IPS - Day 5

Today we transitioned from exploring random events to simulation.

To connect with the idea of what randomness looks like and the Law of Large numbers the class started with a couple of quick assessments of understanding. Students voted on which heads-tails sequence was more likely. Roughly half thought the probability was the same and about 25% went to each individual sequence. The thoughts around the "hot hand" concept was divided roughly the same, half thought there was no such thing, 25% felt like the person was likely to continue with their streak, and 25% thought the streak would end. We discussed the reasoning and students were able to articulate the idea of the Law of Large Numbers.

The One Boy Family Planning activity comes from NTCM's Navigating Through Probability in Grades 9-12. I find the activities in this book to be enlightening and accessible to all levels of students.

I start the activity by asking students to read through the scenario and then develop a hypothesis as to what they think will happen given their current knowledge. We then briefly discuss the simulation aspect of the problem and I ask them to consider other ways that the situation could be simulated. Students came up with flipping a coin, using the random number generator, and mentioned that any device that allowed for two equal outcomes would work.

Students then proceeded to simulate the situation and record their results. Some groups will have misunderstanding of the simulation process so it's a good idea to walk around and make sure everyone is on track.

I ask students to hold off completing the second table as this usually causes problems. I actually have students create two columns: Total Children and Cummulative Children. I give a brief example to illustrate what this looks like. Again, most will get it but some may wander off track. I have students the total number of children and divide it by the number of households and ask what this represents. After a little thought the come up with the average children per household. I then explain how to use the second table to calculate their probabilities.

We discussed their expectations and surprises that came out of the investigation. Many students were somewhat surprised by the average children per household but they did anticipate that roughly half the houses would have a single child since the probability of having a boy was 50%.

I gave students a few minutes to collect their thoughts and summarize their thinking about randomness and to write two questions that they have. I will start class the next class by asking some of their questions.

Next we explored Sounding the Alarm from the Navigating Through Probability in Grades 9-12. In this investigation, I start with providing the scenario and asking what assumptions we would have to make for our simulation. This is always an interesting discussion. I try to push students to consider what they are assuming about each smoke alarm. Some questioning may be necessary to push them to consider the independence of each alarm.

The question of interest is to determine the probability that at least one alarm sounds. I have students consider how the situation could be simulated. In their groups students are very good at determining how to simulate a single alarm. Walking around I ask what they are thinking. Invariably I ask how they would now simulate all three alarms. After additional thought most groups determine a viable way to simulate all three alarms.

Ideas typically include using 4-sided die, spinners, card decks, and random numbers. I explain how students can ignore two sides of a 6-sided die and provide them with their devices of choice to simulate the situation. Again, some students struggle to simulate the results properly but a little guidance quickly gets them on track.

I had each student run 20 simulations and then pool their results within their groups. The idea is to get a probability based upon the entire classes simulation, approximately 500 simulations.

We got as far as completing the simulations and will analyze the data and compare against the theoretical results next class.

Visit my web site to see a student's perspective of the class and to see the lesson slides.