We have spent a lot of time rolling dice in my MDM 4U classes so far this year. One day we played 3 different short games all involving being in pairs and rolling a pair of dice.
Here are the details of the three games.
Game #1:Roll 2 dice and consider the sum of the dice.
Player A wins 1 point if the sum is EVEN. Player B wins 1 point if the sum is ODD.
Game #2: Roll 2 dice and consider the product of the dice.
Player A wins 1 point if the product is EVEN. Player B wins 1 point if the product is ODD.
Game #3: Roll 2 dice and consider the sum of the dice.
Player A wins 2 point if the sum is EVEN. Player B wins 1 point if the sum is greater than or equal to 7.
This lead to a discussion of which player would you choose to be in the game and why.
For Game #1, I had a justification for 3 different outcomes:
1. Choose Player B because the mode sum is 7 so there would be more odd outcomes.
2. Choose either one because Even and Odd are equally likely (but not sure why they are equally likely)
3. Choose Player A because Even outcome more frequent
Even # + Even # = Even #
Even # + Odd # = Odd #
Odd # + Odd # = Even #
It was great that students were willing to share their thought process and we had supporters and opponents of each theory. We then developed a chart to show all possible outcomes of the sums and saw that both Even and Odd outcomes are equally likely. Probability was introduced.
In Game #2, based on their results of playing the game, each student would choose to be player A. Similar to game #1, by examining a chart of the possible products, it was clear that Even outcomes are more likely.
We started to discuss the idea of fair games and mutually exclusive events.
Homework for the class was to determine how to chance Game #2 so that it was equally likely for Player A or Player B to win the game. Students came up with some alternate version of one of the following:
- Player A gets 1 point for an EVEN product and Player B gets 3 points for an ODD product (changing the point value of each player)
- Player A gets 1 point for a product of 1 and Player B gets 1 point for a product of 36. All other products result in no points for either player. (only selecting some of the products)
Further discussions in upcoming classes will focus on fair games.
Monday 26 September 2016
Thursday 8 September 2016
Spiralling MDM 4U - Day 1
I've made it through my first day adventuring down the path of pseudo-spiraling with MDM 4U. And it was actually quite successful and fun! My first disclaimer is that what I am doing may not be true spiraling but I can definitely say that it is activity-based learning. The students have been warned that each class may look disorganized but there is a master plan! My biggest leap of faith was leading a lesson without having a nice handout for students to fill in as we went through the activity.
Today's lesson began with each student being given a dice and asked to roll it 30 times. They were asked to keep a tally of each value rolled in a chart. Students were then asked to chat about their outcomes where they discussed why a few of one number and more of another arose and the discussion led to each value should be represented equally.
We then tallied everyone's responses in one large table and discussed these findings. There was some excitement when the first two columns (# of 1s and # of 2s) were the same and then further intrigue when there were almost double the number of 3s. Again, the discussion led to why each value was not represented equally.
A comment was made about the size of our sample (287 total rolls) and if this was large enough. This is where we distinguished between a population and a sample and started to lead to the idea that the closer the sample represents the population, the better representation it is of the population.
The data was then graphed in a histogram (first individually by each student and then together with the large group). One question that arose was "why can't we draw a scatterplot?" I was really excited that this came from the group and I wasn't telling them right from the start that a histogram was the way to go because of ...
Again, we discussed what should the histogram look like in an ideal situation and why use a histogram over other graphs. The terms discrete random variable and continuous random variable were introduced. Outcomes and events were discussed.
Students were then given a second die that was different than 6-sided and asked how their results would differ if the same activity were repeated with the new die. The idea that the larger the number of sides on the die, the less uniform the histogram would be for 30 rolls. Furthermore, the larger the number of sides in the die, the more rolls would be required to see a uniform distribution.
We conlcuded the lesson by reading a current article and discussing the need to critically assess the numbers we read. In the future, I will give the article for homework before discussing it to ensure all students have enough time to fully read the article.
Key Terms Introduced in the Lesson:
Population; Sample; Discrete Random Variable; Continuous Random Variable; Event; Outcome
Today's lesson began with each student being given a dice and asked to roll it 30 times. They were asked to keep a tally of each value rolled in a chart. Students were then asked to chat about their outcomes where they discussed why a few of one number and more of another arose and the discussion led to each value should be represented equally.
We then tallied everyone's responses in one large table and discussed these findings. There was some excitement when the first two columns (# of 1s and # of 2s) were the same and then further intrigue when there were almost double the number of 3s. Again, the discussion led to why each value was not represented equally.
A comment was made about the size of our sample (287 total rolls) and if this was large enough. This is where we distinguished between a population and a sample and started to lead to the idea that the closer the sample represents the population, the better representation it is of the population.
The data was then graphed in a histogram (first individually by each student and then together with the large group). One question that arose was "why can't we draw a scatterplot?" I was really excited that this came from the group and I wasn't telling them right from the start that a histogram was the way to go because of ...
Again, we discussed what should the histogram look like in an ideal situation and why use a histogram over other graphs. The terms discrete random variable and continuous random variable were introduced. Outcomes and events were discussed.
Students were then given a second die that was different than 6-sided and asked how their results would differ if the same activity were repeated with the new die. The idea that the larger the number of sides on the die, the less uniform the histogram would be for 30 rolls. Furthermore, the larger the number of sides in the die, the more rolls would be required to see a uniform distribution.
We conlcuded the lesson by reading a current article and discussing the need to critically assess the numbers we read. In the future, I will give the article for homework before discussing it to ensure all students have enough time to fully read the article.
Key Terms Introduced in the Lesson:
Population; Sample; Discrete Random Variable; Continuous Random Variable; Event; Outcome
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