1. Middle School Content Shifts
Statistics
Middle School Content Shifts

2. Concerning statistics, what have you usually taught or done
Concerning statistics, what have you usually taught or done? Share with an elbow partner.
Read “Data in Grades K – 5” handout and pages 2 – 3 of “6 – 8 Statistics and Probability Progression Document”. Compare what you read to what you have been teaching.

3. Critical Area for Grade 6
Building on and reinforcing their understanding of number, students begin to develop their ability to think statistically. Students recognize that a data distribution may not have a definite center and that different ways to measure center yield different values. The median measures center in the sense that it is roughly the middle value. The mean measures center in the sense that it is the value that each data point would take on if the total of the data values were redistributed equally, and also in the sense that it is a balance point.
Iowa Core, page 41
Of the three grades, statistics and probability is a critical area only in grade 6. This captures the shift from grades K – 5 where no measures of center are now expected to be done.

4. Finding the Balance Point Adapted from “Teaching Student Centered Mathematics, grades 5 - 8, Van de Walle and Lovin

5. Count the number of letters in your first and last name
Count the number of letters in your first and last name. Place that number on a sticky note and post it on the class number line graph. With a partner, create a number line like the class one, but do not put any data on it.

6. Discuss with your partner….
“Using the sticky notes, how could we find the balance point of the data of our graph?”

7. How could we move some sticky notes to find the balance
How could we move some sticky notes to find the balance? What would be another movement we can do?
Continue until all the data stickies are in one column or cannot be moved any further. Then have students post their line plots in order on the wall so they can observe the progression of the movements to the center.

8. This balance point is called the mean
This balance point is called the mean. How can you use your data to calculate the mean? Will your mean be the same as other partnerships?
What does this final display represent?
Will your set of data have the same balance point?

9. Iowa Core Standards
5.MD.2 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally. 6.SP.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.

10. Softball Team Weekly Study Hours

11. Make a display
Look at the data given and display the data in an appropriate way.
What observations can be made with your display?
Are there strengths and weaknesses with your display?
Core Tools: Univariate Quantitative: Study Time

12. Share your display Which display represented the data the best?
What other displays could have been made?
Core Tools: Univariate Quantitative: Study Time

13. Iowa Core Standards
6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

14. Critical Area for Grade 6
Students recognize that a measure of variability (interquartile range or mean absolute deviation) can also be useful for summarizing data because two very different sets of data can have the same mean and median yet be distinguished by their variability. Students learn to describe and summarize numerical data sets, identifying clusters, peaks, gaps, and symmetry, considering the context in which the data were collected.
Iowa Core, page 41
Of the three grades, statistics and probability is a critical area only in grade 6. This captures the shift that statistics is not only mean, median, mode and graphing data sets.

15. Means & MADs Adapted from “Means and MADs”, Mathematics Teaching in the Middle Grades, NCTM, 1999

16. Launch
In a survey, nine individuals were asked, “How many people are in your family?” One result from the poll was that the average family size for those asked was “5”.

17. If the mean size is “5”, how many members could be in each of those nine families? In your group, determine some possible distributions of the nine families. For this problem, consider family sizes no smaller than 2 and no larger than 11. Display one of your distributions as a dot plot using chart paper and post-its. When your group is finished, post the chart on the wall.

18. Given these 8 distributions with a mean of 5
Explore
Given these 8 distributions with a mean of 5
What are the limitations of only knowing the mean family size?
What additional information about the data could be given to identify which of the distributions matched the results of the survey?
Think-Pair-Share

19. A major goal of statistics is to offer ways to summarize and measure this “spread” (or variability)
Of all these distributions, which distribution shows data values with the least variation? Explain.
Of all these distributions, which distribution shows data values with the most variation? Explain.

20. How would you order, from least variation to most variation, the 8 distributions?
Individually (1 minute)
Group (2 minutes)
Share

21. Because it can be difficult to come to a consensus on this single ordering, we need a number (i.e. metric) to quantify variation in a set of data. In your group determine a method to do this…
….and consider as
many ways as you can

22. Share Out Each group will select a reporter.
Each reporter will describe one of the methods their group created unless it is a duplicate.
Rotation among groups will continue until all methods are presented.

23. Calculate the MAD
Mean Absolute Deviation: The sum of the distances of each piece of data from the mean, divided by the number of pieces of data

24. Distribution Number
MAD A B C D E F G H

25. Does the MAD ordering give you the same ordering your group got?
If yours was the same (or close), given a different set of distributions, do you think you would always be close?
At what number of data points does is become difficult to order visually?

26. Summarize
How can the MAD help you distinguish between Distribution C and Distribution E?
What does a MAD of 1.78 indicate about the set of data?
The mean and median are often referred to as “measures of center”. The notion of center is that of some halfway point, the median is the value at the middle. How is mean a center?

27. Check for Understanding
At exactly the same instant, Mike and Chuck checked the time on the clocks and watches at their homes. The times on Chuck’s ten clocks were:
8:16 8:10 8:14 8:16 8:12 8:15 8:13 8:17 8:15 8:23
Find the MAD (mean absolute deviation) of Chuck’s clocks.
Mike had the same mean on his ten clocks as Chuck did, but his MAD was 10 minutes. Find an example of the ten times that could be the times on Mike’s clocks.
What do the mean and the MAD tell you about how useful it is to look at a clock at Mike’s house and at Chuck’s house?

28. Iowa Core Standards
6.SP.2 Understand 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.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.5 Summarize numerical data sets in relation to their context, such as by:
Reporting the number of observations.
Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
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.
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.

29. Text Messaging

30. Interpreting Box Plots
Box plots are…
Used for organizing and displaying data
Easy to create but not always easy to understand
Used to foster higher order thinking in students
Part of the 6th grade CCSS

31. With a Partner
Work through the activity “Text Messaging”. Ask for assistance as needed. Be prepared to discuss the results of your work.

32. Summarize What are the components of a box plot?
Where is the IQR and how could you calculate it?
Why would you use a box plot to represent data?

33. Iowa Core Standards
6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. 7.SP.4 Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.

34. Make a Match
Use worksheet: Introduction to Statistics and Data Analysis- Quartiles, IQR, and Boxplots.

35. In partnerships….
Look at the 4 histograms on the top of the sheet. Below are 4 box plots. Decide which histogram and which box plot are using the same data. Be prepared to explain your reasoning using specific evidence.

36. Iowa Core Standards
6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
6.SP.5 Summarize numerical data sets in relation to their context, such as by:
Reporting the number of observations.
Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
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.
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.

37. Variability

38. On your own…
For each of the four pairs of histograms, choose the statement that best describes the situation:
A has more variability than B
B has more variability than A
Both graphs are equally variable
Record your choice and thinking on the answer sheet.
Use worksheet: Introduction to Statistics and Data Analysis- Measuring Spread (Variability).

39. With your table group… Variability:
Discuss your thoughts and reasoning about the variability of each pair of graphs.
Variability:
How "spread out" a set of data is
The extent to which data points diverge from the average or mean value.
The extent to which data points differ from each other.
The extent to which data points in a statistical distribution or data set diverge from the average or mean value. Variability also refers to the extent to which these data points differ from each other.

40. Iowa Core Standards
6.SP.2 Understand 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.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.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. 6.SP.5 d Summarize numerical data sets in relation to their context, such as by: 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.

41. The Blue M&M™ Problem

42. Launch
Finley loves to go to the grocery store with her dad, Mike, and help him do the grocery shopping because she gets to pick out a treat for the car ride home. Her treat of choice is usually plain M&M’S™ Milk Chocolate Candies. Finley often dumps the entire bag out on her lap and sorts the candies by their color. She especially likes the blue ones.
Like any other weekend, Finley and Mike made a trip to the grocery store where Finley picked out a treat for the car ride home. Like any other car ride home, Finley dumped her bag of M&M’S™ out on to her lap and sorted them by color.

43. Was Finley’s bag of M&M’S™ “bad”? How could you decide?
However, unlike any other car ride home from the grocery store, Finley was sad. She said “I need a new bag of M&M’s...this bag is bad…there are only 8 blue ones!”
Was Finley’s bag of M&M’S™ “bad”? How could you decide?
Have the students discuss their conjectures with an elbow partner. After a few minutes, have the students share some of their thoughts with the entire group. Record them on the board or chart paper. Do not indicate correctness or lead them in any way.

44. Explore
How can we randomly test bags of M&M’S as a class? Please do not open bag or eat any candies until instructed to do so!
Ask the class: How can we randomly sample bags of M&M’s as a class? What considerations need to be made to make sure it is a valid sampling procedure? Let the students make some suggestions so as to discuss what would be random and what would not

45. Process Open your bag of M&Ms™
Dump out your bag and all the M&Ms™ onto a paper plate (do not eat any at this time)
How many in excess of 50 do you have? Remember that number.
Mix them all up again.
Process
Close your eyes and randomly remove enough M&Ms™ so that you have 50 left. This is your random sample of 50 M&Ms™.
Count the number of blues in your random sample 50 M&Ms™.
Record your “number of blues” on a post-it note.

46. What Do You Think?
Based on your sample, is Finley’s bag “bad”? Explain.
Based on the samples at your table, is Finley’s bag “bad”? Explain.
Hang your post-it at the appropriate spot on the class number line.
Based on the samples of ALL your classmates, is Finley’s bag “bad”? Explain.
Do you have enough information to make this decision now?
When will you have enough information to make this decision?

47. The statistical question we are trying to answer is:
Is it a rare event (meaning an uncommon occurrence) for a “good” bag of M&M’S™(same size as Finley’s) to have 8 blue candies? You will explore some aspects of binomial distributions that will help you answer this question.

48. Each observation falls into one of just two categories, which for convenience we call “success” or “failure”. There is a fixed number n of observations. The n observations are all independent. That is, knowing the result of one observation tells you nothing about the other observations. The probability of success, call it p, is the same for each observation.
We are concerned with observing whether the M&M™ is blue (success) or it is not (failure) We are looking at 500 bags of candies. One bag’s results does not depend upon another bag’s results. M&Ms™ company says 24 percent of the bag should be blue. So the p of success (blue) is .24
Use of this slide is optional depending upon what the teachers wishes to discuss with the students.

49. Use the Technology Open up Core Tools.
Open up Core Math Tools
Choose “Simulation Menu”. Select the “build” pull down menu, then selecting “distribution”, followed by “random binomial”.
This will allow you to build a binomial situation. The defaults of n=100 and p=0.5 can be changed by double-clicking near the text beginning with "Binomial." Two text boxes will appear where you will enter new values to change the value of n (number of trials) or p (probability of success). The number of trials for our problem is the number of M&M’s in our sample (50) and the probability of getting a blue (.24). Place these numbers in the appropriate boxes.
On the right top of the screen will be the command “Conduct” with a changeable value. In this box you can place any number to represent the packages of M&M’s you wish to simulate. This could be the number of students in the class. Under the View pull down menu, check the “model”, “table”, and “graph”.
Perform the simulation by pressing “conduct”. As students look at their results, ask the class if they have enough data to decide if Finley’s package is a “bad” one. Students may decide that they might need to have more bags and can press conduct again (note how the graph will change).
Let your students experiment and discover how they can change the number of simulations or even has continuously run simulations. Visit the “help” menu to get further information on how this program works. As the program runs, students may notice how technology can make their decision more informed
Open up Core Tools.
Work with a partner to simulate opening bags of M & M’s. What do you notice?

50. Summarize
How was the idea of spread displayed in this lesson? Explain.
How was the idea of shape shown in this lesson? Explain.
How was the idea of center used in this lesson? Explain.
Is Finley’s bag of M&M’S™ “bad”? Explain.

51. Check for Understanding
What will happen if we continue to simulate bags of M&M’S™ over a long period of time?
Demo Play Button for continued samples in binomial section of core tools.

52. Iowa Core Standard(s)
7.SP.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.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.
7.SP.7 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.
Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events.
Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process.
Discuss what parts of the standards have been present for this lesson/activity.

53. Scattered

54. Bivariate Data…. is data that has two variables
deals with causes or relationships
analysis purpose is to explain
may have one variable that influences or determines the second variable
can be represented using a scatter plot →

55. Explore some data….
With a partner, work through “Using Bivariate Data”. Be prepared to share your results from numbers 4 and 5 with the group.
Super Bowl Ad 
Description: Cost of a 30-second Super Bowl ad (in $1K) and winning team players’ share.
Source: Associated Press 2/2/2012
Selected Fast Food 
Description: Grams of fat and amount of calories in selected fast food items
Uses: Find a regression line to summarize the linear relationship between two variables; interpret the meaning of the slope and y-intercept of the regression line; and identify a potential outlier (7,360) and test how it influences the equation of the regression line and the correlation.
Movie Running Times 
Description: Running times and gross receipts for thirty of the top movies of 1997.
Source: Navigating Through Data Analysis in Grades 6–8 (NCTM, 2003)
Uses: Plot a scatterplot and compute correlation and determine if there is a relationship between gross receipts and running times of movies.

56. Wrap up
Was your line a “best fit”? Explain. How does knowing the equation of the line and the slope help your analysis? What other bivariate relationships could you explore?

57. Iowa Core Standard(s)
8.SP.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. 8.SP.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 line, and informally assess the model fit by judging the closeness of the data points to the line.
Discuss what parts of the standards have been present for this lesson/activity.

58. Comparing Categorical Data

59. Launch
Data can be organized in many ways. Use the information on your handout and organize it any way you would like. Be prepared to share your organization with the class and describe why you chose to organize it in this way.
Activity adapted from:eucc2011.wikispaces.com/file/view/8.SP.4+-+Two+Way+Tables.docx
As well as: Bridging the Gap Between Common Core State Standards and Teaching Statistics. Retrieved February 27, 2013, from
Have participants share their displays. Share two-way table last(if one exists). Then move forward to the next slide to practice using a two-way table.

60. Complete “Two-Way Table Exploration” to solidify your understanding.
Explore
Complete “Two-Way Table Exploration” to solidify your understanding.

61. Relative Frequency
Relative Frequency is the ratio of the number of observations in a statistical category to the total number of observations. Using “Survey Sheet”, determine the relative frequencies for the table and then complete the questions.

62. Summarize
How can relative frequencies be used to determine any possible associations in the data set you just finished? How is it different than just using the raw numbers? Explain. When and why are two-way tables used? How can two-way tables be manipulated to answer multiple questions?
Simpson’s paradox states that when data is separated it can show different results than when it is combined. For instance, ambulances are more effective at savings lives than rescue helicopters. This is true when all the data is put together, but does not hold true when separated by the seriousness of the accident.

63. Iowa Core Standard(s)
8.SP.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.
Discuss what parts of the standards have been present for this lesson/activity.