1. Statistics and Probability from the High School Core
Building Foundational Skills
Module 1.A
Address personal needs prior to starting.

2. Getting Started… Median = 8 Mode = 8 Mean is between 6 and 10
Create as many data sets as you can having 8 elements with the following characteristics:
Median = 8
Mode = 8
Mean is between 6 and 10
Success Criteria: I can explain the differences between the three measures of center.
Make sure you get participants to a point that they can be successful with this statement.
Note: Mode is not specifically mentioned in the Iowa Core but that does not mean we cannot discuss it as a measure of center.
Have participants work individually, then have them share some of their results. This includes the process or strategy used to create the data set.
Often participants will start will a strategy of 8 eights and then adjust the number of eights in use. If someone is struggling use questioning to start them here and then ask how can we adjust the numbers to make additional data sets.
Questions to consider:
In what ways would this have been easier or harder if there had been an odd number of elements?
For example: the median is easier to find since it is the middle number with an odd number of elements, etc.
What do the mean, median, and mode of a set of data tell you?
Describe a situation in which you would be most interested in knowing the:
mode of a data set.
mean of a data set.
median of a data set.
(This process will bring about MP2: “attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects”, pg. 8 of the Iowa Core)

3. I can explain the differences between the three measures of center.
Iowa Core Standard(s)
Foundation for:
S-ID.2
S-ID.3
6.SP.3 6.SP.5(d)
Covers standards 6.sp.3 and 6.sp.5(d)
Foundation for standards S-ID.2 and S-ID.3
I can explain the differences between the three measures of center.

4. Learning Targets
Understand that data can be displayed in a variety of ways depending on the purpose of the display. Understand that the study of statistics must be grounded in relationships among shape, center, and spread. Understand that data can be compared in a variety of ways using a variety of tools.

5. Success Criteria
I can explain the differences between the three measures of center.
I can list two displays used for categorical data.
I can list two displays used for quantitative data.
I can describe why categorical and quantitative displays differ.
I can list three terms used to describe variability.
I can define the Five Number Summary and interpret its meaning.
These will be reviewed at the end of session 1.A to check for success.
FYI: The Five Number Summary is used to create a boxplot and represents the minimum, quartile 1 (25%), median (50%), quartile 3 (75%), maximum. It does not specify “Five Number Summary” in the Iowa Core but it is implied and illustrated in the Progressions documents.

6. Categorical vs. Quantitative
What types of displays are used for categorical data?
What types of displays are used for quantitative data?
Why are they different?
List specific characteristics and limitations
Do both types of data show variability?
Success Criteria:
(The next two slides cover aspects of all the success criteria. The bolded statements are only covered in these slides.)
I can explain the differences between the three measures of center.
I can list two displays used for categorical data.
I can list two displays used for quantitative data.
I can describe why categorical and quantitative displays differ.
I can list three terms used to describe variability.
I can define the Five Number Summary and interpret its meaning.
Ask participants to answer the first two questions in a group. After completing have participants share with the whole group. If needed, after sharing, fill in any gaps that exist before moving to the following question.
Move to ask participants why they are different and listing specific characteristics/limitations. Then discuss as a whole group.
Finally, have participants consider the final question. This question is designed to keep the flow of statistics moving and not stay stagnant on one feature or skill at a time.
Sample Answers:
Categorical: Bar graph, Pie Chart, Relative Frequency
Quantitative: Histogram, Dot/Line Plot, Box Plot, Stem-and-Leaf
Categorical does not show variability, Quantitative does show variability.
This can be seen in the difference between a bar graph and a histogram. The bar graph is separated because it represents discrete data that is just looking for a total count in each category. The histogram is connected because it represents continuous data that is looking to find the spread of the data. The histogram intervals can be adjusted to fit the data.
There is an additional resource about categorical vs quantitative data attached if needed. (Word document: Categorical Quantitative Data)
An answer key for the additional resource is also available. (Word document: Categorical Quantitative Data-Answer Key)

7. 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?
Share your display with the group.
Which display represented the data the best?
Handout is in the resources. (Word document: Make a Display – Study Time)
Graphical displays are in the resources as well. (Word document: Make a Display – Study Time – graphs large)
(Word document: Make a Display – Study Time – graphs one page)
Retrieved from NCTM Core Tools: Univariate Quantitative: Study Time
Directions: Handout Make Display – Study Time to participants. Make sure participants have access to resources such as graph paper, plain white paper, straight-edge, etc. If participants finish, ask them to create another display. They can start to compare their displays if they make multiple displays, or compare with their tablemates. This activity will take time.
After giving some time to work, ask participants to wrap-up making displays and answer the two questions on their worksheet. Share the answers to these questions as well as the displays with the whole group.
Provide different displays if needed to generate conversation, but only if participants do not create these first. (Stem and Leaf 0-9, Stem and Leaf 0-4 & 5-9, Histograms of different intervals, Dot Plot, Box Plot)
Discussion of Box Plots:
If someone has a box plot, have them share the plot as well as how they created it. If not, you will need to bring an example of a box plot to their attention. Make sure vocabulary words such as Minimum, Quartile 1, Median, Quartile 3, and Maximum occur. Use these vocabulary words in defining the Five Number Summary. Also, discuss IQR (InterQuartile Range) and what it represents, Q3 – Q1. You do not need to define 1.5 * IQR as a way to find outliers, but you will want to discuss why IQR is important (effects the shape of the display, helps understand variability, can show patterns in your data, etc).
Discussion of Stem and Leaf:
This is not in the Iowa Core, but conversation around the display will help participants to see why we focus on box plots and histograms, as they give us a good picture of the data and help show variability. Stem and Leafs do not handle decimals as well as histograms, which is another reason to focus on histograms.
Which display is best?
This will bring about conversation around shape, center, and spread. Make sure participants use statistical reasoning to back up their responses. Which display helps to see outliers? Where is the center? Is it the mean or the median? Can mode be seen? How can I define variability (Quartiles, IQR, standard deviation, range, etc.)? Which measures of variability provide more information about the display than others?

8. Iowa Core Standard(s) 6.SP.4 6.SP.5(c) Foundation for: S-ID.1 S-ID.2
Covers standards 6.sp.4 and 6.sp.5(c)
Foundation for standards S-ID.1, S-ID.2, and S-ID.3
I can list two displays used for categorical data.
I can list two displays used for quantitative data.
I can describe why categorical and quantitative displays differ.
I can define the Five Number Summary and interpret its meaning.

9. Break
10-15 minutes as needed by your group

10. Statistics: Creating a New Way to Think About Mathematics
Statistics is multi-faceted; it is not just making calculations using algorithms. Statistics calls for a transformed way of thinking that requires us to consider what the data is saying, to determine the available evidence, and to communicate the findings.
Use this slide as a transition into exploring statistical reasoning.

11. Statistical Reasoning
Four-Step Investigative Process
Formulating a statistical question- a question that can be answered with data
Designing a plan for collecting useful data, implementing the plan, and collecting relevant data
Analyzing the data- creating and exploring various representations of the distribution to identify and describe patterns in the variability in the data and summarize various features of the distribution with appropriate methods
Interpreting the results- providing a statistical answer to the question posed that takes the variability in the data into account
Referenced in the progression 6-8 Stats and Prob pg 2
Dev Essential Understanding of Statistics 6-8 Pg 8, 93-99
Highlight the verbs to show the level of DOK or Bloom’s that will be reached by doing this process.
As a presenter reference the Iowa Core, specifically standard 6.SP.1:
Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.
Also reference pg 4 of the 6-8 Progressions document to get a better explanation of this idea.

12. Progressions Overview
Read Overview of Progressions. Discuss as a group. Focus discussion on the content as well as the emphasis on shape, center, and spread that will now exist in grades 6-8.
Progressions document available in resources. Just print pages 2 & 3. (pdf: 6 – 8 Stats and Prob Progression)

13. Means & MADs Adapted from “Means and MADs”, Mathematics Teaching in the Middle Grades, NCTM, 1999
Success criteria: I can list three terms used to describe variability.
Original article is in the resources. (pdf: MeansAndMAD)
There is a more in-depth lesson plan also available in the resources. (Word document: MeansAndMADs_LessonPlan)

14. Launch
In a survey, nine people were asked, “How many people are in your family?” One result from the poll was that the average family size for the nine people was five.
Place in groups of three

15. Launch
In your group, determine some possibilities for the distribution of the nine different family sizes utilizing the fact that the mean family size is 5?
Display your distributions as line plots using the chart paper and post-its supplied by your teacher.
For this problem, consider family sizes no smaller than 2 and no larger than 11.

16. Explore Think-Pair-Share
Given these 8 distributions with a mean of 5
Hang larger pictures of distributions up in the classroom. (Word document: Means Mads graphs large)
Handout smaller pictures of distributions to each group, making sure they were cut in advance so they can manipulate the ordering. (Word document: Means Mads graph one page)
A TI-Inspire file with the distributions is also available if needed. (tns: MeansMADs)
Note: The mean is often viewed as the most important piece of information because it is easily calculated. These 8 distributions show us that the mean of 5 doesn’t really tell us much at all, as the distributions can be very different.
Think-Pair-Share
What are the limitations of only knowing the mean family size?

17. A major goal of statistics is to offer ways to summarize and measure variation in data.
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.
This goes back to formulating the question, as the variation helps us define the question we are trying to answer.
Have participants share their responses. Least variation is often easier to see, but most variation can be more difficult. Probe the participants to see why that happens.

18. How would you order, from least variation to most variation, the 8 distributions?
Individually (1 minute) Group (2 minutes) Share
Time should be restricted so that the students can not use a metric (i.e. MAD or standard deviation) to do their ordering
Document the orders from the class for all to see. Discuss how they may differ and what we may do about this.
Record Individually
Come to a group consensus
Share out

19. (consider as many methods as you can)
Because it can be difficult to come to a consensus on this single ordering, we need a number (a metric) to quantify variation in a set of data. In your group determine methods to do this…
(consider as many methods as you can)
This is creating your own numeric system to define the variability. Some participants will have methods they have used before, but guide them to think differently to consider new ideas.
Guiding questions for struggling participants:
Can we use the measure of center as a reference?
Can we use the space between data points as a reference?
Does it matter how many data points exist on each number?

20. 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.
If the process for MAD does not come out of this sharing, it must be introduced.
MAD is defined in the Iowa Core in the glossary pg 89.
Iti is also defined on pg 401 of the Means and Mads article.

21. Calculate the MAD (Mean Absolute Deviation)
Ask participants to calculate the MADs for all 8 displays. You may separate and have each group do a few displays to save time.

22. Distribution Number MAD1 2 3 4 5 6 7 8
Record answers from the class on slide or use answer sheet under the document camera.
Answer key is available in the resources. (Word document: MeansMAD Answer Key)

23. 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?
This is the push to HS when standard deviation is used instead of MAD.

24. Summarize
How can the MAD help you distinguish between Distribution 3 and Distribution 5?
What does a MAD of 1.78 indicate
about the set of data?
C. The mean and median are often
referred to as “measures of center”.
The median is the value in the middle
(the idea of halfway). How is mean a
center?
When two values have the same MAD there is no procedure to distinguish between them. At this point we may be looking for other statistical ideas to differentiate if needed.
MAD helps decide, on average, how far is the data away from the center (mean). So this means, on average, the data points are 1.78 away from the center in either direction.
We often define mean as average, so it takes all data points into consideration. It is important to note that mean will be affected by points that seem extreme in either direction.

25. Extension(s)
Given the Standard Deviation formula. Compare it to the MAD.
Research the development of the Standard Deviation formula. Explain.
This is only done if extra time exists or a group finishes early and needs an alternate task.

26. I can list three terms used to describe variability.
Iowa Core Standard(s)
Foundation for:
S-ID.1
S-ID.2
S-ID.3
S-ID.4
6.SP.2 6.SP.3 6.SP.5
Covers standards 6.sp.2, 6.sp.3, and 6.sp.5
Foundation for standards S-ID.1, S-ID.2, S-ID.3, and S-ID.4
I can list three terms used to describe variability.

27. Variability
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
Success criteria: I can list three terms used to describe variability.
Worksheet is available in the resources. (Word document: Variability Data graphs)
Answer sheet is available if needed to record work. (Word document: Variability Data graphs – answer sheet)
Answer key is also available. Realize many interpretations can be taken and use the responses from students to guide the conversation. Do not focus on matching the answer key exactly. (Word document: Variability Data graphs – answer key)
It is important to note that Pair 4 can be used to discuss how outliers pull the mean and will make the data more variable.

28. I can list three terms used to describe variability.
Iowa Core Standard(s)
6.SP.2 6.SP.3 6.SP.4 6.SP.5(d)
Foundation for:
S-ID.1
S-ID.2
S-ID.3
S-ID.4
Covers standards 6.sp.2, 6.sp.3, p.sp.4, and 6.sp.5(d)
Foundation for standards S-ID.1, S-ID.2, S-ID.3, and S-ID.4
I can list three terms used to describe variability.

29. Success Criteria
I can list the differences between the three measures of center.
I can list two displays used for categorical data.
I can list two displays used for quantitative data.
I can describe why categorical and quantitative displays differ.
I can list three terms used to describe variability.
I can define the Five Number Summary and interpret its meaning.
Review the success criteria to see how participants feel about each. It is recommended to use a thumbs up, sideways, or down to get feedback.

30. Reflection
3 things I can take away from this learning. 2 things I want to implement in my teaching. 1 question I still have.
Document is available in the resources. (Word document: Reflection)
Have participants hand this in as their exit ticket to end the day or to head to lunch.

31. Statistics and Probability from the High School Core
Building Foundational Skills
Module 1.B
Address personal needs prior to starting.
If this PD is one full day, do 1.A and 1.B together and make this a lunch slide.

32. Sampling Techniques
The school food service wants to increase the number of students who eat hot lunch in the cafeteria. The student council has been asked to conduct a survey of the student body to determine the students’ preferences for hot lunch. They have selected three ways to do the survey. Which survey option should the student council use and why?
Obtain a class list of the freshman, number the students starting at 1, then choose every student whose number ends in a 0 (For example: 10, 20, 30, etc.). Repeat this process for the 10th, 11th, and 12th grade classes.
Survey the first 30 students that enter the lunch room.
Write all of the students’ names (9-12) on cards and pull out 25 names to determine who will complete the survey.
Success criteria: I can utilize a random sample to answer questions about a data set.
Adapted from the katm flipbooks 7.sp.1, which is available in the resources. (Pdf: KATM Flipbook 7-SP-1.pdf)
Answer sheet is available in the resources. (Word document: Sampling Techniques)
Answer key is available in the resources. (Word document: Sampling techniques Answer Key)
Please read the answer key carefully, as it explains the goal of this activity.

33. I can utilize a random sample to answer questions about a data set.
Iowa Core Standard(s)
Foundation for:
S-ID.1
S-ID.2
S-ID.3
S-ID.6
7.SP.1
Covers standard 7.sp.1
Foundation for standards S-ID.1, S-ID.2, S-ID.3, and S-ID.6
I can utilize a random sample to answer questions about a data set.

34. Learning Targets
Understand that data can be displayed in a variety of ways depending on the purpose of the display. Understand that the study of statistics must be grounded in relationships among shape, center, and spread. Understand that data can be compared in a variety of ways using a variety of tools.
If done as a one day course, you do not need to cover this slide again.

35. Success Criteria
I can develop a probability model and use it to create relative frequencies.
I can find probabilities of compound events using multiple strategies.
I can utilize a random sample to answer questions about a data set.
I can form comparative inferences using measures of center and spread.
I can determine an appropriate model for bivariate data.
I can compare categorical data using a two-way table.
These will be reviewed at the end of session 1.B to check for success.

36. Comparative Inferences
Compare the random sampling of heights of soccer players and basketball players using the line plots and statistical information given. Make sure to compare the two data sets in terms of shape, center, and spread.
Success criteria: I can form comparative inferences using measures of center and spread.
Adapted from the katm flipbooks 7.sp.3, which is available in the resources. (Pdf: KATM Flipbook 7-SP-3.pdf)
Statistical data and Answer sheet are available in the resources. (Word document: Comparative Inferences)
Answer key is available in the resources. (Word document: Comparative Inferences Answer Key)
Allow participants time to work individually then in small groups. Bring it back together as a large group and share all together.
Standard 7.SP.3 is often not how we think when comparing data sets. The next three slides help participants understand what is meant by this standard.

37. By the standard…
Iowa Core states we compare two data sets by measuring the difference between the centers and expressing it as a multiple of a measure of variability. But why?
This comes from 7.SP.3. If needed, you can guide participants to look at the standard in the Iowa Core document.
Allow participants to share ideas of why they think this is in the document. It may be best to do this in small groups then come back as a larger group.
The next slide should help put everyone on the same page.

38. Our data
The difference between the means is Knowing the difference between centers is nice, but it is more meaningful when it is made relative to the variation in the data. In this case, it is logical to wonder if soccer players are shorter than basketball players. It looks to be likely by the naked eye, but what data is there to support that? Taking the difference and dividing by the Mean Absolute Deviation will give us numerical evidence that this is true. In this case, 7.68/2.53 gives us a value of The larger this number, the less likely that the data sets will share values in common.
It is recommended that the answer key be distributed after conversation so participants can make a tie to the Iowa Core standards.
Answer key is available in the resources. (Word document: Comparative Inferences Answer Key)

39. Check for Understanding
Which pair of data sets is most likely to have the greatest number of values in common? A B Data set 1: mean = 7 Data set 1: mean = 7 Data set 2: mean = 15 Data set 1: mean = 15 MAD for both data sets is 4 MAD for both data sets is 8 C D Data set 1: mean = 10 Data set 1: mean = 10 Data set 2: mean = 15 Data set 1: mean = 18 MAD for both data sets is 1 MAD for both data sets is 2
Answer: B
By comparing the difference in means and dividing by the MAD we get 1 in B, 2 in A, 4 in D, and 5 in C. The smaller the number, the greater the chance that the two data sets will share common values.
*In a classroom setting, it is vital that more examples are used to illustrate 7.SP.3 and 7.SP.4 to show how the visual look of the plots will change when the variability changes. Making strong connections to this both graphically and numerically will impact the understanding of these standards.

40. I can form comparative inferences using measures of center and spread.
Iowa Core Standard(s)
Foundation for:
S-ID.5
S-ID.6
7.SP.3 7.SP.4
Covers standards 7.sp.3 and 7.sp.4
Foundation for standards S-ID.5 and S-ID.6
I can form comparative inferences using measures of center and spread.

41. The Right Fit
Success criteria: I can determine an appropriate model for bivariate data.
This activity is done while using the PowerPoint and using technology, therefore there are no activity sheets for participants to fill out. A lesson plan is available to assist the presenter. (Word document: The RightFit_LessonPlan)
This activity is adapted from:

42. Launch
Fred and Ginger have been given the following coordinates and are asked to find a linear function to model the data. The coordinates include: (0,2), (1,3), (2,5), (3,6), (4,8), and (5,9). Fred believes f(x) = 2x + 1 will be a good model while Ginger believes g(x) = x + 3 will be a better equation to model the data. Which equation is a better model to predict data?
Allow participants to use whatever method they think can help them answer the question, but without the use of technology.
Provide plain paper, graph paper, markers, straight edges, etc.
If participants start to create a metric or use the idea of residuals, thank them for their work and ask them to take a more visual approach for now. This metric will come back soon.
Teacher Follow-Up to Whole Class
Questions:
What method was used to decide on the better fit? Explain.
What comparisons could be made while graphing the lines? Explain.
Will we all make the same conclusion? Explain.

43. Putting it into Context
The JRM Company has decided to start a big ad campaign to increase their customer base. This company currently has 20,000 customers. After 3 months of the campaign, or 1 quarter, the company has grown to 30,000 customers. As their ad campaign wraps up after 5 quarters, they have grown to a total of 90,000 customers.
Puts it in a context to hit 8.SP.3.

44. Putting it into Context
What would be the labels of the x- and y-axis? What do the numerical values of slope and y-intercept represent in Fred and Ginger’s equations? In context, which model seems to fit the data the best?
x- qtrs, y-customers in 10,000’s
Fred: 2x +1 so they start at 10,000 customers and increase by 20,000 per qtr
Ginger: x +3 so they start at 30,000 customers and increase by 10,000 per qtr
3. Sample answer: Ginger’s data makes more sense since we don’t expect customers to continue to grow at such a high rate.
Have participants answer these questions in a small group based on the context provided in the previous slide. Then share as a whole group.

45. Explore
In order to determine whether Fred or Ginger has the better model we will need to develop a way to measure how well their equations fit the data.
Put participants into partnerships.

46. Work Together
As a pair, create a metric that will quantify which equation better fits the data. Record the method that you will use and then test it out to determine if Fred or Ginger had the better model.
This is creating your own numeric system to define which equation is the better fit. Some participants will have methods they have used before, but guide them to think differently to consider new ideas.
Guiding questions for struggling participants:
What numeric data do you have and how can you use it?
Can we compare the sample lines with our data points?
Do we need to use both the x and the y coordinates?
What happens to y as x changes?
This is preparing students to understand the idea of residuals by building some conceptual knowledge. If you need materials to look into residuals, a few resources are offered below:

47. Share Out
Explain your methodology for developing a metric to find the equation with the better fit.
Leave time for discussion. The goal of this section is to build a foundation for work with residuals in the HS Core. There is not a right answer, so bringing forth examples to the class to look at and compare/contrast will help promote thinking.
Possible Probing Questions:
Was there anything in common between the methods chosen? Explain.
Do some measurements seem easier to come by? Explain.
Will some methods be better than others? Explain.

48. Technology Abounds!
Use technology to fit the best line you can to the data. Write down the equation of this line and interpret the slope and y-intercept. Use the metric you created earlier to test your line.
Use a graphing utility for this activity. Examples include: NCTM Core Tools, TI Calculators, etc.
If using NCTM’s Core Tools, model how it could be used. This includes opening the spreadsheet, finding data sets, making scatter plots, fitting a line to the data, and getting the equation of the line. If using a TI-84, you will need to discuss the use of tables with the STAT key, graphing using the STAT PLOTS key as well as the WINDOW and GRAPH keys, using the Y= key to try best fit lines, and finally graphing the possible best fit lines with the original data points.
The goal of this activity is to show than neither Fred or Ginger had good model. By fitting a better line to the data we get a better idea of what will happen in the context of the problem. If the metrics created give a valid metric, it should give better results with this newer line.

49. Summarize
Did the metric created for the “best” fit line verify your visual intuition?
The majority of “best” fit lines did not start at the y-intercept of 2, is this a problem?
Why would we need a line to represent our data?
The metric should help show the best fitting line and work as a numerical verification.
The y-intercept is simply another piece of data, so the “best” fit lines do not necessarily need to go through at that value. It is more necessary to include the majority of points and not to focus on specific intercepts.
Sample answer: We may want to have an idea of what happens over time and a line is a good way to make predictions.

50. I can determine an appropriate model for bivariate data.
Iowa Core Standard(s)
Foundation for:
S-ID.6
S-ID.7
S-ID.8
8.SP.1 8.SP.2 8.SP.3
Covers standards 8.sp.1, 8.sp.2, and 8.sp.3
Foundation for standards S-ID.6, S-ID.7, and S-ID.8
I can determine an appropriate model for bivariate data.

51. Break

52. Comparing Categorical Data
Success criteria: I can compare categorical data using a two-way table.

53. Launch
Data can be organized in many ways. Take the following information and organize it however you would like. Be prepared to share your organization with the class and describe why you chose to organize it in this way.
Answer sheet is available in the resources. (Word document: Two Way Table Launch)
Answer key is available in the resources. (Word document: Two Way Table Launch-Answer Key)
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
At this point we want students to see a few representations but really want them to understand a two-way table. If a participant presents it, make sure to ask clarifying questions so participants know why the numbers go in certain places in the table. Also make sure to include the rows and columns to total the data.
Have participants share their displays. Share two-way table last(if one exists). If no one created a two-way table, share one at the end of this section. Make sure to discuss why a two-way table is appropriate for this information. The Iowa Core is approaching it as using a two-way table to use this information and then can go to a Venn diagram. Venn diagrams are not listed in the Iowa Core. Next, move forward to the next slide to practice using a two-way table.

54. Explore
Complete the Worksheet to solidify your understanding of Two-Way Tables.
Answer sheet is available in the resources. (Word document: Two Way Table Explore)
Answer key is available in the resources. (Word document: Two Way Table Explore-Answer Key)
In the last slide, participants should have been clear about two-way tables. This worksheet should help solidify that thinking. Have participants complete the worksheet individually then discuss as a group. Make sure participants state how they chose the appropriate numbers.

55. Summarize
When and why are two-way tables used? How can two-way tables be manipulated to answer multiple questions?
The second question helps push students towards the ideas of marginal and conditional probabilities.
Possible follow-up question: How many different questions could you ask about a two-way table?

56. Check for Understanding
Complete the worksheet.
Answer sheet is available in the resources. (Word document: Two Way Table Check for Understanding)
Answer key is available in the resources. (Word document: Two Way Table Check for Understanding-Answer Key)
This worksheet will continue to push the thinking of the many ways a two-way table can be used to gain information. Debriefing this worksheet with the participants will provide conversation to clarify the ideas.

57. I can compare categorical data using a two-way table.
Iowa Core Standard(s)
Foundation for:
S-ID.5
S-CP.4
8.SP.4
Covers standard 8.sp.4
Foundation for standards S-ID.5 and S-CP.4
I can compare categorical data using a two-way table.

58. Calculate the Probability…
Getting an even number when rolling a fair dice Getting one head and one tail when tossing two coins Landing in the circle
Success criteria: I can develop a probability model and use it to create relative frequencies.
3 even numbers out of a total of 6 numbers on a dice, so 3/6 or 1/2. (simple event)
There is a 1/4 chance for HH, HT, TH, and TT. That leaves 1/4 + 1/4 = 1/2 for one of each side. (compound event)
The area of the circle is πr2, or 25π, or approximately The area of the square is 10x10, or So the probability of landing in the circle is 25π/100, or π/4, or approximately 78.54%. (probability model)
This is a launch to get participants thinking about probability.

59. IT TAKES TIME
Probability uses theory to create calculations, but those values show what will happen over many, many trials. In the short-run, the chance process to produce sample outcomes is not as predictable. The question then becomes: “How long does it take to feel confident in a probability prediction?”
The following activity is designed to show students that data can be variable, but over the long run, will move closer to theoretical probabilities. Make sure students and staff do not get frustrated if some samples do not move as quickly as others. Data is variable, and students need to experience this. This activity should develop a sense of variation in sampling and an ability to accepts slight variations with each additional trial versus big swings that may occur early in the

60. “In the Bag” http://nrich.maths.org/6016
Direction sheet as well as answer sheet are available in the resources as a single document. (Word document: In the Bag)
No answer key is given, as the answers will vary as the participants experience the web activity. The hope is that students experience that probability is not as predictable in a short run.
Have students play this individually or in partnerships. It is recommended that the direction sheet is read aloud to students before going to the web activity. It is also recommended to sample the activity with the participants first.

61. Iowa Core Standard(s) 7.SP.6 7.SP.7 Foundation for: S-MD.7
Covers standards 7.sp.6 and 7.sp.7
Foundation for standard S-MD.7
I can develop a probability model and use it to create relative frequencies.

62. One and One Equals Win
Success criteria: I can find probabilities of compound events using multiple strategies.

63. The Situation
The basketball team is down by one with one second on the clock but a foul may save them. Can a 60% free throw shooter win the game for her team? How often?

64. Simulate the Situation
Using technology, perform a simulation that will determine how often a 60% shooter can win the game.
Directions for possible simulation ideas for NCTM’s Core Tools as well as a TI-84 is available in the resources. (Word document: One and One Technology Guide)
Answer sheet to be used after the simulation activity is available in the resources. (Word document: One and One Answer Sheet)
Answer key for an organized list, table, and tree diagram is available in the resources. (Word document: One and One Answer Key)
It is recommended that you walk the participants through the use of the technology. After that, hand out the answer sheet and ask for the desired number of results from participants and have a place for them to be recorded that everyone can access. They will record this on their answer sheet then move forward to the second question. They may not have much background in calculating the theoretical probability using organized lists, tables, or tree diagrams. Using a 10x10 table is the easiest way to get a handle of a visual that represents losing the game, getting a tie, and winning the game.

65. The Summary
What connection is there between the free throw shooter’s percentage and the percent of wins? How was this seen in the organized list, table, or tree diagram? What percentage of times did the shooter lose the game? What does that have to do with the shooter’s percentage?
The summary is designed to verify an understanding of the various displays and how a probability of .36 of winning can be created. This must be based on the visual and not just on multiplying .6 and .6. Students need something to connect the idea to, so the table and/or organized list are vital.
Extension: Ask how the students could have changed their simulation to look for losses. Change the p to .40 or have the program count “0” instead of “2”

66. I can find probabilities of compound events using multiple strategies.
Iowa Core Standard(s)
Foundation for:
S-CP.1-3
S-CP.6-9
7.SP.8
Covers standard 7.sp.8
Foundation for standards S-CP.1-3and S-CP.6-9
I can find probabilities of compound events using multiple strategies.

67. Success Criteria
I can develop a probability model and use it to create relative frequencies.
I can find probabilities of compound events using multiple strategies.
I can utilize a random sample to answer questions about a data set.
I can form comparative inferences using measures of center and spread.
I can determine an appropriate model for bivariate data.
I can compare categorical data using a two-way table.
Review the success criteria to see how participants feel about each. It is recommended to use a thumbs up, sideways, or down to get feedback.

68. Thank you Your hard work and great participation was much appreciated!
New guidelines are trying to get students to look at data sets of 30 or more. As you continue work back in the classroom, focus on trying to get larger data sets in front of students. This will most likely happen as the use of technology increases and data sets can be entered into a calculator or computer.
The following website is a good additional resource: