MDM4U - Activity: introducting non-uniform probability distributions

Students were given time to play "Crossing Toronto Harbour". In my de-cluttering of my filing cabinet this summer, I re-discovered a few past editions of the OAME Gazette and found this activity by Kelly Young in the June 2010 Edition.

The main idea: students are paired up and each given 10 "boats" and a 2 dice. Each student rolls the dice and if they have a boat in the dock with that sum, they get to cross their boat to the other side. If that dock is empty, they do nothing and pass the dice to the other player. The game continues until the first player has crossed all their boats to the other side.

My challenge was to find enough items of different colours for students to use as "boats" that were cheap to find but small enough to fit on the game board. I was at the dollar store this weekend and found these packages of decorative pompoms to use.

Students quickly discovered boats in the #1 dock would never cross and that sums of 6, 7 and 8 occurred most frequently.

This lead nicely to a discussion of that the probability of any event lies between 0 and 1 and that not all events have the same probability of occurring. Unlike a standard die that has a uniform probability, looking at the sum of two dice has a probability distribution where a sum of 2 and 12 are least likely to occur while a sum of 7 has the highest probability of occurring. Students were introduced a probability distribution chart where all outcomes are listed along with their probabilities. A histogram or bar graph could be used to graphically display a probability distribution depending on the event used.

I also asked students to keep track of what numbers they rolled so that we could continue to distinguish between theoretical probability and experimental probability. Here is a chart of the outcomes rolled in this class:

We will refer to this data to start next class to chat about why it is not an identical match to the theoretical probability distribution.
Homework was assigned from the question bank focusing on probability distributions as well as determining the probability of different sums when rolling 2 dice.

MDM4U - day 2

Today's class had a two main goals:
1. Set up a Google Community for the course.
2. Further investigate the relationship between theoretical and experimental probability.

Task 1:
One of the overall expectations of the course is "demonstrate an understanding of the applications of data management used by the media and the advertising industry and in various occupations." In the past, I addressed this expectation by presenting current articles I have found or asked students to bring in interesting articles and we have had class discussions around these articles throughout the year.

I decided to go digital with this requirement this year. In today's class, students were asked to join a Google Community for this course. This space will be used to collect and store articles students find interesting or have statistical significance. Also, this platform also allows students to post comments on articles their peers have posted to the community. I started by posting an article myself (it was based on what conditions will be required to run a sub 2 hour marathon) and had students post a comment on either a graph or statistic they found interesting in the article. Students were also asked to post, by the Thanksgiving break) an article they find interesting. I envision this being an avenue to generate peer conversations as well as student-teacher conversations in this course. Also, my ultimate goal is to have students post one article a term for a total of 3 in the year.

Task 2:
We played Dice Bingo - with 1 standard dice. Students created a bingo board with only 5 entries similar to this:

Numbers can be repeated.

The teacher rolls the dice. If the number rolled appears on their board, they cross off one occurrence of it (so if they have three 5s and a 5 is rolled, the student only crosses off one of the 5 and leaves the other 5s in play until another 5 is rolled). If the number rolled does not appear, they do nothing. If the number rolled appears on their board but all occurrences of that number have already been crossed off, students add that value to their points total. The game ends when one student has crossed off all the numbers on their board.

At the end of the game, students add up all their points including numbers that have not been crossed off their board and the goal of the game is to get the least number of points.

We played three rounds of the game and then talked strategy.
- What numbers should you use to ensure you get the least number of points?
- What is the max number of points you can get if the game is won in 5 rolls?
- What is the minimum number of points you can get if the game is won in 5 rolls?

We will revisit the game when the teacher rolls 2 standard die and the entries on the board are the sum of the die.

Homework:
Students were asked to post a comment on the article in Google Communities. Students were also asked to determine the max points possible in Dice Bingo if the game was won in 6 rolls; 8 rolls; and 10 rolls. And finally students were given practice questions determining the probability of given events using dice and cards.

MDM 4U - Day 1

The course had a short introductory lesson yesterday where we reviewed the course outline and I explained to the students that this course would be spiralled. 4 main strands (Collecting Data; Organizing Data; Probability; and Statistics) would be incorporated into each lesson and developed throughout the year.

Today was our first day of full lessons. The goal of today's class was to get students used to the format of the class (away from being used to filling out worksheets) and have students begin to problem solve together.

We played a game of "Skunk" with one die. After the winner was declared, we discussed strategies used in the game. Most students identified that if a 1 hadn't occurred in 5 consecutive rolls, then they felt the need to step out of the round because a 1 was more likely to occur next. This lead to the introduction of the idea of theoretical probability. We defined theoretical probability and the idea of patterns that occur in the long run.

Students were then each given a die and asked to roll it 20 times recording the outcome of each roll. We collected this data as a group and noticed that though we should get each number rolled 3.3 times in 20 rolls, not a single student has this occur. However, when we summed out data together (instead of looking at 20 rolls at a time, we now were looking at 220 rolls) we saw that the distribution of rolls was closer to equal for each value.

Finally, students were randomly paired and asked to solve two problems on our VNPSs. The first problem was "The sum of 15 consecutive numbers has an average of 27. Find the average of the first five numbers." I was surprised at how quickly they came up with the answer of 22 but had some challenges explaining their thinking. We discussed how important it is, especially in a statistics based course, to ensure you are using the proper terminology.

The second question was "A band of 10 pirates are going to disband. They have divided up all of their gold, but there remains one giant diamond that cannot be divided. To decide who gets it the captain puts all of the pirates (including himself) in a circle. Then he points at one person to begin. This person steps out of the circle, takes his gold, and leaves. The person on his left stays in the circle, but the next person steps out. This continues with every second pirate leaving until there is only one left - who gets the giant diamond. Who should the captain point at if he wants to make sure he gets to keep the diamond for himself? What if there were 11 pirates?"

This second question was great because it was unique enough that it leveled the playing field for every student in the class - it was not based on past course material so students who may have a stronger math background did not have an advantage to solving this problem. There was lots of problem solving and collaboration in groups. Each group provided an answer (not all the same answer) and we will verify solutions as a group in our next class.

It was great first class and I hope it has set the stage for a great year!