the period of my graph and then look at the period of her graph.  Click…the light bulb goes on and she’s off and running again. Okay, after 10-15 minutes of “play time” I announce that we need to debrief so we can proceed with the days lessons. NO NO NO NO…how dare I make such a statement. They’re asking me for more time because they are soooooo close.  I give them a 5 minute extension.  One kid has been sitting there a while with a content look on his face.  He got it, he’s ready to move on so I walk over, celebrate his knowledge, and then offer him the next challenge.  Now, find another equation that makes the same graph – use sine instead of cosine.  He looks at me with a confused but confident face and he’s back on task.

A debrief with classroom discussion followed. Before we start I asked them to take out a notebook to jot down anything that would be helpful for them.  Nothing was forced as they are now taking notes for themselves, not just “copying” notes off the board.  I just asked some questions and it became a student led discussion.  I have up on the screen and they still have chrome books on their desk.  Someone got the equation to work using an “a” value of 2 and another student got the equation to work using an “a” value of -2.

What?  Is that possible?  Prove it. Why does that work?
Are we convinced that there are 2 answers here?  Are there more?  How many more?
How do you know? Play lawyer and convince us that you’re correct.

I’m changing numbers in the equation and so are they making great observations and connections along the way.  The homework assignment goes something like this…Click the link on my website for another graph.  Use to write as many equations as you can that will lead to the same graph.  Good luck, I’m curious to see which one of you will get the most equations.  See you Monday.

The Thinker Statue

I came across this Thinker Statue at an antique shop and bought it for about $10.  The use I’ve gotten from it in my classroom is priceless.  Here’s why.


As teacher’s, we’ve all witnessed those times when a student has that magical moment in class, when everything comes together at the perfect time and things make sense.  They have a great thought, a great idea, and if they don’t share it with someone, they are going to burst like an overfilled water balloon.  And when they share their thought with the class, It’s almost as if the scene was rehearsed the night before.  They hit their cue perfectly and nail it.  Sometimes the contribution they make to the class ends up getting applause from other students.  I’ve even witnessed an standing ovation a time or two.  Moments like these are the ones that I just want to bottle up and save for another day. 

A thought like that, so powerful, so insightful deserves some sort of reward.  That’s why I have “The Thinker Award”.  It takes just a moment to recognize the student for their outstanding contribution to class and the Thinker Statue that sits on their desk for the remainder of the period reminds them that their contribution was appreciated by all.  After that, they’re in the club and they are the envy of others.  Other students want to earn the Thinker Statue, but they will just have to wait until they have that magical moment.  It will come, they just need to be patient and let it happen naturally.  

How Twitter Changed My Classroom (in a matter of seconds)

And just like that my students’ tweets were answered!  Running with a tweet I saw from @bobloch, I gave my students an assignment to write a product review for the online calculator at  The reviews they wrote were absolutely fantastic.  They wrote about what features they liked, what they didn’t like, what changes they would make and all sorts of other neat things.  I tweeted some of their opinions with the team @desmos and then we received their reply.


I had it up on the screen as they walked into the room and my students went absolutely nuts!  Their opinions were actually being read by people (who really matter) other than me.  Will they be a little more motivated on the next product review assignment?  You bet!