7.1
WELCOME TO COMMON CORE HIGH SCHOOL
MATHEMATICS LEADERSHIP
SUMMER INSTITUTE 2014
SESSION 7 • 24 JUNE 2014
RESIDUALS: HOW GOOD IS A MODEL?
7.2
TODAY’S AGENDA
 Homework review and discussion
 Grade 9, Lessons 12 & 13: Relationships Between Two Numerical Variables
 Grade 9, Lesson 15: Interpreting Residuals from a Line
 Reflecting on CCSSM standards aligned to lessons 12, 13 & 15.
 Break
 Group presentations:
 Homework and closing remarks
7.3
ACTIVITY 1
HOMEWORK REVIEW AND DISCUSSION
Table discussion: Day 6 homework tasks
 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.
7.4
LEARNING INTENTIONS AND SUCCESS CRITERIA
We are learning to…
 Recognize and describe linear and nonlinear relationships in
bivariate numerical data
 Model these relationships with functions
 Estimate how accurately a given model fits the statistical data
7.5
LEARNING INTENTIONS AND SUCCESS CRITERIA
We will be successful when we can:
 Recognize linear, quadratic, and exponential patterns in bivariate
numerical data, and describe those patterns using correct
mathematical and statistical language
 Find the least-squares regression line for a data set that appears
to show a linear association
 Use residuals to estimate the strength of an hypothesized linear
association
7.6
ACTIVITY 2
LESSONS 12 & 13: RELATIONSHIPS BETWEEN TWO
NUMERICAL VARIABLES
MODELING ASSOCIATIONS IN NUMERICAL DATA
ENGAGENY/COMMON CORE GRADE 9, LESSONS 12 & 13
7.7
ACTIVITY 2
LESSONS 12 & 13: RELATIONSHIPS BETWEEN TWO NUMERICAL VARIABLES
Prior knowledge
Already in Grade 8, students encountered scatter plots, and modeled data that
showed a linear association by informally fitting a line to the data.
In Grade 9, Module 1, students have encountered linear, quadratic, and
exponential functions and their graphs.
7.8
ACTIVITY 2
LESSONS 12 & 13: RELATIONSHIPS BETWEEN TWO NUMERICAL VARIABLES
(From Grade 8) Three questions to ask when you look at a scatter plot:
 Does it look like there is a relationship between the two variables used to
make the scatter plot?
 If there is a relationship, does it appear to be linear?
 If the relationship appears to be linear, is the relationship a positive linear
relationship or a negative linear relationship?
7.9
ACTIVITY 2
LESSONS 12 & 13: RELATIONSHIPS BETWEEN TWO NUMERICAL VARIABLES
 In pairs, discuss Lesson 12, Exercises 1, 2 & 3.
 Discuss Lesson 13, Example 1 and Exercises 1-6.
 Discuss Lesson 13, Example 2 and Exercises 7-9.
7.10
ACTIVITY 3
LESSON 15: INTERPRETING RESIDUALS FROM A LINE
ESTIMATING THE DEGREE OF ACCURACY OF A LINEAR MODEL
ENGAGENY/COMMON CORE GRADE 9, LESSON 15
7.11
ACTIVITY 3
LESSON 15: INTERPRETING RESIDUALS FROM A LINE
Individually study the table and scatter plot in Example 1
 Informally (as in Grade 8), find a trend line on the scatter plot
 Find (or estimate) an equation for your trend line
How would you compare your trend line to those of other at your table.
Specifically, how could you decide whether your trend line was a better or worse
fit to the data than the others?
7.12
ACTIVITY 3
LESSON 15: INTERPRETING RESIDUALS FROM A LINE
 At your tables, complete Exercises 1-4, using graphing calculators (or other
technology if any is available) to find the least-squares regression line
 What advantages might there be in using the least-squares line?
 What disadvantages might there be in using this line?
7.13
ACTIVITY 3
LESSON 15: INTERPRETING RESIDUALS FROM A LINE
 At your tables, complete Exercises 5 & 6
7.14
ACTIVITY 3
LESSON 15: INTERPRETING RESIDUALS FROM A LINE
 At your tables, study Example 2, and complete Exercises 7 & 8
7.15
ACTIVITY 4
REFLECTING ON STANDARDS ALIGNED TO LESSONS 12, 13 & 15
Review the following CCSSM School content standards:
S-ID.6
S-ID.7
 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?
7.16
ACTIVITY 4
REFLECTING ON STANDARDS ALIGNED TO LESSONS 12, 13 & 15
S-ID.6: Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the
context of the data. Use given functions or choose a function suggested by the
context. Emphasize linear, quadratic, and exponential models.
b. Informally assess the fit of a function by plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear association.
S-ID.7: Interpret the slope (rate of change) and the intercept (constant term) of a linear
model in the context of the data.
7.17
ACTIVITY 4
REFLECTING ON STANDARDS ALIGNED TO LESSONS 12, 13 & 15
Read MP5, the fifth 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 MP5?
 Are there other standards for mathematical practice that were prominent as
you and your groups worked on the tasks?
7.18
ACTIVITY 4
REFLECTING ON STANDARDS ALIGNED TO LESSONS 12, 13 & 15
CCSSM MP.5
Engageny MP.5
MP.5 Use appropriate tools strategically
MP.5 Use appropriate tools strategically
Mathematically proficient students consider the available tools when solving a mathematical
problem. These tools might include pencil and paper, concrete models, a ruler, a protractor,
a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic
geometry software. Proficient students are sufficiently familiar with tools appropriate for their
grade or course to make sound decisions about when each of these tools might be helpful,
recognizing both the insight to be gained and their limitations. For example, mathematically
proficient high school students analyze graphs of functions and solutions generated using a
graphing calculator. They detect possible errors by strategically using estimation and other
mathematical knowledge. When making mathematical models, they know that technology
can enable them to visualize the results of varying assumptions, explore consequences, and
compare predictions with data. Mathematically proficient students at various grade levels
are able to identify relevant external mathematical resources, such as digital content located
on a website, and use them to pose or solve problems. They are able to use technological
tools to explore and deepen their understanding of concepts.
Students visualize data distributions and
relationships between numerical variables using
graphing software. They select and analyze
models that are fit using appropriate technology
to determine whether or not the model is
appropriate. Students use visual
representations of data distributions from
technology to answer statistical questions.
7.19
ACTIVITY 4
REFLECTING ON STANDARDS ALIGNED TO LESSONS 12, 13 & 15
Closing questions for lessons 12, 13 & 15
 How can you distinguish between linear and nonlinear associations in data by
looking at a scatter plot?
 What are the characteristics of linear, quadratic, and exponential relationships
and/or associations in data?
 How can you tell whether a linear model is a good fit to data?
7.20
LEARNING INTENTIONS AND SUCCESS CRITERIA
We are learning to…
 Recognize and describe linear and nonlinear relationships in
bivariate numerical data
 Model these relationships with functions
 Estimate how accurately a given model fits the statistical data
7.21
LEARNING INTENTIONS AND SUCCESS CRITERIA
We will be successful when we can:
 Recognize linear, quadratic, and exponential patterns in bivariate
numerical data, and describe those patterns using correct
mathematical and statistical language
 Find the least-squares regression line for a data set that appears
to show a linear association
 Use residuals to estimate the strength of an hypothesized linear
association
Break
7.23
ACTIVITY 5
GROUP PRESENTATIONS
Today’s group presentations:
Grade 6, Lesson 3
Grade 6, Lesson 4
7.24
ACTIVITY 5
GROUP PRESENTATIONS
Norms for analyzing teaching
 Make notes to remind yourself of important moments in the lesson that you
would like to comment on
 Base your observations on observed and experienced facts rather than
opinions
 Use non-evaluative language
I noticed…
I wondered…
7.25
ACTIVITY 5
GROUP PRESENTATIONS
Giving feedback
 Warm feedback
Something you would like to see again in the teacher’s next lesson
 Cool feedback
Something you would like the instructor to change and that they could change
in the next lesson
 Challenging feedback
Something you would like the instructor to think about that might change over
a longer period of time
7.26
ACTIVITY 6
HOMEWORK AND CLOSING REMARKS
 Complete the problem set for Lesson 15.
 Extending the mathematics:
In Lesson 15, we saw how residuals can be used to estimate the degree to
which a linear model fits a given data set. How could residuals be used to
estimate the fit of a linear model? How might you use residuals to help you
determine which type of model (linear, exponential, quadratic, …) is most
appropriate for a given data set?
 Reflecting on teaching:
Does your current curriculum include as much statistics content in Grade 9 as
the Engageny materials? If not, how might you arrange to align your
curriculum more closely to the CCSSM standards?