Titanic Tragedy-Part I: A Develop Understanding Task Part I Data from http://en.wikipedia.org/wiki/RMS_Titanic Art by Willy Stower The RMS Titanic had 2224 people on board ship (including the children), yet there was lifeboat space for only 1178. Various factors on that fateful day also made it so many lifeboats were not full when they were launched. Women Person category 1st Class 2nd Class 3rd Class Crew Total Number aboard 144 93 165 23 425 Number saved 140 80 76 20 316 Men Number lost 4 13 89 3 109 Person category 1st Class 2nd Class 3rd Class Crew Total Number aboard 175 168 462 885 1690 Number saved 57 14 75 192 338 Number lost 118 154 387 693 1352 Note: in addition to the adults there were 109 children, undifferentiated by gender, of which 56 died and 53 survived. Create a variety of graphical representations to show relationships between survival and gender. Historically when populations are in danger, societies protect women. Does the Titanic data support this priority? In other words, is survival rate independent from gender? Use your graphical representations and numerical evidence to support your conclusions. Titanic Tragedy-Part I: – Teacher Notes A Develop Understanding Task Purpose: Develop student understanding around the concept of testing the independence of possibly related variables. In the process, students will also solidify their understanding of 2-way Tables, Venn Diagrams and conditional probabilities, as they refine Surfaces ideas such as: ● ● ● ● ● finding conditional probabilities and using those probabilities to make conjectures about independence between categories recognizing the features of a two-way table indicates that a relationship is independent (proportionality of a category to the total is maintained for conditions within the population) explaining independence relationship in context of events understanding that variability and sample sizes from random sampling might impact probability decisions giving evidence to support conclusions Core Standards Focus: S.CP.2: Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. S.CP.3: Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. Related Standards: S.CP.1, S.CP.4, S.CP.5 Launch (Whole Class): Ask: What do you know about the Titanic tragedy? (Janet: “Well, it sank.”) Provide additional background about the Titanic tragedy. Explore (Small Group): Distribute the task sheets and allow participants to explore the data. Choose some individuals or groups to share their graphical representations, numerical calculations, and conclusions about conditional probabilities. ● ● We can expect to see Venn diagrams and 2-way tables because of the prior tasks. Participants may also choose to compare data using bar graphs or circle graphs. Point out that a bar graph does not show the relationship between survival and gender. Compare and contrast different representations and ask participants to find similarities and parallels. Look for representations that differ because counts were used rather than percentages in comparing categories. ● ● ● ● ● ● Segmented bar graphs can lead to rich discussions about representing data, what kinds of information can be obtained by displaying in counts vs. percents, and how the graphs change. This task uses categorical data. It is important to define displays of data carefully so that the diagrams don’t have cases where an individual both dies and survives, or is both male and female. This quality makes it difficult when thinking about how to construct a Venn diagram, or how to label columns and rows in 2-way tables. There are two variables represented in this task. If you are creating a two way table, one variable represents the rows of data, and one represents the columns. The idea of showing independence between categories is difficult to grasp and typically takes exposure to multiple data sets, including both examples and non-examples, to gain a solid understanding. Look for evidence of understanding that if survival rate is dependent on gender (not independent) then the percent of surviving women will be different than surviving men, and in fact the percent of surviving women or men will be different than the overall survival rate. Ask participants “What is the probability of survival?” Discuss how you decide how to determine that probability and record on sticky the different probabilities of survival, P(Survive), P(Survive|Female), P(Survive|Male) and compare. (If time allows other probabilities could be compared.) Discuss (Whole Class): 1. Ask participants to summarize the characteristics of data that would indicate independence between categories, ie, that proportionality between categories of subgroups will equal each other, and that of the overall population. 2. What would a segmented bar graph showing proportions look like for this data? (The bars would be of equal length, and the “survive” portions would be equal at about 30%) 3. Formalize the proportionality relationship of independent events with the probability equation, P(A)=P(A|B) [i.e., P(Survive) = P(Survive|Female)=P(Survive|Male)] 4. Have participants formalize a definition of independence both in a sentence and using probability notation. Write these on a poster for display. 5. Relate the equation back to the Titanic data and verify conclusions. 6. Time permitting, you may productively use Titanic Tragedy-Part II together with this task (Part I).