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Welcome to Common Core High School Mathematics Leadership

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### Welcome to Common Core High School Mathematics Leadership

Summer Institute 2014

Session 6 • 23 June 2014

Two-way tables and association

Today’s Agenda

- Activity 1: Homework review and discussion
- Activity 2: Grade 9, Lesson 9: Summarizing Bivariate Categorical Data and Lesson 10: Summarizing Bivariate Categorical Data with Relative Frequencies
- Reflecting on CCSSM standards aligned to lessons 9 and 10
- Break
- Activity 3: Grade 9, Lesson 11: Conditional Relative Frequencies and Association
- Reflecting on CCSSM standards aligned to lesson 11
- Developing a poster presentation
- Activity 4: Final preparation for group presentations
- Activity 5: Homework and closing remarks

Activity 1 homework review and discussion

Table discussion

Discuss your write ups for the Day 5 homework tasks:

- Compare your strategies with others at your table
- Reflect on how you might revise your own solution and/or presentation

Learning Intentions and Success Criteria

We are learning to…

- Summarize data on two categorical variables using a two-way frequency table.
- Relate relative frequency tables and two-way frequency tables.
- Evaluate conditional relative frequencies as a possible indication of association.

Learning Intentions and Success Criteria

We will be successful when we can:

- Construct a two-way frequency table from data on two categorical variables collected from a sample.
- Construct and interpret a relative frequency table given a two-way frequency table.
- Calculate and then interpret conditional relative frequencies from two-way frequency tables as a possible indication of association between two categorical variables.
- Explain why association does NOT imply causation.

Activity 2 Lessons 9 -10: Summarizing Bivariate Categorical Data

Constructing a two-way table frequency table from data on two categorical variables and interpreting relative frequencies

EngageNY/Common Core Grade 6, Lesson 20

Activity 2 Lessons 9 -10: Summarizing Bivariate Categorical Data

450 surveys were randomly selected from high school students who completed an online survey about superheroes.

The data collected represent categorical data. Bivariate categorical data results from collecting data from two categorical variables.

Why is work with categorical data challenging?

What can you do with categorical data? What can you not do with categorical data?

How is a statistical study generally performed with categorical data?

Activity 2 Lessons 9 -10: Summarizing Bivariate Categorical Data

Read Example 1 from Lesson 9. Reflect on the breakdown of the data by gender of the 450 surveys randomly selected from the completed surveys:

• 100 students indicated their favorite power was “to fly”. 49 were females.

• 131 students selected the power to “freeze time” as their favorite. 71 were males.

• 75 students selected “invisibility” as their favorite power. 48 were females.

• 26 students indicated “super strength” as their favorite. 25 were males.

• 118 students indicated “telepathy” as their favorite. 70 were females.

Activity 2 Lessons 9 -10: Summarizing Bivariate Categorical Data

What is the most popular superpower?

What is the least popular?

Why do you think the survey includes gender responses?

Do you think gender plays a role in superhero power preference?

Activity 2 Lessons 9 -10: Summarizing Bivariate Categorical Data

In small groups, complete exercises 1 – 12 of Lesson 9.

After you have complete the exercises, discuss as a whole group the exercises. What challenges do you anticipate students in grade 9 would have with the exercises and their work with categorical data?

Activity 2 Lessons 9 -10: Summarizing Bivariate Categorical Data

Also complete the Problem Set for lesson 9. Discuss as a whole group when completed.

Do you think there is a difference in the responses by males and females in the Rufus King data (question 5)? Why or why not?

How can we examine the possible connection between two categorical variables more clearly? (Lesson 10)

Activity 2 Lessons 9 -10: Summarizing Bivariate Categorical Data

Continue to work with the Superhero data by competing exercises 1 to 10 in Lesson 10.

After you have completed your answers to the exercises, discuss as a who group some of the challenges high school students might have with this lesson.

Where is this going? The story is still not complete!

Activity 2 Lessons 9 -10: Summarizing Bivariate Categorical Data

Summary questions:

- Why would several students argue against Scott’s initial suggestion of choosing super strength? Why might Scott have made this suggestion?
- Why do you think Jill would say the school should use telepathy?
- Do you think there is a difference in the superpowers selected by males and those selected by females?

Activity 2Lessons 9 -10: Summarizing Bivariate Categorical Data

Review the following high school content standards for these lessons:

S-ID.5

S-ID.9

Reflecting on the high school CCSSM standards aligned to lessons 9 - 10

Activity 2 Lessons 9 -10: Summarizing Bivariate Categorical Data

S-ID.5: Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible association and trends in the data.

S-ID.9: Distinguish between correlation and causation.

- Where did you see these standards in the lessons you have just completed?
- What would you look for in students’ work to suggest that they have made progress towards these standards?

Activity 2 Lessons 9 -10: Summarizing Bivariate Categorical Data

Read MP.2, second 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 MP4?
- Are there other standards for mathematical practice that were prominent as you and your groups worked on the tasks?

Reflecting on CCSSM Standards for Mathematical Practice aligned to lessons 9 - 10

Activity 2 Lessons 9 -10: Summarizing Bivariate Categorical Data

CCSSM MP.2

engageny MP.2

MP.2 Reason abstractly and quantitatively..

Students pose statistical questions and reason about how to collect and interpret data in order to answer these questions. Students form summaries of data using graphs, two-way tables, and other representations that are appropriate for a given context and the statistical question they are trying to answer. Students reason about whether two variables are associated by considering conditional relative frequencies.

MP.2 Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize – to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents – and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibility using different properties of operations and objects.

Activity3Lesson 11: Conditional Relative frequencies and association

Constructing a two-way table frequency table from data on two categorical variables and interpreting conditional relative frequencies

EngageNY/Common Core Grade 6, Lessons 21-22

Activity 3 Lesson 11: Conditional Relative frequencies and association

Students continue their analysis of the bivariate categorical data started in Lessons 9 and 10. Lesson 9 summarized the data in a two-way frequency table. Relative frequencies were then introduced in Lesson 10. In each case, however, the question posed about whether there is a difference in the favorite superpower responses of males and females remands unclear.

This lesson defines conditional relative frequencies and association that are used to describe the possible connection of bivariate categorical data.

Activity 3 Lesson 11: Conditional Relative frequencies and association

What reasoning could be used to decide if the superpower responses of males and not the same as the superpower responses of females? Examining each row of the table can help determine whether or not there is a connection.

Activity 3 Lesson 11: Conditional Relative frequencies and association

Recall the two-way table from the previous lessons:

Activity 3 Lesson 11: Conditional Relative frequencies and association

Use the row totals to create a conditional relative frequencies. Discuss how this table (conditional relative frequencies) helps us examine a possible condition of the selection of superpowers and gender.

Activity 3 Lesson 11: Conditional Relative frequencies and association

Complete exercises 6 to 10 of Lesson 11. Discuss as a whole group your responses to these exercises. From these exercises comes emerges the following important definition:

Two categorical variables are associated if the row conditional relative frequencies (or column relative frequencies) are different for the rows (or columns) of the table. The evidence of an association is strongest when the conditional relative frequencies are quite different. If the conditional relative frequencies are nearly equal for all categories, then there is probably not an association between variables.

Activity 3 Lesson 11: Conditional Relative frequencies and association

Summary questions for exercises 1 – 10:

How is relative frequency calculated?

How could we determine a relative frequency for only the female students?

How do we interpret the conditional relative frequencies for all students in the table?

Activity 3 Lesson 11: Conditional Relative frequencies and association

Complete Lesson 11 by working in groups and discussing as a whole group exercises 11 – 16.

Discuss the difference between association and causation.

Be sure that students understand through exercises of this type that association does not mean there is a cause and effect relation that is generally described as causation.

Review the Lesson Summary for Lesson 11 (p. S.76)

If time permits, discuss as a whole group the Exit ticket questions for this lesson.

Activity 3Lesson 11: Summarizing Bivariate Categorical Data

Review the following high school content standards for this lesson:

S-ID.5

S-ID.9

Reflecting on the high school CCSSM standards aligned to Lesson 11

Activity 3 Lesson 11: Summarizing Bivariate Categorical Data

S-ID.5: Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible association and trends in the data.

S-ID.9: Distinguish between correlation and causation.

- Where did you see these standards in the lessons you have just completed?
- What would you look for in students’ work to suggest that they have made progress towards these standards?

Activity 3 Lesson 11: Summarizing Bivariate Categorical Data

Read MP.2, second 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 MP4?
- Are there other standards for mathematical practice that were prominent as you and your groups worked on the tasks?

Reflecting on CCSSM Standards for Mathematical Practice aligned to Lesson 11

Activity 3 Lesson 11: Summarizing Bivariate Categorical Data

CCSSM MP.2

engageny MP.2

MP.2 Reason abstractly and quantitatively..

Students pose statistical questions and reason about how to collect and interpret data in order to answer these questions. Students form summaries of data using graphs, two-way tables, and other representations that are appropriate for a given context and the statistical question they are trying to answer. Students reason about whether two variables are associated by considering conditional relative frequencies.

MP.2 Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize – to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents – and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibility using different properties of operations and objects.

Activity 3 Lesson 11: Summarizing Bivariate Categorical Data

Read over the Standards for Statistical Practice. Do you see other practice standards that emerged with your work in Lessons 9 – 11? If yes, identify the standards and summarize how they might be evident in students’ behaviors.

Learning Intentions and Success Criteria

Review of our learning intentions and success criteria.

We are learning to…

- Summarize data on two categorical variables using a two-way frequency table.
- Relate relative frequency tables and two-way frequency tables.
- Evaluate conditional relative frequencies as a possible indication of association.

Learning Intentions and Success Criteria

We will be successful when we can:

- Construct a two-way frequency table from data on two categorical variables collected from a sample.
- Construct and interpret a relative frequency table given a two-way frequency table.
- Calculate and then interpret conditional relative frequencies from two-way frequency tables as a possible indication of association between two categorical variables.
- Explain why association does NOT imply causation.

Activity 4 group presentation and updates

- Update instructors on the progress of your lesson. Instructors will meet with each group and assist with your questions. Provide instructors a general outline of your planning and development of the selected lessons. A tentative schedule of presentations will be considered.

Activity 5 Homework and Closing Remarks

- Consider the survey questions and data set provided as a result of an extensive survey project at Rufus King High School in 1990. Use the template and the data to to investigate if two variables are likely to be associated or not for this sample. Be prepared to use your summary to develop a poster for indicating how a your statistical study involving categorical variables.
- Reflection on teaching:Association is defined by a difference in conditional relative frequencies, yet how large the difference needed to hypothesize a connection is still not clear to students. This becomes more important as we move to the next level of students’ work in statistics (inferential statistics). At this point, how might you help students put into perspective when an association suggests that two categorical categories are connected.

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