1. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Grade 8 – Module 6 Linear Functions

2. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Session Objectives 1.Discuss the key ideas of what students learn in the module Linear Functions. 2.Examine sample lessons to become familiar with the main objectives of the module

3. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Agenda

4. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Curriculum Overview of A Story of Ratios

5. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Module topics Topic A: Linear Functions 5 Lessons 8.F.B.4 and 8.F.B.5 Topic B: Bivariate Numerical Data 4 lessons 8.SP.A.1 and 8.SP.A.2 Mid-Module Assessment

6. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Module topics Topic C: Linear and Nonlinear Models 3 Lessons 8.SP.A.1 and 8.SP.A.2 and 8.SP.A.3 Topic D: Bivariate Categorical Data 2 lessons 8.SP.A.4 End of -Module Assessment

7. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Module’s Foundation Connection between the Function and Statistics and Probability Domains Focus Standards: Functions Domain 8.F.B.4 Rate of Change 8.F.B 5 Describe functional relationship Statistics and Probability 8.SP.A.1 Construct scatter plots 8.SP.A..2 Informally fit a line to data 8.SP.A.3 Equation of a line 8.SP.A.4 Patterns of Association

8. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios G8-M6: Vocabulary and Representations

9. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios G8-M6: Vocabulary and Representations

10. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Agenda

11. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Topic A: Linear Functions

12. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Topic A: Linear Functions In Lesson 1, students are given a verbal description of a linear relationship between two variables, and then must describe a linear model. Students graph linear functions using a table of values and by plotting points. They recognize a linear function given in terms of the slope and initial value or y- intercept. In Lesson 2, students interpret the rate of change and the y- intercept, or initial value, in the context of the problem. They interpret the sign of the rate of change as indicating that a linear function is increasing or decreasing and as indicating the steepness of a line.

13. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Topic A: Linear Functions In Lesson 3, students graph the line of a given linear function. They express the equation of a linear function as y=mx+b or an equivalent form when given the initial value and slope. In Lessons 4 and 5, students describe and interpret a linear function given two points or its graph.

14. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Key Idea from Lessons 1, 2 and 3 Lesson 1 Modeling Linear Relationships Students determine a linear function given a verbal description of a linear relationship between two quantities. Rate of change and initial value are emphasized Lesson 2 Interpreting Rate of Change and Initial Value Students interpret the rate of change and initial value of a line in context. Given the equation of a line – interpret the slope in context

15. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Lesson 3: Representations of a Line Outcomes: Students graph a line specified by a linear function. Students graph a line specified by an initial value and rate of change of a function and construct the linear function by interpreting the graph. Students graph a line specified by two points of a linear relationship and provide the linear function. Lesson 3

16. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Lesson 4: Increasing and Decreasing Functions Student Outcomes Students describe qualitatively the functional relationship between two types of quantities by analyzing a graph. Students sketch a graph that exhibits the qualitative features of a function based on a verbal description.

17. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Lesson 4 and 5: Increasing and Decreasing Functions Organize in small groups. Read through Lesson 4 of this module starting on page 19 of the student edition. Answer questions 2 to 4. After you have completed the problems, we will discuss the solutions and student expectations.

18. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Agenda

19. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Topic B: Bivariate Numerical Data Lesson 6 Scatter Plots Lesson 7 Patterns in Scatter Plots Lesson 8 Informally Fitting a Line Lesson 9 Determining the Equation of a Line Fit to Data

20. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Topic B: Bivariate Numerical Data In Lesson 6, students construct scatter plots and focus on identifying linear versus non-linear patterns. In Lesson 7 students distinguish positive linear association and negative linear association based on the scatter plot. Students describe trends in the scatter plot, along with clusters, and outliers (points that do not fit the pattern). In Lesson 8, students informally fit a straight line to data displayed in a scatter plot by judging the closeness of the data points to the line.

21. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Topic B: Bivariate Numerical Data In Lesson 9, students interpret and determine the equation of the line they fit to the data and use the equation to make predictions and to evaluate possible association of the variables. Based on these predictions, students address the need for a “best-fit” line, which is formally introduced in Grade 9

22. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Big Ideas in Lessons 6 and 7 Lesson 6 Scatter plots Students construct scatter plots and investigate relationships. Lesson7 Patterns in Scatter plots Students describe positive and negative trends in a scatter plot.

23. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Lesson 8 Student Outcomes Students informally fit a straight line to data displayed in a scatter plot. Students make predictions based on the graph of a line that has been fit to data.

24. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Determining the Equation of a Line Fit to Data Organize in small groups. Discuss and complete problems 1 to 6 from Lesson 8 of this module. (S60) After you have completed the problems, we will discuss the idea of development of informally fitting a line. Lesson 9

25. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Exit Ticket for Lesson 8

26. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Mid- Module Assessment

27. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Agenda

28. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Topic C: Linear and Nonlinear Models Lesson 10: Linear Models Lesson 11: Using Linear Models in a Data Context Lesson 12: Nonlinear Models in a Data Context (Optional lesson)

29. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Topic C: Linear and Nonlinear Models In Lesson 10, students identify applications in which a linear function models the relationship between two numerical variables. In Lesson 11, students use a linear model to answer questions about the relationship between two numerical variables by interpreting the context of a data set. Students use graphs and the patterns of linear association to answer questions about the relationship of the data. In Lesson 12, students also examine patterns and graphs that describe nonlinear associations of data.

30. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Student Outcomes in Lessons 10 Student Outcomes Students identify situations where it is reasonable to use a linear function to model the relationship between two numerical variables. Students interpret slope and the initial value in a data context.

31. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Lesson 10: Linear Models Organize in small groups. Discuss and complete problems 1 and 2 (S81)and problem 11 (S86) from Lesson 10 of this module. After you have completed the problems, we will discuss the answers Lesson 10

32. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Lesson 11: Using Linear Models in a Data Context Student Outcomes Students recognize and justify that a linear model can be used to fit data. Students interpret the slope of a linear model to answer questions or to solve a problem. Lesson 11

33. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Determining the Equation of a Line Fit to Data Organize in small groups. Discuss and complete exercise 2 from Lesson 11 (S91)of this module. After you have completed the problems, we will go through the solutions. Lesson 11

34. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Agenda

35. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Topic D: Bivariate Categorical Data Lesson 13: Summarizing Bivariate Categorical Data in a Two-Way Table Lesson 14: Association between Categorical Variables

36. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Topic D: Bivariate Categorical Data In Lesson 13, students organize bivariate categorical data into a two-way table. They calculate row and column relative frequencies and interpret them in the context of a problem. In Lesson 14, students informally decide if there is an association between two categorical variables by examining the differences of row or column relative frequencies. They interpret association between two categorical variables as meaning that knowing the value of one of the variables provides information about the likelihood of the different possible values of the other variable.

37. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Lesson 13: Summarizing Bivariate Categorical Data in a Two-Way Table Student Outcomes Students organize bivariate categorical data into a two-way table. Students calculate row and column relative frequencies and interpret them in context. Lesson 13

38. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Summarizing Bivariate Categorical Data in a Two-Way Table Organize in small groups. Discuss and complete exercises 11 to 22 from Lesson 13 of this module. (S106) After you have completed the problems, we will discuss the set up of the two-way table and the vocabulary. Lesson 11

39. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Agenda

40. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios End of Module Assessment

41. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Module Summary: Key Ideas Grade 8 Module 6 is a key. More use of functions and preparation for Grade 9 Algebra and Functions Conceptual Categories. Big theme is ASSOCIATION Key ideas: Patterns in Scatter plots (Association between two numerical variables) Equation of a line Fitting a line to data Analyzing categorical data from two-way frequency tables. Association between two categorical variables

42. © 2012 Common Core, Inc. All rights reserved. commoncore.org NYS COMMON CORE MATHEMATICS CURRICULUM A Story of Ratios Final Comments and/or Questions