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  1. The New Illinois Learning Standards for Algebra I / Math IStatistics and Probability Julia Brenson

  2. The Four Components of a Statistical Investigation* 1) Formulate a question 2) Design and implement a plan to collect data 3) Analyze the data by measures and graphs 4) Interpret the results in the context of the original question *Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report American Statistical Association http://www.amstat.org/education/gaise/GAISEPreK-12_Full.pdf

  3. The New Illinois Learning Standards Algebra I & Math I

  4. Statistics Standards for Algebra I/Math IShape, Center and Spread

  5. Statistics Standards for Algebra I/Math IShape, Center and Spread BIG IDEAS: • When describing distributions, we talk about Shape, Center, and Spread in the context of the data. • Try to use real life data rather than made up data sets whenever possible.

  6. Statistics Standards for Algebra I/Math IShape of the Distribution Skewed Right (Positive) Skewed Left (Negative) Approximately Symmetrical Bimodal possible outlier

  7. Statistics Standards for Algebra I/Math IShape of the Distribution What would the shape be for the distribution of salaries of the 2013 Chicago Cubs? The distribution of salaries for the 2013 Chicago Cubs is skewed right. Most players made less than $2 million. There are two players that made an exceptionally large salary that are possible outliers. (Alfonso Soriano made $19 million and Edwin Jackson made $13 million.)

  8. Statistics Standards for Algebra I/Math IMeasures of Central Tendency BIG IDEAS: • The median is a better measure of center when the data is skewed or an outlier is present. • The mean will be greater than the median when the distribution is skewed right (positive), less than the median when the distribution is skewed left, and approximately equal to the median when the distribution is approximately symmetrical. Mean = Median = the center most value when observations in the data set are ordered

  9. Statistics Standards for Algebra I/Math IMeasures of Central Tendency What was a typical salary for a baseball player on the 2013 Chicago Cub Team? Median = $1,550,000.00 Mean = $3,485,024.20 What is the better measure of center for this data? Why? Which is greater: the mean or the median?

  10. Statistics Standards for Algebra I/Math IMeasures of Central Tendency Demonstration: Comparing the Mean and Median with Fathom OR NCTM Illuminations Mean and Median Applet http://illuminations.nctm.org/Activity.aspx?id=3576 Fathom is currently owned by McGraw-Hill Education

  11. Statistics Standards for Algebra I/Math IMeasures of Variability Range = maximum value – minimum value Interquartile Range = Quartile3 – Quartile1 Standard Deviation = (sample std. deviation) Big Idea: A standard deviation is the typical distance (deviation) from the mean that we expect the majority of the data values to fall within.

  12. Statistics Standards for Algebra I/Math IMeasures of Variability Example: A Brightest Bulb company manufactures light bulbs that last 800 hours on average with a standard deviation of 50 hours. We would expect the majority of Brightest Bulb light bulbs to last between 750 and 850 hours. If I have a Brightest Bulb light bulb that lasts 750 hours, I am not too surprised.

  13. Statistics Standards for Algebra I/Math IMeasures of Variability Standard Deviation = 50 hours 675 50 hrs 50 hrs 925 Mean = 800 hours If I have a Brightest Bulb light bulb that lasts 675 hours, that is 2 ½ standard deviations below the mean. I would be surprised and, at the same time, disappointed. If I have a Brightest Bulb that lasts 925 hours, that is 2 ½ standard deviations above the mean. I am pleasantly surprised.

  14. Statistics Standards for Algebra I/Math IShape, Center and Spread Activity: Identifying Distributions

  15. Statistics Standards for Algebra I/Math IShape, Center and Spread Activities: • Shape, Center and Spread – A little practice with S.ID.1-3 • More Shape, Center and Spread with Real-Life Data

  16. Airline P Airline Q -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 Statistics Standards for Algebra I/Math IShape, Center and Spread Activity: The dot plots below compare the number of minutes 30 flights made by two airlines arrived before or after their scheduled arrival times. • Negative numbers represent the minutes the flight arrived before its scheduled time. • Positive numbers represent the minutes the flight arrived after its scheduled time. • Zero indicates the flight arrived at its scheduled time. Activity adapted from a Smarter Balanced Assessment Consortium task. (http://www.smarterbalanced.org)

  17. Statistics Standards for Algebra I/Math IShape, Center and Spread Part I Write a paragraph comparing arrival times for Airline P and Airline Q. Part IIBased on this data, from which airline will you choose to buy your ticket? Use the ideas of center and spread to justify your choice.

  18. Statistics Standards for Algebra I/Math IShape, Center and Spread Comparison of Airline Arrival Times Example Paragraph: There are similarities and differences between the distributions of the arrival times of 30 flights made by Airline P and Airline Q. The distribution for Airline P appears to be approximately symmetrical while the distribution for Airline Q is skewed right (positive). Airline Q appears to have 2 potential outliers: one that arrived 60 minutes late and another that arrived 45 minutes late. Both Airlines have a median arrival time of 0 minutes. This means that we would typically expect for a flight from either airline to arrive on time. However, there is a great deal of variability in the arrival times for both airlines. Airline P’s arrival times have a range of 50 minutes with flights arriving as early as 25 minutes ahead of schedule and flights also arriving as late as 25 minutes behind schedule. Airline Q has even more variability with a range of 95 minutes. Flights arrived as early as 35 minutes ahead of schedule and as late as 60 minutes behind schedule.

  19. Statistics Standards for Algebra I/Math IShape, Center and Spread Possible Activity: Weight of Backpacks 1) Formulate Question - Do ninth grade girl’s backpacks weigh less than ninth grade boy’s backpacks? 2) Collect Data - Who will we collect data from? - How will we collect data? 3) Analyze Data - Select a type of comparative graph. - Select and calculate appropriate statistics. 4) Interpret Results - What can we conclude? - Can we make inferences about the weights of backpacks for all ninth graders in our school?

  20. Statistics Standards for Algebra I/Math IShape, Center and Spread Activities: • Identifying Distributions – Algebra I/ Math I • Shape, Center and Spread – A little practice with S.ID.1-3 • More Shape, Center and Spread With Real-Life Data • Airline Activity (www.smarterbalanced.org)

  21. Statistics Standards for Algebra I/Math IBivariate Categorical Data

  22. Statistics Standards for Algebra I/Math I Bivariate Categorical Data Summarize categorical data in two categories Big Ideas: • Review the difference between numeric data and categorical data. • Explain that frequency refers to the count of the data. Relative frequency is a proportion. • Analyze relative frequencies and assess possible associations and trends in the data. • Recognize that association does not imply causation.

  23. Statistics Standards for Algebra I/Math I Bivariate Categorical Data Example: A random sample of 100 eleventh graders who took the ACT twice were selected. 52 of these students took a test prep class before taking the ACT a second time. The shaded cells are marginal frequencies and the unshaded cells within the table are joint frequencies.

  24. Statistics Standards for Algebra I/Math I Bivariate Categorical Data Relative Frequency by Table What is the joint relative frequency for students who took a test prep class and increased their score? What proportion of students in this sample took a test prep class and the score did not increase?

  25. Statistics Standards for Algebra I/Math I Bivariate Categorical Data Conditional Relative Frequency by Row Of the students who took a test prep class, what proportion increased their score? If a randomly selected student from the sample did not take a test prep class, what would you predict the student’s change in scores would be from the first ACT to the second?

  26. Bivariate Categorical DataTwo-way Frequency Tables Conditional Relative Frequency by Column Given that a randomly selected student from this sample increased their score, would you predict that the student did or did not take a test prep class? Explain. What proportion of students whose scores decreased did not take a test prep class?

  27. Statistics Standards for Algebra I/Math I Bivariate Categorical Data Association of two categorical variables • There is an association between two categorical variables if the row (or column) conditional relative frequencies are different from row to row (or column to column) in the table. • The greater the difference between the conditional relative frequencies, the stronger the association. • An association between two categorical variables does not mean that we can infer that there is a cause-and-effect relationship between the two categorical variables!

  28. Statistics Standards for Algebra I/Math I Bivariate Categorical Data Association versus Causation Do you think that there is an association between taking a test prep class and seeing an increase in ACT scores? According to this data, can we conclude that taking a test prep class causes scores to increase? Explain.

  29. Statistics Standards for Algebra I/Math I Bivariate Categorical Data Activities: • “Do high school males have different preferences for superhero powers than high school females?” Engage NY Algebra I, Module 2 http://www.engageny.org/mathematics • Census at School http://www.amstat.org/censusatschool/

  30. Statistics Standards for Algebra I/Math IInterpret Linear Models

  31. Statistics Standards for Algebra I/Math IInterpreting Scatterplots Which is the best interpretation of the scatterplot? A. As heights go up, weight increases. B. As heights go up, weight tends to increase. C. If you get taller, you will get heavier. D. Taller football players tend to be heavier football players. Below is a graph of the heights and weights of currently rostered Chicago Bears (April 2014). http://chicagosports.sportsdirectinc.com/football/nfl-teams.aspx?page=/data/nfl/teams/rosters/roster16.html

  32. Statistics Standards for Algebra I/Math IFit a Function to Data Activity: Tootsie Pop Lab Can you predict how many Tootsie Pops you can pick up in your hand?

  33. Statistics Standards for Algebra I/Math IFit a Function to Data Tootsie Pop student graph with line of best fit and r Num_Pops = 2.07 Hand_Span – 24 r = 0.68

  34. Statistics Standards for Algebra I/Math ICorrelation When interpreting the correlation coefficient there are FOUR things that should be discussed: 1. The strength of the linear relationship: strong, moderate, weak, or no linear relationship.

  35. Statistics Standards for Algebra I/Math ICorrelation 2. Whether the relationship between x and y is positive or negative. If the slope of the best fit line is positive, then the r value is also positive. If the slope of the best fit line is negative, then the r value is negative.

  36. Statistics Standards for Algebra I/Math ICorrelation 3. The relationship we are evaluating is a linear relationship between x and y. For example, the two graphs below show a relationship between x and y that is something other than linear.

  37. Statistics Standards for Algebra I/Math ICorrelation 4. CONTEXT! Be sure to interpret the correlation coefficient in the context of the problem. The linear association is between what two variables? Interpretation: r = 0.7 There is a moderate, positive linear relationship between the height (in inches) and the weight (in pounds) for the currently rostered Chicago Bears.

  38. Statistics Standards for Algebra I/Math ICorrelation Discovery Activity: Correlation Sorting scatterplots by their r-value

  39. Statistics Standards for Algebra I/Math ICorrelation vs. Causation Example: Do fresh lemons cause a lower highway fatality rate? www.grossmont.edu/johnoakes/s110online/Causation%20versus%20Correlation.pdf

  40. Statistics Standards for Algebra I/Math IResiduals • The residual is the vertical distance from the data point to the line of best fit. (5, 28) actual (5, 21) predicted Residual = Actual – Predicted Residual = 28 – 21 Residual = 7

  41. Statistics Standards for Algebra I/Math IResiduals • To create a residual plot, graph each x-coordinate with its corresponding residual. (x-value, residual) (x-value, residual) (5, 7) Residual = 7 This horizontal axis corresponds to the regression line.

  42. Statistics Standards for Algebra I/Math IResiduals • Patterns of a residual plot are used to assess the fit of a function to the data. • Patterns should appear scattered with no discernible pattern. Outlier Curved pattern Larger residuals for larger values of x

  43. Statistics Standards for Algebra I/Math I Residuals Why do we need to look at residual plots? Here is an example from Engage NY: The temperature (in degrees Fahrenheit) was measured at various altitudes (in thousands of feet) above Los Angeles. The scatter plot (next slide) seems to show a linear (straight line) relationship between these two quantities. http://www.engageny.org/sites/default/files/resource/attachments/algebra_i-m2-teacher-materials.pdf

  44. Statistics Standards for Algebra I/Math I Residuals The graph on the left of altitude vs. temperature appears to be linear. However, look at the residual plot: The residuals indicate there is a non-linear relationship between altitude and temperature. http://www.engageny.org/sites/default/files/resource/attachments/algebra_i-m2-teacher-materials.pdf

  45. Statistics Standards for Algebra I/Math I Residuals Example – Snakes! (www.insidemathematics.org) Snake 1 Snake 1 Rita catches 5 more snakes. She wants to know whether they belong to species A or to species B. The measurements of these snakes are shown in the table below.

  46. Statistics Standards for Algebra I/Math I Assessing the Fit of a Linear Function To assess the fit of a linear function to a scatterplot consider all of the following: 1. Look at the scatterplot. Does the data appear linear? 2. Look at r-value. What is the strength of the linear relationship between x and y? 3. Look at the residuals. Are they scattered with no discernible pattern?

  47. Statistics Standards for Algebra I/Math I Fit a Function to Data and Interpret Linear Models Activities: • Tootsie Pops and Hand Span (This lab allows students to make connections between what they learned in 8th grade about fitting a linear function to data and the Algebra I standards.) • Correlation Activity • If the Shoe Fits This webinar, presented by Daren Starnes, is available at http://www.amstat.org/education/msss/. (The webinar focuses on an activity from Making Sense of Statistical Studies. Permission to share given by Daren Starnes.)

  48. Statistics Standards for Algebra I/Math I Fit a Function to Data and Interpret Linear Models Additional Activities: • Crickets (http://www.smarterbalanced.org/wordpress/wp-content/uploads/2012/09/performance-tasks/crickets.pdf) • Snakes – A residual activity from Inside Mathematics (http://www.insidemathematics.org/common-core-math-tasks/high-school/HS-S-2003%20Snakes.pdf) • Iris Activity – An extension activity for residuals

  49. Acknowledgements and Resources Chance, B. & Rossman, A. (Preliminary Edition). Investigating Statistical Concepts, Application and Methods. Duxbury Press. Chance, B., et al. Rossman/Chance Applet Collection. Retrieved from http://www.rossmanchance.com/. Chicago Tribune. (2014, April). Chicago Bears. Retrieved from http://chicagosports.sportsdirectinc.com/football/nfl-teams.aspx?page=/data/nfl/teams/rosters/roster16.html Daily Mail. (2012, December 2). What are the odds? New study shows how guessing heads or trails isn’t really a 50-50 game. Retrieved from http://www.dailymail.co.uk/news/article-2241854/What-odds-New-study-shows-guessing-heads-tails-isnt-really-50-50-game.html. Duggan, M. & Brenner, J. (2013, February 14). The Demographics of Social Media Users – 2012. Retrieved from http://www.pewinternet.org/2013/02/14/the-demographics-of-social-media-users-2012/. Focht, D., Spicer, C, and Fairchok, M. (2002). The Efficacy of Duct Tape vs Cryotherapy in the Treatment of Verruca Vulgaris (the Common Wart). 156 (10) pp. 971-974. Retrieved from http://archpedi.jamanetwork.com/article.aspx?articleid=203979&resultClick=1.