Transcript Document

NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
A Story of Ratios
Grade 8 – Module 6
Linear Functions
© 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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
Introduction to the Module
Concept Development
Module Review
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Curriculum Overview of A Story of Ratios
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
G8-M6: Vocabulary and Representations
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
G8-M6: Vocabulary and Representations
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
Introduction to the Module
Concept Development – Topic A
Module Review
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Topic A: Linear Functions
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
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 yintercept.
In Lesson 2, students interpret the rate of change and the yintercept, 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.
© 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.
© 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
© 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
© 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.
© 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.
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
Introduction to the Module
Concept Development – Topic B
Module Review
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
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
© 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.
© 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
© 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.
© 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.
© 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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Exit Ticket for Lesson 8
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Mid- Module Assessment
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
Introduction to the Module
Concept Development – Topic C
Module Review
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
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)
© 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.
© 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.
© 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
© 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
© 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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
Introduction to the Module
Concept Development – Topic D
Module Review
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
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
© 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.
© 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
© 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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
Agenda
Introduction to the Module
Concept Development
Module Review
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
NYS COMMON CORE MATHEMATICS CURRICULUM
End of Module Assessment
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Ratios
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
© 2012 Common Core, Inc. All rights reserved. commoncore.org
NYS COMMON CORE MATHEMATICS CURRICULUM
A Story of Ratios
Final Comments and/or Questions
© 2012 Common Core, Inc. All rights reserved. commoncore.org