# Introducing Probability Distributions with “Clear the Board” in MDM4U

Time is flying by in my Grade 12 Data Management class! I can’t believe we are already on our last unit, which is also my favourite unit: Probability Distributions.

I draw a lot of inspiration from other math teachers I’ve discovered on Twitter and through the MTBoS database. One of them is Sarah Carter, better known as Math Equals Love. If you’ve never heard of her, go check out her blog now! Sarah has lots of great lesson and classroom ideas and I borrow from her regularly (thanks Sarah!).

Last year was my first year of teaching and also my first time teaching Data Management (a discrete math, statistics, and probability course in Ontario, Canada). When I was first teaching the Probability unit, I used one of Sarah Carter’s lessons, Blocko! That day one of the teachers in my department and her student teacher decided to stop by my classroom to see what I was up to with the linking cubes. I had them join in the lesson and gave them some linking cubes to play the game with. After I explained the instructions, I was walking around the room to make sure everyone knew what to do, and my colleague asked me, “This is about Probability Distributions, right?” Actually it wasn’t. It was about theoretical and experimental probability. That comment got me thinking, and the more I thought about it, the more it made sense in the context of probability distributions. This semester, I am teaching Data Management again, and I decided to try out Blocko! for probability distributions! Here’s what my lesson looked like:

We started with this warm up question:

We had done questions like this before in the probability unit. Most students used a tree diagram. I encouraged students to think about how they could use counting techniques to answer the question (combinations and permutations).

This was the first day of a new unit, so I reminded students of some terminology and introduced the idea of a random variable: a variable that has a single value for each outcome of an experiment. I had students create a Probability Distribution Table and Probability Distribution Graph for the event of rolling one fair six-sided die.

After that I gave students a modified version of the Blocko game board with spaces for numbers 1 through 6. Each group of 2 or 3 got one game board and 12 linking cubes.

The modified Blocko! game board.

I called the game “Clear the Board” and told my students to place the 12 cubes wherever they wanted on the board, so long as each cube falls in only one section of the board. I explained that I would be rolling one die 12 times, and whichever group is left with the fewest cubes on the board at the end of the game would win. Now that they knew what the game was about, some groups quickly decided to rearrange the cubes. I instructed each group to take a picture of their original game board so that if they won, we would know what the winning game board looked like.

After playing a couple of rounds, I chose one of the winning game boards and drew it on the board at the front of the room (I purposely chose a game board where the cubes were distributed fairly evenly).

Next I introduced the full version: 2 dice, 12 rolls, new board for the sums of the rolls:

This game ran the same way as Sarah Carter’s. Experimental probability is unpredictable. In the first round I didn’t roll any 7s! After a few rounds of playing I chose a winning board and drew it at the front of the room underneath the drawing of the winning board for one die. I then asked students to discuss with a partner: what are some of the differences between the winning boards in the one-die and two-die games? They quickly realized that, as they remembered from the previous unit, when you roll one die all outcomes are equally likely, but when you roll two dice, the sums are not all equally likely.

The winning game boards, overlaid with the probabilities of rolling each value or sum.

I then had students create a probability distribution table and graph for the two-dice game. We compared the probability distribution graphs for one die and two dice: in the first graph all outcomes had the same probability, while in the second they didn’t. I introduced the terminology of a Uniform Distribution and Non-Uniform Distribution to describe the 2 different patterns.

We then did a similar example to the warm up where students practiced creating a probability distribution table and graph.

I like to end my lessons with a summary of the learning goals we covered. I find that announcing the learning goals at the beginning ruins the fun!

Questions? Feedback? Hit up the comments!

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