Transcript 6.1 WELCOME TO COMMON CORE HIGH SCHOOL MATHEMATICS LEADERSHIP SUMMER INSTITUTE 2014 SESSION 6 • 23 JUNE 2014 TWO-WAY TABLES AND ASSOCIATION.

6.1
WELCOME TO COMMON CORE HIGH SCHOOL
MATHEMATICS LEADERSHIP
SUMMER INSTITUTE 2014
SESSION 6 • 23 JUNE 2014
TWO-WAY TABLES AND
ASSOCIATION
6.2
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
6.3
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
6.4
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.
6.5
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.
6.6
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
6.7
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?
6.8
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.
6.9
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?
6.10
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?
6.11
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)
6.12
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!
6.13
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?
6.14
ACTIVITY 2
LESSONS 9 -10: SUMMARIZING BIVARIATE CATEGORICAL DATA
Reflecting on the high school CCSSM standards aligned to lessons 9 - 10
Review the following high school content standards for these lessons:
S-ID.5
S-ID.9
6.15
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?
6.16
ACTIVITY 2
LESSONS 9 -10: SUMMARIZING BIVARIATE CATEGORICAL DATA
Reflecting on CCSSM Standards for Mathematical Practice aligned to
lessons 9 - 10
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?
6.17
ACTIVITY 2
LESSONS 9 -10: SUMMARIZING BIVARIATE CATEGORICAL DATA
CCSSM MP.2
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.
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.
Break
6.19
ACTIVITY 3
LESSON 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
6.20
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.
6.21
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.
6.22
ACTIVITY 3
LESSON 11: CONDITIONAL RELATIVE FREQUENCIES AND ASSOCIATION
Recall the two-way table from the previous lessons:
To Fly
Freeze
time
Invisibility
Super
Strength
Telepathy
Total
Females
49
60
48
1
70
228
Males
51
71
28
25
48
222
Total
100
131
75
26
118
450
6.23
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.
Females
49/228
60/228
48/228
1/228
70/228
228/228
Males
51/222
71/222
27/222
25/222
48/222
222/222
Total
100/450
131/450
75/450
26/450
118/450
450/450
6.24
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.
6.25
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?
6.26
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.
6.27
ACTIVITY 3
LESSON 11: SUMMARIZING BIVARIATE CATEGORICAL DATA
Reflecting on the high school CCSSM standards aligned to Lesson 11
Review the following high school content standards for this lesson:
S-ID.5
S-ID.9
6.28
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?
6.29
ACTIVITY 3
LESSON 11: SUMMARIZING BIVARIATE CATEGORICAL DATA
Reflecting on CCSSM Standards for Mathematical Practice aligned to
Lesson 11
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?
6.30
ACTIVITY 3
LESSON 11: SUMMARIZING BIVARIATE CATEGORICAL DATA
CCSSM MP.2
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.
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.
6.31
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.
6.32
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.
6.33
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.
6.34
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.
6.35
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.