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The New Illinois Learning Standards for Sixth Grade Statistics and Probability

The New Illinois Learning Standards for Sixth Grade Statistics and Probability. Julia Brenson. 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

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The New Illinois Learning Standards for Sixth Grade Statistics and Probability

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  1. The New Illinois Learning Standards for Sixth GradeStatistics 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 Sixth Grade

  4. Statistics Standards for 6th Grade

  5. Statistics Standards for 6th GradeRecognize a Statistical Question A statistical question is one that can be answered by collecting data and where we anticipate that there will be variability in the data.

  6. Statistics Standards for 6th GradeRecognize a Statistical Question Which of the following are statistical questions? • How tall are you? • How tall are your class mates? • What is your favorite color? • How many books have you read so far this school year? • How many books has each student in the school read this year? • How many pairs of shoes does your teacher own? • What is the favorite ice cream flavor for each of the 6th grade students at your school?

  7. Statistics Standards for 6th GradeTypes of Graphs Gaps Categories No Gaps

  8. Statistics Standards for 6th GradeShape, 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.

  9. Statistics Standards for 6th GradeDot Plots Dot Plots • Create a number line that covers the full range of the data from the minimum value to the maximum value. • For each data value, place a dot above the corresponding number on the dot plot. • Each dot should be the same size.

  10. Statistics Standards for 6th GradeShape of the Distribution Approximately Symmetrical Skewed unusually large value

  11. Statistics Standards for 6th GradeShape of the Distribution What would the shapebe for the distribution of total days absent in a semester for each student at your school? The graph at left shows the number of days absent in one semester for a random sample of 50 middle school students. The distribution of absences for these students is skewed. Most students had four or fewer absences. There is one value that seems to be very different from the rest. One student missed 23 days of school.

  12. Statistics Standards for 6th GradeMeasures of Center Mean = Median = the center most value when observations in the data set are ordered BIG IDEA: The median is a better measure of center when the data is skewed.

  13. Statistics Standards for 6th GradeMeasures of Center What is a typical number of days absent in one semester for this sample of 50 students? Median (1 day) Mean (1.98 days) What is the better measure of center for this data? Why?

  14. 6th Grade & Algebra I / Math IMeasures of Center Demonstration: Comparing the Mean and Median NCTM Illuminations Mean and Median Applet http://illuminations.nctm.org/Activity.aspx?id=3576

  15. Statistics Standards for 6th GradeMeasures of Center The Mean as Fair Share Dave, Sandy, Javier, and Maria have 12 cookies. How many cookies will each student have if each student receives a fair share? 3 3 12 3 3

  16. Statistics Standards for 6th GradeMeasures of Center The Mean as Fair Share What would each student’s fair share be if there are: 14 cookies? 9 cookies? 7 cookies? ? ? ? ?

  17. Statistics Standards for 6th GradeMeasures of Center From the PARCC Grade 6 EOY Evidence Table Evidence Statement Key 6.SP.3 Rate the following statement as True/False/Not Enough Information. “The average height of trees in Watson Park is 65 feet. Are there any trees in Watson Park taller than 65 feet?”

  18. Statistics Standards for 6th GradeMeasures of Spread Range = maximum value – minimum value Mean Absolute Deviation (MAD) = sum of the distances of each data value from the mean divided by the total number of observations. Big Idea: The mean absolute deviation (MAD) is the average distance (deviation) of data values from the mean.

  19. Statistics Standards for 6th GradeShape, Center and Spread Activity Human Dot Plot

  20. Statistics Standards for 6th GradeMeasures of Spread The Mean as a Balance Point (An Introduction to MAD) From Engage NY Grade 6 Module 6 Lesson 7 Sabina wants to know how long it takes students to get to school. She asks two students how long it takes them to get to school. It takes one student 1 minute and the other student 11 minutes. She thinks the mean is the balance point. What do you think? http://www.engageny.org/sites/default/files/resource/attachments/math-g6-m6-teacher-materials.pdf

  21. Statistics Standards for 6th GradeMeasures of Spread Introducing Deviations A deviation is the distance of a piece of data from the mean. A value that is below the mean has a negative deviation. A value above the mean has a positive deviation. The deviation of 1 to the mean is 1 – 6 = - 5 The deviation of 11 to the mean is 11 – 6 = 5 Questions: 1) What is the deviation from the mean for each of the pennies? 2) What is the sum of these two deviations?

  22. Statistics Standards for 6th GradeMeasures of Spread Introducing Deviations Sabrina wants to know what happens if there are more than two data points. Suppose there are three students. One student lives 2 minutes from school, and another student lives 9 minutes from school. If the mean time for all three students is 6 minutes, she wonders how long it takes the third student to get to school. She tapes pennies at 2 and 9. - 4 + 3 +1 Questions: 1) Where should the third penny be placed to balance the ruler? 2) How can we use deviations to check this answer?

  23. Statistics Standards for 6th GradeMeasures of Spread Introducing Mean Absolute Deviation (MAD) Activity: School Night Sleep How many hours of sleep do sixth graders get on a school night? Let’s make some predictions: • Typically, how many hours of sleep do you think a sixth grader gets? • How much will the number of hours of sleep vary if we asked a group of ten sixth graders? • What do you predict will be the fewest hours? • What do you predict will be the most hours?

  24. Statistics Standards for 6th GradeMeasures of Spread On Monday morning, Carlos asked ten of his sixth grade classmates how many hours of sleep they usually get on school nights. He then created a dot plot of their answers. Questions: Looking at the dot plot above, typically how much sleep did the ten sixth graders get on a school night? How much did the amount of sleep vary? What is the shape of this distribution?

  25. Statistics Standards for 6th GradeMeasures of Spread Let’s look at another method of measuring the spread of the data. Mean Absolute Deviation (MAD) The mean absolute deviation (MAD) is the average distance of the data from the mean. We find MAD by doing these steps: • Calculate the mean. • Find the deviation for each data value. • Take the absolute value of each deviation. • Find the average of these absolute deviations (distances).

  26. Statistics Standards for 6th GradeMeasures of Spread Calculating Mean Absolute Deviation (MAD) Mean = = = 8.75 hours 1.25 0.25 2.25 2.25 0.25 0.25 -0.75 0.75 MAD = = = 1.05 hours -2.25 2.25 1.25 1.25 -0.75 0.75 -0.75 0.75 -0.75 0.75 0.00 10.5

  27. Statistics Standards for 6th GradeMeasures of Spread Interpreting Mean Absolute Deviation (MAD) Mean = 8.75 hours MAD = 1.05 hours mean + MAD mean - MAD mean The number of hours of sleep on a school night for these ten sixth graders varies1.05 hours, on average, from the mean of 8.75 hours.

  28. Statistics Standards for 6th GradeMeasures of Spread Mean Absolute Deviation (MAD) Ten sixth graders are asked to report the number of hours of sleep they typically get on a school night. Their hours of sleep are shown on the dot plot below. Questions: What is the mean number of hours of sleep on a school night for these ten sixth graders? What is the median? How much variability is there amongst the ten sixth graders? What is the value of MAD for this data?

  29. Statistics Standards for 6th GradeHistograms Histograms are always graphed with a scaled number line for both the horizontal and vertical axes. Step 1 Determine the number of intervals that will be used. Step 2 Tally the number of observations in the data that fall on each interval. This gives us the count (frequency) of observations in that interval. Step 3 Draw and scale the horizontal and vertical axes. Be sure that the horizontal axis is scaled so that it goes at least as low as the minimum and at least as high as the maximum. If the minimum observation is far from 0, a may be used to indicate that we are skipping over values that are below the minimum. Step 4 Draw bars for each interval, with the horizontal sides marking the start and end of the interval, and the top edge of the bar corresponding to the count (or proportion) of observations on the interval.

  30. Statistics Standards for 6th GradeHistograms Creating Histograms Example Josie and her father went fishing on Lake Michigan. Josie is curious about the size of the fish that they are catching, so she and her dad decide to measure the length of each fish before releasing it back into the water. Below are the lengths rounded to the nearest centimeter for the 22 fish that they caught. How many observations are in this set of data? n = ____ What is the unit of measure that was used? ____________ 24,28,32,21,56,38,27,18,32,28,36, 43,27,25,18,16,31,34,37,29,41,42 22 centimeters

  31. Statistics Standards for 6th GradeHistograms Creating Histograms 16-<24 represents fish lengths from 16 cm up to, but not including 24 cm. (16≤ fish length<24). Questions: What fraction of the fish were less than 32 cm? What fraction of the fish were at least 32 and less than 48 cm?

  32. Statistics Standards for 6th GradeHistograms Creating Histograms Relative frequency for an interval is the frequency divided by the total observations. Questions: What is the sum of all the relative frequencies? What percent of the fish that were caught were less than 40 cm?

  33. Statistics Standards for 6th GradeBox Plots Box plots are always graphed on a scaled number line. Step 1 Arrange the numbers in the data set in order from lowest to highest. Step 2 Draw a number line. Be sure that it is scaled so that it goes at least as low as the minimum and at least as high as the maximum. Step 3 Plot points above the number line for the minimum and maximum values. Step 4 Calculate the median and then quartile 1and quartile 3. Step 5 Draw vertical lines above the number line to represent the values of quartile 1, the median, and quartile 3. Connect these three vertical lines to make the box. Step 6 Draw a horizontal line from Q1 to the minimum and a horizontal line from Q3 to the maximum.

  34. Statistics Standards for 6th GradeBox Plots Creating Box Plots Example 1 How many observations are in this set of data? n = ____ 5, 6, 9, 12, 14, 15,15, 17, 20 9 (odd) Minimum Maximum Median Q 2 = 7.5 Q 1 = 16 Q 3

  35. Statistics Standards for 6th GradeBox Plots Creating Box Plots Example 2 How many observations are in this set of data? n = ____ 2, 4, 5, 5, 6, 8, 12,14,15,16, 16,18 12 (even) Minimum Maximum = 10 Median = 5 Q 1 = 15.5 Q 3

  36. Statistics Standards for 6th GradeBox Plots - Shape of the Distribution Approximately Symmetrical Skewed unusually large value

  37. Statistics Standards for 6th GradeBox Plots – Measures of Center & Spread Measure of Center: Median Measures of Spread: Range = maximum value – minimum value Interquartile Range = Quartile3 – Quartile1 Interquartile Range (iqr) is the spread of the middle 50% of the data.

  38. Statistics Standards for 6th GradeBox Plots Creating Box Plots Back to Example 1 25% 25% 25% 25% Questions: What percent of values in this data set are 14 or above? What percent of values are 16 and below? What percent of values are above 16? What percent of values are between 7.5 and 16?

  39. Statistics Standards for 6th GradeBox Plots - Describing Distributions Example: The distribution of the number of years of professional experience for the 2013-2014 Chicago Bears is shown in the box plot below. What does the shape, center and spread of the box plot tell us about the number of years of professional experience for last season’s Chicago Bears? iqr = 7 – 2 = 5 Chicago Bear data retrieved from http://sportsillustrated.cnn.com/football/nfl/rosters/bears/byLAST_NM.html.

  40. Statistics Standards for 6th Grade Shape, Center and Spread Activity What’s Your Age?

  41. Statistics Standards for 6th Grade Census at School (http://www.amstat.org/censusatschool/) Statistics Education Web (http://www.amstat.org/education/stew/)

  42. Statistics Standards for 6th Grade Activities: • Human Dot Plots • Mean as a Fair Share - Cookie Page • Mean, Median, Mode, and Range (http://map.mathshell.org/materials/download.php?fileid=1360) • Candy Bar (http://map.mathshell.org/materials/download.php?fileid=1178) • Census at School ( http://www.amstat.org/censusatschool/) • How Long is 30 Seconds Statistics Education Web (STEW) (http://www.amstat.org/education/stew/pdfs/HowLongis30Seconds.pdf) • The Mean as a Balance Point Engage NY Grade 6 Module 6 (http://www.engageny.org/sites/default/files/resource/attachments/math-g6-m6-teacher-materials.pdf) • What’s Your Age?

  43. 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.

  44. Acknowledgements and Resources Franklin, C., Kader, G., Mewborn, J. M., Peck, R., Perry, M. & Schaeffer, R. (2007) Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report: A Pre-K-12 Curriculum Framework. Alexandria, VA: American Statistical Association. McCallum, B., et al. (2011, December 26). Progressions for the Common Core State Standards in Mathematics (draft) 6-8 Statistics and Probability. Retrieved from http://commoncoretools.files.wordpress.com/2011/12/ccss_progression_sp_68_2011_12_26_bis.pdf. McCallum, B., et al. (2012, April 21). Progressions for the Common Core State Standards in Mathematics (draft) High School Statistics and Probability. Retrieved from http://commoncoretools.me/wp-content/uploads/2012/06/ccss_progression_sp_hs_2012_04_21_bis.pdf. Moore, D. & McCabe, P. (1989). Introduction to the Practice of Statistics. New York, NY: W. H. Freeman. Oakes, J. “Causation verses Correlation” Grossmont. Retrieved July 7, 2013, from www.grossmont.edu/johnoakes/s110online/Causation%20versus%20Correlation.pdf Peck, R., Gould, R., & Miller, S. (2013). Developing Essential Understand of Statistics for Teaching Mathematics in Grades 9-12. Reston, VA: The National Council of Teachers of Mathematics, Inc.

  45. Acknowledgements and Resources Peck, R., Olsen C. & Devore J. (2005). Introduction to Statistics and Data Analysis. Belmont, CA: Brooks/Cole. Peck, R. & Starnes, D. (2009). Making Sense of Statistical Studies. Alexandria, VA: American Statistical Association. Ramsey, F. & Schafer, D. (2002). The Statistical Sleuth: A Course in Methods of Data Analysis. Boston, MA: Brooks/Cole, Cengage Learning. Rossen, J. (2014, January 15). Taste Test Pits Fine Chocolate Against Cheaper Brands. Retrieved from http://www.today.com/video/today/54076112#54301611. Rossen, J. (2014, February 26). Underage Alcohol Buys. Retrieved from http://www.today.com/video/today/54076112#54515111. Rossman, A. (2012). Interview With Roxy Peck. Journal of Statistics Education, 20(2). pp. 1 – 14. Retrieved from http://www.amstat.org/publications/jse/v20n2/rossmanint.pdf. Rossman, A., Chance, B., & Von Oehsen, J. (2002). Workshop Statistics Discovery With Data and the Graphing Calculator. New York: Key College Publishing.

  46. Acknowledgements and Resources Scheaffer, R., Gnanadesikan, M., Watkins, A., & Witmer, J. (1996). Activity-Based Statistics. New York: Springer-Verlag. Stickgold, R., James, L. & Hobson, J. (2000). Visual discrimination learning requires sleep after training. 3(12) pp. 1237-1238. Retrieved from http://www.nature.com/neuro/journal/v3/n12/pdf/nn1200_1237.pdf. Strayer, D. and Johnston, W. (2001, November 6) 12(6). Driven to Distraction: Dual-Task Studies of Simulated Driving and Conversing on a Cellular Telephone. Pp. 462-466Retrieved from http://www.psych.utah.edu/AppliedCognitionLab/PS-Reprint.pdf.

  47. Online Resources Census at School. http://www.amstat.org/censusatschool/ Consortium for the Advancement of Undergraduate Statistics Education. http://causeweb.org/ Engage NY. http://www.engageny.org/mathematics Illustrative Mathematics. http://www.illustrativemathematics.org/ Inside Mathematics. http://www.insidemathematics.org Mathematics Assessment Project. http://map.mathshell.org/ Math Vision Project. http://www.mathematicsvisionproject.org/ NCSSM Statistics Institutes. http://courses.ncssm.edu/math/Stat_Inst/links_to_all_stats_institutes.htm

  48. Online Resources PARCC Model Content Frameworks. http://www.parcconline.org/sites/parcc/files/PARCCMCFMathematicsNovember2012V3_FINAL.pdf PARCC Mathematics Evidence Tables. https://www.parcconline.org/assessment-blueprints-test-specs Smarter Balanced Assessment Consortium. http://www.smarterbalanced.org/ Statistics Education Web (STEW).  http://www.amstat.org/education/STEW/ The Data and Story Library (DASL). http://lib.stat.cmu.edu/DASL/ The High School Flip Book Common Core State Standards for Mathematics. http://www.azed.gov/azcommoncore/files/2012/11/high-school-ccss-flip-book-usd-259-2012.pdf

  49. The New Illinois Learning Standards for Sixth GradeStatistics and Probability Julia Brenson Lyons Township High School jbrenson@lths.net

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