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WELCOME BACK TO CAMP! What’s golden? Agenda • Warm-Up for statistical and probabilistic thinking • Norms for our PD • Recapping from lesson study • NCTM (2007) Teaching Standards: Worthwhile task • Break • Statistics Task • Lunch • Exploring Statistics and • • • • Probability Content Standards Standards for Mathematical Practice Synthesizing from the Day Student evaluation Closure Warm-up Activity in Stats and Prob Are you a Good Timer? •Quick Experiment: – Close your eyes – When you hear the “START”, begin counting off seconds in your head – When you hear the “STOP”, write down the number you reached Are you a Good Timer? • Graph the results-univariate and bivariate options • What do we see? • Can you guess what was the exact number of seconds? • How consistent were the estimates? • What does the y-intercept of the regression line tell us? Slope? Are you a Good Timer? •Experimental Design Options: 1. Announce a #seconds; everyone counts until the timer goes off / we say Stop. 2. Everyone counts until they reach 30, then opens their eyes and looks at a clock and writes down how many seconds actually elapsed. 3. Like #1 but don’t announce #seconds ahead of time. Everyone counts until we say Stop •Use the same #seconds the 2nd time? •Tell what the #seconds was after the 1st time? Evolving Norms for this PD • We will persist with every problem and examine it from multiple perspectives. • We will be ready for class and use our class time effectively. • We will keep our focus on learning and use technology for personal reasons during breaks. • We will be respectful of each other’s time and space and work efficiently. • We will actively participate by (a) listening to each other, (b) giving others our attention, (c) not speaking when someone else is talking, and (d) regularly sharing our ideas in class. • If we disagree with someone or are unclear, we will ask a question about his or her idea and describe why we disagree or are confused. • We will ask questions when we do not understand something. We will comment on others’ ideas rather than the person. Evolving Norms for this PD • We will be ready for class and use our class time effectively. • We will keep our focus on learning and use technology for personal reasons during breaks. • We will be respectful of each other’s time and space and work efficiently. • We will actively participate by (a) listening to each other, (b) giving others our attention, (c) not speaking when someone else is talking, and (d) regularly sharing our ideas in class. • If we disagree with someone or are unclear, we will ask a question about his or her idea and describe why we disagree or are confused. • We will ask questions when we do not understand something. We will comment on others’ ideas rather than the person. Evolving Norms for this PD O We will take advantage of opportunities to share ideas and gather feedback through presentations. O We will encourage one another to share ideas. O We will show our appreciation to one another after a presentation by applause. O If we disagree with someone or are unclear about their ideas related to mathematics content and pedagogy, we will ask a question about his or her idea and describe why we disagree or are confused. O We will ask questions when we do not understand something about mathematics content and pedagogy. O We will comment on others’ ideas about mathematics content and pedagogy rather than the person. Evolving Norms for this PD • We will always look for another approach to solve problems. • We will use pictures, graphs, tables, symbols, numbers, manipulatives, and/or words to assist us while doing mathematics. • We will persist with every problem and examine it from multiple perspectives. • We will be mathematically precise whenever possible. • We will explain and justify our ideas in a way that everyone can understand. Expectation for technology use • Please limit the use of technology for the use of chatting, phone calls, and texts strictly to break times as well as before and after class out of respect for the nature of our collaboration and thinking together. Lessons learned from Lesson Study • • • • What did you learn from the experience? What surprised you? What did you like about the experience? How would improve upon those aspects that you did not like? • …thoughts for moving forward. NCTM Standards for Teaching and Learning as Related to the Common Core State Standards (2007) 1. Knowledge of Mathematics and General Pedagogy 2. Knowledge of Student Mathematical Learning 3. Worthwhile Mathematical Tasks 4. Learning Environment 5. Discourse 6. Reflection on Student Learning 7. Reflection on Teaching Practice Frayer models: Worthwhile task Break STATISTICAL ASSOCIATION 2 TYPES WE WILL FOCUS ON: •BIVARIATE CATEGORICAL: Is there an association between gender and whether you have a part-time job? •BIVARIATE QUANTITATIVE: Is there an association between a golf ball’s drop height and bounce height? Categorical Association • CCSS.Math.Content.8.SP.A.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores? Categorical Association What numerical analysis could be done to explore the data? Categorical Association Job Experience Male Female Total Never had a part44% 60% 52% time job Had a part-time job 31% 25% 28% during summer only Had a part-time job 25% 15% 20% but not only during summer Total 100% 100% 100% Categorical Association What graphical representations could we make to display our numerical analyses? Segmented Bar Chart Categorical Association Is there an association between gender and job experience for the students in this sample? Justify your response. How does the knowledge needed to approach these tasks connect with other topics in the mathematics curriculum? Teaching Categorical Association • 2 tasks: smoking and drug • Sample student responses Teaching Categorical Association In summary, these are the common student misconceptions students have when analyzing categorical data for associations: • Lack of proportional reasoning • Deterministic: absolute, everyone has to follow • Unidirectional: direct only • Localist: look at only one cell or one conditional distribution • Use of intuition and ignoring the data LUNCH Quantitative Association 8.SP.A.2: Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. 8.SP.A.3: Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. Quantitative Association: Graphing Quantitative Association STEW Lesson plan “What Fits?” http://www.amstat.org/education/stew/ •Using the piece of spaghetti, determine the line of best fit for the data shown in the scatterplot. •Be cognizant of your thoughts as you decide where to place the line. Quantitative Association • What things did you consider when you were deciding where to place it? • Why did you choose to put the line there? • What is your definition of the line of best fit? Quantitative Association Student conceptions study Quantitative Association Handout: Part Two Instructions Some sample student responses to this task are on the following page. For each student’s response, analyze his/her criterion. Will the criterion always work to produce a line that accurately models any data set? If it will, explain why. If it won’t, draw at least one example of a scatterplot with the line placed using that criterion and explain why the criterion produces a poor line of best fit. Quantitative Association: Slope The equation of the regression line for the golf ball task is: Bounce height = 0.7 Drop Height -3.4 Interpret the slope of this line in context. Quantitative Association: Slope y 3 x2 5 Bounce height = 0.7 Drop Height -3.4 Quantitative Association: Slope No association case Quantitative Association: Slope • Algebra: slope = how much y will change for a 1-unit change in x • Statistics: slope of regression line = AVERAGE DIFFERENCE in y per 1-unit DIFFERENCE in x • AVERAGE: no guarantee that y will change exactly that much. • DIFFERENCE: saying “change” might give the impression that we are changing the x value of a data point (putting someone on a stretching machine) instead of comparing two different x values (two people of different heights) Quantitative Association: y-intercept • Predicted value for response variable (y) when predictor variable (x) is 0 Bounce height = 0.7 Drop Height -3.4 • Huh? How can the bounce height be -3.4 cm? Quantitative Association: Slope • Statistical model: usually not of interest; fitting to the data at hand which aren’t necessarily close to y-axis; when fitting line of best fit, it is not the starting point • Mathematical function: usually of interest; often starting point when graphing Quantitative Association Learning mathematical lines and statistical lines of best fit Students’ previous study of lines and slope with mathematical functions can create cognitive obstacles to •Plotting statistical data •Making lines that don’t go through all of the points •Understanding what the line of best fit is •Developing criteria for the line of best fit •Correctly interpreting the slope and y-intercept of best fit lines in a statistical setting. Synthesizing from the Day 1. Describe the SMPs that you engaged in during today’s tasks. 2. What evidence might you notice when reviewing tasks that suggests it meets expectations for a worthwhile task? Student Evaluation PSM6, PSM7, and PSM8 Take Care • Next meeting is Friday January 16 @ Lima HS. • Please ask any questions to Dr. Stats and/or Dr. Math. • Most importantly, keep in touch!