1. Welcome to Common Core High School Mathematics Leadership
Summer Institute 2014
Session 2 • 17 June 2014
Comparing distributions and beginning with statistical questions

2. Today’s Agenda Homework review and discussion
Grade 9, Lesson 8: Comparing Distributions
Reflecting on CCSSM standards aligned to lesson 8
Break
Grade 6, Lesson 1: Posing Statistical Questions
Reflecting on CCSSM standards aligned to lesson 1
Group presentation planning time
Homework and closing remarks
Timing for our purposes:
Homework review and discussion 3:30
Grade 9, Lesson 9: content 4:15, reflection 5:15
Dinner & Break 5:30
Grade 6, Lesson 1 content 6:00, reflection 7:00
Group presentation planning time: 7:15
Homework and closing remarks 8:15

3. Activity 1 homework review and discussion
Table discussion
Discuss your write ups for the Day 1 homework tasks:
Compare your strategies with others at your table
Reflect on how you might revise your own solution and/or presentation

4. Learning Intentions and Success Criteria
We are learning to…
• Describe how two distributions compare using center, variability, and shape.
• Interpret the center of a data population distribution as the typical value of that distribution.
• Interpret the interquartile range (IQR) as description of variability, specifically for data distributions that are skewed.

5. Learning Intentions and Success Criteria
We will be successful when we can:
• identify a nearly symmetrical distribution or a skewed distribution using the mean and the median.
• identify how two data distributions are similar or different by describing their shapes, comparing their centers (means and medians), and describing their variability in general terms.
• pose statistical questions that involve collecting and interpreting data.

6. Activity 2 Lesson 8: comparing distributions
Review of data distributions and representations
EngageNY/Common Core Grade 9, Lesson 8

7. Activity 2 Lesson 8: comparing distributions

8. Activity 2 Lesson 8: comparing distributions
What percent of people in Kenya are younger than 5? How do you find the answer to this question?
What does the first bar in the U.S. histogram mean?
Describe the differences in the shape of the two histograms.

9. Activity 2 Lesson 8: comparing distributions

10. Activity 2 Lesson 8: comparing distributions
What information is displayed in a box plot?
What do you think the (*) represents in the box plot for Kenya?
Does the box plot tell us the same or different information from the histogram?

11. Activity 2 Lesson 8: comparing distributions
Reflecting on CCSSM standards aligned to lesson 8
Review the following CCSSM High School content standards:
S-ID.1
S-ID.2
S-ID.3
Where did you see these standards in the lesson you have just completed?
What would you look for in students’ work to suggest that they have made progress towards these standards?

12. Activity 2 Lesson 8: comparing distributions
S-ID.1: Represent data with plots on the real number line (dot plots, histograms, and box plots). S-ID.2: Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. S-ID.3: Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliners).

13. Activity 2 Lesson 8: comparing distributions
Reflecting on CCSSM standards aligned to lesson 8
Read MP6, the sixth CCSSM standard for mathematical practice.
Recalling that the standards for mathematical practice describe student behaviors, how did you engage in this practice as you completed the lesson?
What instructional moves or decisions did you see occurring during the lesson that encouraged greater engagement in MP6?
Are there other standards for mathematical practice that were prominent as you and your groups worked on the tasks?

14. Activity 2 Lesson 8: comparing distributions
CCSSM MP.6
engageny MP.6
MP.6 Attend to precision
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
MP.6 Attend to precision.
Students interpret and communicate conclusions in context based on graphical and numerical summaries. Students use statistical terminology appropriately.

15. Activity 2 Lesson 8: comparing distributions
Closing questions for lesson 8
What does a histogram show about the general shape of a data distribution?
What does a box plot show about the general shape of a data distribution?
What does a “typical” value indicate about the data distribution?

16. Break

17. Activity 3 Lesson 1: posing statistical questions
Measures that summarize data distribution shape
EngageNY/Common Core Grade 6, Lesson 1

18. Activity 3 Lesson 1: posing statistical questions
Statistics is about using data to answer questions. In this module, the following four steps will summarize your work with data: Step 1: Pose a question that can be answered by data. Step 2: Determine a plan to collect the data. Step 3: Summarize the data with graphs and numerical summaries. Step 4: Answer the question (the statistical question) posed using the data and the summaries.

19. Activity 3 Lesson 1: posing statistical questions
What is a statistical question?
Identify what you think are the characteristics of a statistical question based on your work with the previous lessons.
What were possible statistical questions investigated in Lesson 2?
What were possible statistical questions investigated in Lesson 8?

20. Activity 3 Lesson 1: posing statistical questions
A statistical question is a question that can be answered with data and for which it is anticipated that the data (information) collected to answer the question will vary.

21. Activity 3 Lesson 1: posing statistical questions
Could the following questions be answered by collecting data? Would you consider them to be statistical questions?
How tall is your 6th grade teacher?
How tall is a typical 6th grade teacher?
What is your hand span (in inches)?
What is the hand span of 6th graders?

22. Activity 3 Lesson 1: posing statistical questions
Reflecting on CCSSM standards aligned to lesson 1
Read the following CCSSM Grade 6 content standards:
6.SP.A.1, 6.SP.A.2, 6.SP.B.4, 6.SP.B.5b
Where did you see these standards in the lesson you have just completed?
What would you look for in students’ work to suggest that they have made progress towards these standards?

23. Activity 3 Lesson 1: posing statistical questions
6.SP.A.1: Recognize a statistical question as one that anticipates variability in the data related to the question and takes account of it in the answers. 6.SP.A.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.B.4: Display numerical data on plots on a number line, including dot plots, histograms, and box plots. 6.SP.B.5b: Summarize numerical data sets in relation to their contexts by describing the nature of the attribute under investigation, including how it was measured and its units of measurement.

24. Activity 3 Lesson 1: posing statistical questions
Reflecting on CCSSM standards aligned to lesson 1
Compare the Grade 6 standards with the High School standards you read earlier.
What similarities and differences do you see between the standards at these two grade levels?
How can you ensure that the study of statistical distributions in High School builds on, but is not a repeat of, that in the middle grades?

25. Activity 3 Lesson 1: posing statistical questions
Reflecting on CCSSM standards aligned to lesson 1
Read MP1, the first CCSSM standard for mathematical practice.
Recalling that the standards for mathematical practice describe student behaviors, how did you engage in this practice as you completed the lesson?
What instructional moves or decisions did you see occurring during the lesson that encouraged greater engagement in MP1?
Are there other standards for mathematical practice that were prominent as you and your groups worked on the tasks?

26. Activity 3 Lesson 1: posing statistical questions
CCSSM MP.1
engageny MP.1
MP.1 Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
MP.1 Make sense of problems and persevere in solving them.
Students make sense of problems by defining them in terms of a statistical question and then determining what data might be collected in order to provide an answer to the question and therefore a solution to the problem.

27. Activity 3 Lesson 1: posing statistical questions
Closing questions for lesson 1
What makes a question a statistical question?
How is a statistical question answered?
Consider the question: “What do participants want to eat during this class?”
Is this a statistical question? Why or why not?
If data is collected, how would you describe the data?
How would we answer the question?

28. Learning Intentions and Success Criteria
We are learning to…
• Describe how two distributions compare using center, variability, and shape.
• Interpret the center of a data population distribution as the typical value of that distribution.
• Interpret the interquartile range (IQR) as description of variability, specifically for data distributions that are skewed.

29. Learning Intentions and Success Criteria
We will be successful when we can:
• identify a nearly symmetrical distribution or a skewed distribution using the mean and the median.
• identify how two data distributions are similar or different by describing their shapes, comparing their centers (means and medians), and describing their variability in general terms.
• pose statistical questions that involve collecting and interpreting data.

30. Activity 4 group presentation planning time
During Week 2 of the institute, you will present (in groups of no more than three) one of the following Engage NY lessons:
Grade 6, Lessons 2, 3, 4, 5, 16
Grade 8, Lesson 6
Grade 9 (Algebra 1), Lessons, 3, 17
For the rest of our time today, you should study these lessons, decide which one you wish to present, and find a group with which you will present.

31. Activity 5 Homework and Closing Remarks
Complete the problem set problems in Algebra I, Lesson 8 and Grade 6, Lesson 1 that are assigned in our discussion.
Extending the mathematics: The data distribution of ages from a sample from Kenya had a noticeable difference in the median age and the mean age. Summarize what makes a data distribution have a noticeable difference in the median and mean. Describe at least two other data sets that you think will have a noticeably different mean and median. Explain why you think they will have noticeable differences.
Reflecting on teaching: In middle and high school, work on statistics often focuses on calculating measures of center and spread and creating data displays. In what ways might the consideration of a statistical question support the development of students’ understandings of statistics?