On the second slide of today's PowerPoint are two basic probability questions, and these serve as Today's Openers. As students enter the room, I tell them to try to answer both questions. They should feel free to work together or alone.
After a few minutes, we have a brief class discussion.
Given six different colors, what are your chances of guessing the color I'm thinking about?
What if I'm thinking about two different colors, then what is the probability that you can guess both of them?
Am I allowed to repeat colors: for example, can I say that I'm thinking of "red and red"?
If that's allowed, how is the probability different from if there are no repeats allowed? And finally, does the order of my choices matter?
Depending on how much my students remember about basic probability, I may try to explore all of these questions. Compound probability, combinations, and permutations are not really the point of today's lesson, so I won't dwell for too long on the latter questions. I'm curious about who will know how to deal with the question of whether or not the order matters, and as I circulate while student are working, I am learning about the level of mathematical background knowledge they bring to the class.
What's most salient is this point that I make to students:
If there's only one color, this problem is pretty easy, and my guessing the color is just based on luck. If there are two colors, the problem instantly gets more complicated, but it also gets more interesting.
I tell students to keep this in mind, because we're about to choose sets of four colors, and then to see if others can guess them.
Today is the first of three classes that are dedicated to the Mastermind Project. Please take a look at the Project Overview. My main goal for today is to teach all students how to play the code-breaking game Mastermind. For many of my students this is a new game.
How to Play Mastermind
Look at slides #3 through #15 of Intro to Mastermind to see how I introduce the game. I show students the purpose and basic rules of the game, I narrate how the flow of a game works, and I carefully demonstrate how the "judge" will use black and white dots to provide the "codebreaker" with feedback. We practice a round of feedback, and I give students the chance to ask questions. Until they actually play, some students don't realize that they might have a hard time providing feedback, but I'll be able to help with that when they start to play.
I provide all students with a copy of the MM Game Sheet and I explain that this is a "primary source document," just like they use in History class. The difference between this and a history Document Based Question (DBQ), I explain, is that students are producing and documenting their own data, instead of reading from someone else's. At the end of this project, however, it's going to be just the same as in history class. You'll have to refer to you primary source document, produce some secondary sources in the form of data representations, and then give a written interpretation of these sources.
Without further adieu, it's time to play. While the real game of Mastermind is played using plastic pegs and game board, we will use colored pencils and the Game Sheet. The codebreaker writes a guess in the first column, and the judge provides feedback in the second column as the partners pass the page back and forth.
Some students will ask if they need to use colored pencils at all, suggesting that they could do just as well using letters to represent each color. I tell them that this is fine - indeed, this is an example of moving to a more abstract representation of colors - but they should still use the standard form of feedback (black and white dots) so there's no confusion if someone else wants to understand what's happening on the game sheet.
Probably the most common mistake I find judges making is that they think the positioning of black and white dots should change based on the feedback they're trying to give. I explain that there is not meant to be any information encoded in the position of the feedback dots. Everyone should simply provide black dots first, then white dots, without giving away which of the codebreaker's dots are in the right places and which are not.
Data Collection and Representation
As in the previous class, when students played the game of "Greed", I distribute colored sticky notes to all students. For each pair of student partners, one color is for the first code-breaker and the other color is for the student who judged first, then played code-breaker second. I ask students to write the number of guesses it took for them to crack the code, and then to place these sticky notes on sheet of poster paper at the front of the room.
When everyone is done playing and all scoring stickies are posted, I again show everyone the first learning target (see slide #18) and ask how we can best organize this data. It's very interesting to compare this Mastermind data set to the Greed data set that we produced during the previous class. Please see the next section of this lesson my notes about comparing the two data sets.
How to Make a Dot Plot
When it comes time to choose a representation, dot plots make sense this time. And sticky notes do a good job of producing one, because all we have to do is draw a simple number line running from the minimum number of guesses to the greatest, counting by 1's. Then, we can "stack" repeated numbers on top of the number line, quickly producing our first dot plot.
Here is a key contrast between the Greed data and this Mastermind data: the Greed data has a much wider spread than the Mastermind data, and few, if any, repeated values (for more on this see my Choosing a Data Set reflection). The dot plot is a very simple idea, and it works for the Mastermind data, but another key takeaway for students is that the dot plot is limited in scope when compared to box plots and histograms.
One of two primary types of homework assigned in this class is the weekly Problem Set. Stats Problem Set 1 (from survey) is generated from data I collected on the First Day Survey that students completed at the end of our first class. On it, students are asked to create a box plot, a dot plot, and a histogram.
Teacher's Note: I include the survey data from my class in Stats Problem Set 1, but I recommend using real data from your class if you can: I find that this is a great to build investment from my students. It shows them that I take the time to actually use the results from the survey they took the time to complete, and it serves as an example of how successive experiences will build upon each other as we move through the class.
When I distribute the problem set, I remind students that they've already seen a box plot and a dot plot during the first two days of class, and I tell them that they'll have a chance to practice with histograms next time we meet.
Record Sheets are a place for students to write down some of their thoughts at the end of each lesson.
To end today's class, I show students how to start their first Record Sheet of the semester and complete the sentence on slide #21. Instructions for students are also on slide #21. Here is today's prompt:
Today when we played Mastermind, I felt _____________ because...
I make sure to introduce the idea that Record Sheets will serve as another primary source document to which students can refer when they write their papers for this project.