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The New Illinois Learning Standards for Grades 6 - 8 Statistics and Probability

The New Illinois Learning Standards for Grades 6 - 8 Statistics and Probability. Dana Cartier Illinois Center for School Improvement Julia Brenson Lyons Township High School Tina Dunn Lyons Township High School. The New Illinois Learning Standards. Agenda Resources Available Through ISBE

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The New Illinois Learning Standards for Grades 6 - 8 Statistics and Probability

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  1. The New Illinois Learning Standards for Grades 6 - 8Statistics and Probability Dana Cartier Illinois Center for School Improvement Julia Brenson Lyons Township High School Tina Dunn Lyons Township High School

  2. The New Illinois Learning Standards Agenda • Resources Available Through ISBE • Sixth Grade – Shape, Center, Spread • Seventh Grade – Random Sampling for Inference and Simulation for Probability • Eighth Grade – Bivariate Data

  3. The New Illinois Learning Standards ILStats http://ilstats.weebly.com/ • All materials from this session are available at this website. • This website is currently under construction, but please keep checking back for more information about the Statistics Standards.

  4. The New Illinois Learning Standards Sixth Grade

  5. Statistics Standards for 6th Grade

  6. Statistics Standards for 6th Grade • 6.SP.A.2Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. • 6.SP.A.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. • 6.SP.B.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

  7. Statistics Standards for 6th Grade • 6.SP.B.5 Summarize numerical data sets in relation to their context, such as by: • 6.SP.B.5a Reporting the number of observations. • 6.SP.B.5b Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. • 6.SP.B.5c Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. • 6.SP.B.5d Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered

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

  9. 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.

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

  11. 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. Most players made less than $2 million. There are two players that made an exceptionally large salaries. (Alfonso Soriano made $19 million and Edwin Jackson made $13 million.)

  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 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?

  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. 6th Grade & Algebra I / Math IMeasures of Center Comparing the Mean and Median

  16. 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

  17. 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? ? ? ? ?

  18. 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?”

  19. Statistics Standards for 6th GradeMeasures of Spread Range = maximum value – minimum value Interquartile Range = Quartile3 – Quartile1 Interquartile Range (iqr) is the spread of the middle 50% of the data. 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.

  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 Grade Shape, Center and Spread Activity What’s Your Age?

  30. Statistics Standards for 6th Grade Activities: • 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) • 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?

  31. The New Illinois Learning Standards Seventh Grade

  32. Statistics Standards for 7th Grade * See Evidence Statement 7.D.3 Micro-models from the PARCC Evidence Table – Grade 7 PBA

  33. Statistics Standards for 7th Grade • 7.SP.A.1 Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. • 7.SP.A.2 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.

  34. Statistics Standards for 7th GradeRandom Sampling to Draw Inferences About a Population Activity: Gettysburg Address Part I Judgment Sample Part II Simple Random Sample  Sampling Distribution

  35. Statistics Standards for 7th GradeRandom Sampling to Draw Inferences About a Population Gettysburg Address Judgment Sample First ask students to take a quick look at the population of 268 words and select 5 words that they think form a representative sample of the length of words found in the Gettysburg Address. This is a judgment sample. Students record the five words and the number of letters in each word in the table provided. After calculating the mean of the sample, each student records his mean on the class dot plot on the chalkboard.

  36. Statistics Standards for 7th Grade Random Sampling to Draw Inferences About a Population How do we ensure that we select a sample that is representative of the population? We choose a method that eliminates the possibility that our own preferences, favoritism or biases impact who (or what) is selected. We want to give all individuals an equal chance to be chosen. We do not want the method of picking the sample to exclude certain individuals or favors others. One method that helps us to avoid biases is to select a simple random sample. If we want a sample to have n individuals, we use a method that will ensure that every possible sample from the population of size n has an equal chance of being selected.

  37. Statistics Standards for 7th GradeRandom Sampling to Draw Inferences About a Population Which of the following would produce a simple random sample of size 6 from the population of all students in our classroom? A. Select the first 6 students that enter the classroom. B. Put every student’s name in a hat, mix and draw 6 names. C. The classroom has 6 tables with three students per table. Randomly select two tables. The students at these two tables are the sample. D. The classroom has 6 tables of students. Randomly select one student from each table.

  38. Statistics Standards for 7th GradeRandom Sampling to Draw Inferences About a Population Back to Gettysburg Address Simple Random Sample Use a random number generator or a random digits table to select a simple random sample of size 5 from the population of 268 words.

  39. Statistics Standards for 7th Grade Random Sampling to Draw Inferences About a Population Random Digits Table Suppose, for example that we wanted a sample of size 5. There are 268 words. First select a row to use in the table. Select three digits at a time, letting 001 represent 1, 002 represents 2, and so on. Skip 000 and numbers that are greater than 268. Skip repeats. Our Sample: 32, 148, 238, 128, 104

  40. Statistics Standards for 7th GradeRandom Sampling to Draw Inferences About a Population Random Number Generator Random sample of 5 numbers representing the 5 words to be selected. The random number generator above is shared with permission from Beth Chance and Allan Rossman. This applet can be found at http://www.rossmanchance.com/applets/RandomGen/GenRandom01.htm

  41. Statistics Standards for 7th GradeRandom Sampling to Draw Inferences About a Population Gettysburg Address Sampling Words – Permission to share this applet was given by Beth Chance and Allan Rossman. Number of Letters for all Words in the Population Population Mean Last random sample of size 5 that was selected. Sample Mean http://www.rossmanchance.com/applets/GettysburgSampleE/GettysburgSample.html

  42. Statistics Standards for 7th GradeRandom Sampling to Draw Inferences About a Population Gettysburg Address 100 random samples of size 5 mean = 4.46

  43. Statistics Standards for 7th GradeRandom Sampling to Draw Inferences About a Population Gettysburg Address 500 random samples of size 5 Sampling Distribution mean = 4.313

  44. Statistics Standards for 7th Grade • 7.SP.C.8 Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. • 7.SP.C.8a Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. • 7.SP.C.8b Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event. • 7.SP.C.8c Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

  45. Statistics Standards for 7th GradeChance Processes and Probability Models Example: Tree Diagram Michael and Gita would like to have three children. What is the probability that all three children will be boys? Third Child Possible Outcomes BBB (0.5)(0.5)(0.5) = 0.125 BBG BGB BGG GBBGBGGGBGGG Second Child First Child B 0.5 B 0.5 0.5 G B 0.5 0.5 B 0.5 G 0.5 G 0.5 B 0.5 B 0.5 0.5 G G 0.5 0.5 B G 0.5 G

  46. Statistics Standards for 7th GradeChance Processes and Probability Models Three Children Continued… Another way to look at this problem is to create a list of all possible outcomes (the sample space). (B, B, B) (G, B, B) (B, B, G) (G, B, G) (B, G, B) (G, G, B) (B, G, G) (G, G, G) This is a uniform distribution in which every outcome has an equal chance of occurring. There are 8 outcomes and each outcome has a 1/8 chance of occurring. We can now answer questions like: What is the probability of the couple having 3 boys? What is the probability of having one boy?

  47. Statistics Standards for 7th GradeChance Processes and Probability Models Activity: Blood Type A If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

  48. Statistics Standards for 7th GradeChance Processes and Probability Models Activity:Blood Type A • Using a random digits table, let 1, 2, 3, 4 represent having type A blood. 0,5,6,7,8,9 represent not having type A blood. • Select a row. • Count how many digits it takes to reach a 1,2,3, or 4. • Record this count with a tally mark in a table. • Repeat many times to determine the long-run behavior.

  49. Statistics Standards for 7th GradeChance Processes and Probability Models Row 14 58842 81316 30021 29902 35106 87744 89832 15 89104 07798 63824 84546 52699 12394 59894 16 32161 26081 81678 46319 40588 24581 51397 17 43757 41089 36430 92049 88555 90515 64921 18 56847 26072 30263 70043 29892 48430 11287 12341 12111 21231 11234 11212 34561 12341 Continue on to simulate the long run behavior or combine results with classmates.

  50. Statistics Standards for 7th GradeChance Processes and Probability Models Blood Type A - Part II Tree Diagram Let A = the event that a donor has blood type A Let O = the event that a donor has some other blood type.

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