IPS - Day 8

Today we finished our exploration of simulations and experimental versus theoretical probability. We discussed the Sounding an Alarm worksheet. Most students did not have any idea how to list out the sample space. They also had no recall of calculating probabilities from previous classes they took.

We discussed the sample space and came up with the 8 possible outcomes. We then discussed how to calculate the probability for each outcome. As we calculated the probabilities, some students started to get a better sense of what they should be doing. The theoretical probability of having at least one alarm sound was calculated to be approximately 98%. This compared favorably to the 96% we calculated from a relatively small number of simulations. I discussed how sometimes we can calculate theoretical probabilities but that running enough simulations would get us close to the theoretical results.

The first portfolio problem was assigned today. Portfolio problems are designed to have students fully explain their reasoning and justify their results. I allow students to make revisions on their work until the get the results correct. Portfolio problems become mentor text that show students what they need to do when responding to questions. Students do not receive credit for a portfolio problem until it is 100% correct.

I also use a grading standard of Essential Correct (E), Partially Correct (P), and Incomplete (I). An E says that students "get it" and know what they are doing and can communicate their results and reasoning. There may be minor issues but that these are things that a student could, in essence, self-correct. A P indicates a student knows what they are doing but gets stuck, does not explain their reasoning, or would need some prodding or help through questioning to move on. An I indicates the student cannot proceed without a lot of assistance or provides an answer without any explanation of their thinking or justification of their results.

All graded work is scored using E, P, and I. On a test, a student getting all P's would receive a grade of C. A student getting all E's would receive an A. A student receiving a mix of E's and P's would fall somewhere in between.

The first portfolio problem asks students to assign random integers to simulation situations and explain why those assignments work. Since simulations will be used throughout the semester, I want to be sure students are comfortable with how to assign appropriate values for simulations. This is the first time I have used this particular version of the worksheet and I know that I want to modify it to reflect not just the assignment of digits but to include a description of the simulation process as well.

After a brief summary in their notes about what to remember when simulating situations we moved on to probability rules.

To get things going I have students work through an activity involving the Monty Hall problem. We use three cards (2 black and 1 red or vice versa) and have students work in groups of three. Each student rotates through a roll of player, host, and data recorder. The objective is to determine the probability of staying and winning versus the probability of switching and winning.

This took more time than expected as students did not understand the written instructions. I guess a whole class demonstration would help here. Once they understood what they were doing the activity moved along. We calculated probabilities of 36% and 55% for the two situations, which clearly show there is an advantage to switching. I use this to illustrate the probability and intuition do not mix well and that situations need to be thought through carefully to identify an appropriate sample space.

I asked students to consider any probability rules or properties that they remembered from previous classes. Very little came out of this discussion other than probabilities sum to 100%. We then went through some rules which I related to the Sounding an Alarm work. I asked students to review the rules in a textbook and then consider two examples and how the probability rules were applied to the situations. As one of the examples involved rolling a pair of dice, we discussed how the sample space was depicted and then I asked students to calculate some probabilities for some events: rolling a 7, rolling a pair, rolling a 3, and rolling a 4.

We'll work on applying probability rules next class.

Visit the class summary for a student's perspective of the class and to view the lesson slides.