0 Probability Bingo

Foam Counting Blocks to Make Dice (From Dollar Tree)

Over Thanksgiving Break, I picked up some foam counting blocks from Dollar Tree for $1.  I don't remember where I saw the idea originally, but someone had suggested that these be purchased and used to make your own dice.  They've been sitting under my podium, just waiting to be used.

I'm not positive if I ever actually blogged about it, but I attended an AP Summer Institute for AP Statistics this summer at The University of Tulsa (my alma mater) with Dave Ferris.  Dave is an AP Stats teacher in Noblesville, IN.  He has a website that is chock full of stats resources.  This is my first time to ever teach statistics, so I have referenced it frequently.

My statistics students have been working really hard since we came back from Christmas Break.  We've been dealing with the probability of random variables.  I was in the mood to postpone the next section in our textbooks by doing a fun, thought-provoking activity.  (Don't worry.  This is a non-AP stats class.)  

On Dave's probability resource page, I found a file called "Probability Bingo."  I opened the file, and I was instantly intrigued.  (By the way, the file gives credit to Brian Mehmed, 2011 WV APSI.)  

Here are the instructions at the top of the bingo card:

Each of two die has colored faces, 3 green, 2 blue and 1 red. The two dice will be rolled. The outcome will be considered to be one “bingo call.” If you have this outcome on your bingo card, mark it off. The winner will be the student who gets a bingo card completely marked off (all 25 squares). Mark each square on your bingo card (use BG for “blue green,” BB for “blue blue,” etc.) so that you have the best chance of winning.

Probability Bingo Dice

Two of my students volunteered to make the necessary dice for this activity.  At first, they colored the sides of the foam cubes.  But, the colors were hard to differentiate since the foam was orange underneath.  I suggested that they take a sharpie and write the first letter of the name of the color on each face of the die.  They decided we should have just done that in the first place.  

After the dice were made, I handed out blank bingo cards (two-sided) to my students.  We discussed how the dice were made, and they got to work filling out their bingo cards.  I decided I wanted to play along.  I quickly calculated the probability of each combination of colors.  I multiplied each probability by the 25 bingo squares to determine how many squares I should label with each color.  I had one square left at the end, so I went ahead and labeled it RR even though the probability of rolling a RR is approximately 0.03.  

The students took turns rolling the two dice.  With each roll, we marked off the combination once.  This is not your typical bingo game.  You're not looking for five in a row.  You're looking to be the first to fill up your entire bingo card.  At first, I was off to a good start.  With every roll, I got to mark off a square.

Probability Bingo Card - Game 1 (Beginning)

Then, the dark time began.  I quickly ran out of Blue/Green squares.  So, did my students.  We all agreed that we should have put more Blue/Green squares on our bingo cards.  The GG combination was not rolled as much as I would have liked.

Probability Bingo Card - Game 1 (Middle)

I didn't take a picture of it, but I sat for a long time with only my RR square unmarked.  I wasn't the only person waiting for an RR.  Eventually, an RR was rolled, and I won!  

Probability Bingo Card - Game 1 (End)

My students instantly wanted to play again.  I had anticipated this, hence the double-sided bingo cards.  Based on our first round of bingo, my students set out to create a better bingo card.  One of my students decided to calculate the probability like I had.  She accidentally left the BB combination off of her card.  She was not happy about this!

Just for fun, I decided to leave my card exactly the same since I had won the first round.  Here's the start of Round 2.

Probability Bingo Card - Game 2 (Beginning)

Pretty soon, history repeated itself.  Once again, I found myself waiting and waiting and waiting for RR.

Probability Bingo Card - Game 2 (End)

That Red/Red combination never came because another student filled up her card first.  The moral of the story?  If there's only a 3% chance that something will be rolled, don't put it on your bingo card.

In order to give my students a grade for the day, I asked them to calculate the probability of rolling each color combination.  Then, they had to critique the strategy of designing your card to reflect these probabilities.

Colored Dice Probabilities 

I would definitely recommend this activity!  It was thought-provoking and fun.  If time allows, I would love to play this with my Algebra 1 students when we review probability for the EOI.

Detail: Probability Bingo

0 The Probability of Marriage

I'm gonna be honest.  My students and I have way too many conversations about my current relationship status.  I'm not married, and that bothers them.  Students have stopped me in the Wal-Mart parking lot to tell me who they think I should be dating.  They interrupt class to give me dating advice.  I'd love to think that this is because they care so much about me.  But, sometimes I think they'll talk about anything if it gets them out of talking about math.

Last week, I was walking down the hall during my planning period when a student in the office called out my name.  This was a student I had last year for Algebra 1.  I have students I get along with extraordinarily well.  This student is not one of them.  We tolerate each other, and that's about as far as it goes.  Last year, he made it very clear, through his actions, that he did not want to be in my class.  I walked in to the office, and this is the conversation that followed:

Student: Do you still eat lunch with the science teacher every week?
Me: Ummm... I've never eaten lunch with the science teacher.
Student:  Yes, you have.  I've seen you.
Me: No, I haven't.  I promise.
Student:  Well, maybe you should start.  Where do you eat lunch?
Me: I eat lunch in my classroom by myself.
Student: Why?
Me: I've got things to do.
Student: Well, you're never going to find a boyfriend that way!

The conversation took a turn after this.  Much to my surprise, this student expressed excitement over the fact that he was going to get to have me for Algebra 2 next year.  "You are going to teach Algebra 2 next year, right?"  "I am going to get to have you next year, right?"  "You are going to still be here next year, right?"  "If you're not here next year, I'm going to hunt you down and find you."  I'm going to take that threat as a compliment.  This is proof that students do come around.  He isn't the first student this year to come to me and tell me, in retrospect, how much he enjoyed my class last year.  This is important because it seems like I have a lot more haters this year.  There's hope, though.  They may just come around, too.  

So, enough of that tangent.  Let's talk about some math.  Remember the cuboctahedrons?  In the same document that I found the cuboctahedron net in, I also found a reference to a Russian fable that predicted whether a young lady would get married in the next year.  It seemed like an interesting probability problem.

Last week, I posed the problem to my statistics students.  Apparently, in some Russian villages, they use a certain method to determine which girls will be married in the next year.  Fold three long blades of grass in two.  Have someone hold them so he loose ends are hanging down.  Tie the ends together in three knots.  The person holding the blades of grass lets go.  If your knots have led to the formation of one large loop, you will supposedly be married in the next year.

Is this true?  I have no clue.  Either way, it still makes for an interesting problem.  First off, what is the probability that you will end up with a large loop?  I wish I had had my students write down their gut instinct regarding the probability before making any calculations.  To me, the likelihood that a large loop would be formed was quite slim.

Predicting the Future...

I set my students loose on this problem, and they were quickly frustrated.  During this time, I had been cutting some pieces of clothesline rope (the only thing I could find in my cabinet that would work!) to model this scenario.  A pair of students acted out the ritual.  And, amazingly, a large loop was formed!  Wedding bells would be ringing!  Another pair of students acted out the ritual.  Wow, another large loop!  Maybe this is more likely than we thought...

I only have five statistics students.  After five trials, we had 3 successes and 2 failures.  It was my turn.  A student held the pieces of rope for me.  I tied the ends together in three separate knots.  By this time, I had made a probability tree model and knew that the probability was over 50 percent that I would be married in the next year.  (I won't give away the actual probability in case you want to try out this problem on  your own.)

What was my result?  I know you are DYING to know.

My Future

Not surprisingly, I am supposed to be wed in the next 51 weeks according to this piece of rope.

We'll see...

I ended up having to walk my students through how to make the tree diagram for this problem.  We haven't done much work with tree diagrams because they aren't presented in our textbook.  If I ever teach statistics again, I will introduce tree diagrams and then have students walk through this problem with the instruction of making a tree diagram.

One student wondered if this is where the term "tying the knot" originated.  I did some googling, and it doesn't seem so.  But, I thought that was very creative thinking!

Detail: The Probability of Marriage

0 Cuboctahedrons: A Perplexing Polyhedron Probability Problem

Due to the Thanksgiving holiday, we only had two days of school this week.  Needless to say, my students were not very excited about having to come to school on Monday and Tuesday.  Several of the schools around us took the entire week off, so that made the week seem even more torturous to my students.

My statistics students were especially restless.  We're in the middle of our unit on probability.  Monday, we looked at some probability word problems.  On Tuesday, I wanted to do something fun and interesting but still related to probability.  I did a quick google search of probability activities, and I ran across a net of a cuboctahedron.  Isn't that just a fun word?  Cuboctahedron.  Cuboctahedron.  Cuboctahedron.  It just makes me smile.

The activity instructed students to assemble their own cuboctahedron.  (The net is on page 4 of the linked PDF document.  I'm also intrigued by the probability activity on page 5 that involves acting out a Russian fable that predicts who will get married within the next year.)  Then, they were to toss the cuboctahedron 100 times and count how many times it landed on a square face and how many times it landed on a triangular face.

Assembled Cuboctahedron and Net Pattern

I let my students each pick a sheet of cardstock from the cabinet, and I quickly ran off nets for them to cut out and assemble.  The cutting and gluing process was more time intensive than I realized.  This activity took the entire 50-minute period.  Since it was the day before Thanksgiving break, this was perfectly fine.  Most of my students ended up opting for tape because the net was so hard to put together.  I used glue, and it works fine if you have enough patience to let the glue dry a little between steps.

My Class' Finished Cuboctahedrons 

The cuboctahedron consists of 6 square faces and 8 triangular faces.  Students were asked to predict the probability that the cuboctahedron would land on each type of face BEFORE tossing it 100 times.

As a class, they decided that 6/14 of the faces were squares.  Therefore, the probability of landing on a square face was approximately 0.43.  8/14 of the faces were triangles.  Thus, the probability of landing on a triangular face was approximately 0.57.  '

Our Class Data 

In 299 trials, the cuboctahedron landed on a square face.  In 91 trials, the cuboctahedron landed on a triangular face.  So, the experimental probability of landing on a square face was approximately 0.77, and the experimental probability of landing on a triangular face was approximately 0.23.

My students were intrigued by this data.  I'm not sure what the authors' motivation was in writing this activity.  Were we supposed to get these surprising results?  We had a discussion of the difference between theoretical and experimental probability.  What is the reason behind this discrepancy?  Is it related to the differing areas of the faces?  Or, is it as one student suggested related to the way that the cuboctahedron lands?  It often hits on a corner, and when this happens it almost always favors the square faces for landing.

I liked this activity because it got my students thinking and talking about math on a day when they didn't feel like doing any math.  It's a rare thing when I give my students a problem I don't already know the answer to.  I need to do this more often!  Does anyone know more about this perplexing polyhedron probability problem?  

Detail: Cuboctahedrons: A Perplexing Polyhedron Probability Problem