Sunday, January 24, 2016

Analyzing M&Ms

Statistics is hands down my favorite unit of all time in either Algebra class. It's the one unit where I feel like you can't not let students make connections and problem solve on their own. So in Statistics for Algebra 2 I always introduce distributions, normal curves, and standard deviation the same way every year: with the M&M lab.

Here's how it works. I give every student a bag of fun size M&Ms and ask them to 1. Predict how many M&Ms the bag has, how many M&Ms the bag shouldn't have, and how many of each color they should have. 2. Record how many of each color in their bag as well as the bags of their group members. 3. Record the number of M&Ms in every bag for our class. Now, at this point the students work through creating a histogram, a probability distribution, calculating mean, and reading through various vocabulary in action. 

But what's really important is what I'm doing and not doing. As the groups work together I'm not answering questions directly, instead encouraging them to help each other and seek the answers from their group members. I'm also recording any questions that casually come up as the students count and record data. And lastly, I'm recording results from the M&M bags for the lesson follow up and the introduction to standard deviation and normalizing data. 

I'm impressed with the questions my students asked, and I'm excited to encourage them to follow up and investigate the results. Here's a picture I took of my board as I added as I recorded the questions.

These are the actually questions my student s asked, just worded nicely to fit on the screen. Because I collected all of the data myself from each group I'm going to use it to follow up and answer some of these questions. Students will learn normal curves and see what the distributions for the bags and each color look like. Then I might bring in a normal bag and a big bag to see how they compare. We might also connect this back to probability to answer some other  questions like, "what percent of bags will only have 4 of the 6 colors?". I'm interested to see where this takes us.

Saturday, January 23, 2016

Permutations and Survivor Season 8

My husband and I enjoy lounging around our living room with our 10 month old son while reruns of old Survivor seasons play in the background. Although we aren't fans of the drama and politics we both really enjoy watching the challenges (yes, we've seen every episode of Ninja Warrior). Before I continue I feel obligated to warn you of potential spoilers for a season that ended many years ago...

I'm watching Season 8: All Stars Episode 10 when the castaways are prompted to "Drop Their Buffs". To set the stage I'm going to provide a quick synopsis of the season so far. There are currently two tribes Mogo Mogo and Chapira. Chapira has 6 members while Mogo Mogo has 4. At this point in the game Survivor almost always forces a switch up (usually the merging of the two tribes). So, each of the 10 participants are asked to draw buffs from a jar because instead of a merge the two teams are going to be redistributed evenly, potentially "mixing up" the game. Imagine every ones surprise when every member of the previous Mogo Mogo tribe are end up together on the "new" Chapira tribe. I look at my husband as my jaw drops and exclaim, "What're the odds of that happening?!" And realize I've got the perfect hook for my students for permutations and combinations.

Now for some classroom background: My Honors Algebra 3-4 students are working at a brutal pace to include all of the material in Algebra 3-4 as well as all of the material in Pre Calculus (this is how we get our students into Calculus before they graduate). This is my first year teaching Honors Algebra 3-4 and although I'm keeping my eye on the state standards and doing my best to align I'm really just following the scope and sequence as it's been done for the past two decades with only a few changes (like the addition of ONE WEEK of Statistics to satisfy common core). That means I had 60 minutes and potentially one homework assignment to teach my students how to find probabilities using Permutations and Combinations. Surprise surprise, the students didn't get it and averaged a 50% for that learning target. What's a teacher to do? Assign practice and offer a retake of course! Unfortunately, I've seen this episode one day too late so now the question is, do I show this to my students for fun? Use it as a test question? Or file it away for next year? 

The plan: show students a clip of the tribe "swap" and then provide the prompt (which hopefully some of them are already thinking) "What are the chances of all four members staying together?"

Work through the problem in groups and collect questions/ideas/suggestions to share out. Two connections I want the students to make and will be listening for: the fifth member of the tribe can be any one member from the opposing group. (Which increases the possible combinations by a factor of 6) and 2. The four members could have ended up on either tribe (which doubles the possible combinations).

The solution as I've interpreted the problem: 2*4C4*6C1/10C5 = 4.8%

Of course I had to pause the show to crunch these numbers immediately on paper. After hearing the result my husband concluded: "Proof that the show is rigged!" But is it? Or is this just how random works? I wonder if my students will think the same.

About Me

This is my 4th year teaching Algebra 1, Algebra 2, and AP Computer Science A. This is my first year as the Algebra 1 PLC leader for my school as well as my third year as Algebra 1 representative in district collaborations (we call the cadre). I'm in pursuit of a Masters in Math Education and have previously completed a BA in Mathematics as well as a BAE in Secondary Education: Mathematics.