You may want to look at the data chart before starting this section of the lesson. I begin this class by giving each student a penny and asking them to each flip the coin 4 times and record the number of times they get heads. This will be a noisy and somewhat chaotic activity but is well worth the effort because students are engaged from the get-go both by the activity of flipping a coin and also by wondering where it will lead. As my students finish I have them post their data on the front board on a chart I've drawn. (See resources for "Data Chart"). I ask my students to reflect on what the data chart looks like, then select random students to share what they've observed. (MP7) At this point you may have some students who want to critique or edit what their classmates are saying, but I've found it more effective to let students speak without this interruption for this activity so that more students stay engaged. If I allow critiquing during the initial information sharing phase of an activity there are students who begin to shut down out of fear of say something wrong. I expect observations about the measures of center and spread and also about the general shape of the chart. As they share, I summarize their comments on the board, editing only to eliminate repetition.
When we're done, I tell my students that their coin flips were a "simulation" of a real-life situation; the gender distribution in a family with four children, with heads representing girls. This usually gets some interesting comments about the numbers of families with all boys or all girls. That often leads to discussion about whether or not the gender of a baby can really be represented by a coin flip, and whether gender is truly a 50-50 proposition. If that question doesn't come up I ask leading questions to get my students there because it takes us to the next work we'll be doing. I acknowledge that there may not be exactly equal chances of a baby being a boy or girl, then ask what we would need to do to change the simulation if the real proportion is 51% boys to 49% girls (according to the National Longitudinal Study of Youth conducted by the US Dept of Labor). I tell my students to pair-share their ideas, then ask for volunteers to share what they've discussed. I expect a few ideas but have found that since my students have never had to come up with ways to simulate any kind of data their ideas are usually not practical or may not actually model the real ratios. (MP2,4) For example, students may suggest that we roll the coin 51 (or 49) times to represent boys (or girls). This lets me know that they don't understand that the 50-50 ratio comes from having two equally possible options not from the number of flips. Another possible misunderstanding might come as the idea that we could flip 100 coins at the same time and count until we have 51 (or 49). Both of these misconception can be addressed by emphasizing that the number of coins or flips does not determine the ratio of boys to girls; that is taken from the possible outcomes (50% heads and 50% tails). I explain that there are lot of ways to simulate data if we have an idea about what the population data looks like and that's what we'll be doing today.
For this section of the lesson you will need copies of the "Challenges" handout and an assortment of simulation tools. I include coins, dice (6-sided and others), popsicle sticks, paper bags, decks of cards, beans and/or beads, straws and/or toothpicks, and colored paper. Your students will also need calculators with the ability to generate random numbers and/or a random number list (a copy is included in my resources) I've also included in my resources a brief summary, simulation tools.docx of some of the ways I use these tools for simulations and directions for using the random number table or a TI caculator, Random Numbers.docx that you can share directly with your students or use it to help guide your discussion.
Class Discussion (5 minutes): I begin this section by asking how the coin flips we did are a simulation. I'm looking for an understanding that getting heads up represents a girl being born and tails up represents a boy being born. I then ask why the coin flip isn't completely accurate (MP6). Here I'm looking for recognition that heads or tails is equally likely while gender is not. Finally, I ask again if anyone has suggestions for modeling birth if the actual numbers are 51% boys to 49% girls. If there are no new ideas, I give a simpler example using the distribution of students wearing jeans in my class. Instead of telling them how I can represent the jeans/non-jeans students, I hand each student a bead/bean colored to represent their status. For example students wearing jeans get a brown bean and those not in jeans get a white bean. I then collect all the beans in a clear container so they can see that there are different numbers of brown and white beans. This example is generally enough to prompt someone to suggest we do the same thing for the gender problem, using 51 brown for boys and 49 white for girls. (MP8) I could save a little time and tell my students how to do this but then they have less ownership of the solution.
Teamwork-Practice (15 minutes): I tell my students that there many tools for creating simulations and that they will be working in teams on several simulation challenges. I distribute the "Challenges" handout and advise them to work through each challenge and make conjectures about the patterns they see, then return any materials they've borrowed before moving on to the next one. (MP1) As the teams are working I walk around giving encouragement and redirection or assistance as needed.
Teamwork-Creating (15 minutes): As teams finish their "Challenges" I have them draw (select) a scenario from the class problems handout randomly and work together to create a simulation, including an explanation of why their simulation is appropriate. (MP1, MP3, MP4).