Welcome to Common Core High School Mathematics Leadership

7.1

WELCOME TO COMMON CORE HIGH SCHOOL MATHEMATICS LEADERSHIPSUMMER INSTITUTE 2014

SESSION 7 • 24 JUNE 2014RESIDUALS: 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 dataModel these relationships with functionsEstimate how accurately a given model fits the statistical

data

7.5

LEARNING INTENTIONS AND SUCCESS CRITERIAWe 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 VARIABLESMODELING 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 LINEESTIMATING 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

Review the following CCSSM School content standards:S-ID.6S-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?

ACTIVITY 4REFLECTING ON STANDARDS ALIGNED TO LESSONS 12, 13 & 15

7.16

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.

ACTIVITY 4REFLECTING ON STANDARDS ALIGNED TO LESSONS 12, 13 & 15

7.17

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?

ACTIVITY 4REFLECTING ON STANDARDS ALIGNED TO LESSONS 12, 13 & 15

7.18

ACTIVITY 4REFLECTING ON STANDARDS ALIGNED TO LESSONS 12, 13 & 15

CCSSM MP.5MP.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.

Engageny MP.5MP.5 Use appropriate tools strategically

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

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?

ACTIVITY 4REFLECTING ON STANDARDS ALIGNED TO LESSONS 12, 13 & 15

Closing questions for lessons 12, 13 & 15

7.20

LEARNING INTENTIONS AND SUCCESS CRITERIA

We are learning to… Recognize and describe linear and nonlinear relationships

in bivariate numerical dataModel these relationships with functionsEstimate how accurately a given model fits the statistical

data

7.21

LEARNING INTENTIONS AND SUCCESS CRITERIAWe 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

Today’s group presentations:

Grade 6, Lesson 3Grade 6, Lesson 4

ACTIVITY 5 GROUP PRESENTATIONS

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 feedbackSomething you would like the instructor to think about that might change over a longer period of time

7.26

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?

ACTIVITY 6 HOMEWORK AND CLOSING REMARKS