Enduring understanding (Big Idea): Recognize linear and nonlinear patterns in tables and graphs; describe data patterns using words and symbols; write equations to express patterns appearing in tables, graphs, and problems; solve linear equations; model situations with inequalities; write equations to describe inverse variations; use linear and inverse variation equations to solve problems and to make predictions and decisions.

Essential Questions: What are the key variables in this situation? What is the pattern relating the variables? What kind of equation will express the relationship? How can I use the equation to answer questions about the relationship?

Unit Plans

Common Core Standards Alignment

Connection to 2003 Standards

Investigation 1
Exploring Data Patterns
Problems 1.1,1.2, & 1.3
Math Reflections

Use functions to model relationships between quantities. 8.F.2- Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

8.F.4- Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

8.F.5-Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally

Investigate patterns of association in bivariate data.8.SP.3- Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.

Goal 4.02, 5.01 a-d

Investigation 2
Linear Models and Equations
Problems 2.1, 2.2, 2.3, 2.4
Math Reflections

8.EE.6- Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

Analyze and solve linear equations and pairs of simultaneous linear equations. 8.EE.7- Solve linear equations in one variable.

8.EE.7.b - Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

Use functions to model relationships between quantities. 8.F.3-Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear.

8.F.5-Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. ==Investigate patterns of association in bivariate data. 8.SP.2 - Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.== 8.SP.3 -Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.

Goal 4.01, 4.02, 5.01b

Prior Knowledge: variables and patterns; finding slopes of lines and investigating parallel lines; formulating, reading, and interpreting symbolic rules; solving problems in geometric and algebraic contexts; and modeling situations with linear equations.

Mathematical Practices Standards for Common Core
1-Make sense of problems and persevere in solving them 2-Reason abstractly and quantitatively 3-Construct viable arguments and critique the reasoning of others 4-Model with mathematics 5-Use appropriate tools strategically 6-Attend to precision 7-Look for and make use of structure 8-Look for and express regularity in repeated reasoning

Essential Terms Developed in This Unit

Useful Terms Referenced in This Unit

Terms Developed in Previous Unit

Inverse variation
Linear model
Mathematical

Additive Inverse

Breaking weight

Demand

Direct variation

Multiplicative Inverse

Supply

area profit
average speed proportion
coefficient rate
constant term rate of change
coordinate graph ratio
coordinate pair rise
dependent variable run
fact families scale
independent variables scatter plot
inequality slope
length surface area
linear equation table
linear relationship variables
patterns of change width
point of intersection y-intercept
prism

Thinking With Mathematical Models is a review of the seventh grade book titled Moving Straight Ahead. Students are exposed to situations that can be represented as a linear model. Students perform three experiments and record results in a table. When they graph the results, they discover a pattern that can be expressed as a linear equation. The second focus of this unit is a week-long review of solving linear equations.

Example of a linear function: y = 2x + 7

For an in-depth explanation of unit goals, specific questions to ask your student and examples of core concepts from the unit, go to Thinking With Mathematical Models.

Online resources for Thinking With Mathematical Models*

LessonLab Online Courses Unit Technology Tips Goals of the Unit • Recognize linear and nonlinear patterns from verbal descriptions, tables, and graphs and describe those patterns using words and equations

• Write equations to express linear patterns appearing in tables, graphs, and verbal contexts • Write a linear equation when given specific information, such as two points or a point and the slope• Approximate linear data patterns with graph and equation models• Solve linear equations• Develop an informal understanding of inequalities• Write equations describing inverse variation• Use linear and inverse variation equations to solve problems and to make predictions and decisions

Unit Title: Thinking with Mathematical ModelsSuggested Time: 11 Days (75 to 90 minute Blocks)Enduring understanding (Big Idea): Recognize linear and nonlinear patterns in tables and graphs; describe data patterns using words and symbols; write equations to express patterns appearing in tables, graphs, and problems; solve linear equations; model situations with inequalities; write equations to describe inverse variations; use linear and inverse variation equations to solve problems and to make predictions and decisions.Essential Questions:What are the key variables in this situation? What is the pattern relating the variables? What kind of equation will express the relationship? How can I use the equation to answer questions about the relationship?Unit PlansCommon Core Standards AlignmentConnection to 2003 StandardsInvestigation 1Exploring Data Patterns

Problems 1.1,1.2, & 1.3

Math Reflections

Use functions to model relationships between quantities.8.F.2-Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).8.F.4-Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.8.F.5-Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verballyInvestigate patterns of association in bivariate data.8.SP.3- Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.Investigation 2Linear Models and Equations

Problems 2.1, 2.2, 2.3, 2.4

Math Reflections

8.EE.6- Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equationy=mx+bfor a line intercepting the vertical axis atb.Analyze and solve linear equations and pairs of simultaneous linear equations.8.EE.7- Solve linear equations in one variable.8.EE.7.b- Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.Use functions to model relationships between quantities.8.F.3-Interpret the equationy = mx + bas defining a linear function, whose graph is a straight line; give examples of functions that are not linear.8.F.5-Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.==Investigate patterns of association in bivariate data. 8.SP.2 - Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.==

8.SP.3-Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.Prior Knowledge:variables and patterns; finding slopes of lines and investigating parallel lines; formulating, reading, and interpreting symbolic rules; solving problems in geometric and algebraic contexts; and modeling situations with linear equations.Mathematical Practices Standards for Common Core1-Make sense of problems and persevere in solving them 2-Reason abstractly and quantitatively 3-Construct viable arguments and critique the reasoning of others 4-Model with mathematics 5-Use appropriate tools strategically 6-Attend to precision 7-Look for and make use of structure 8-Look for and express regularity in repeated reasoning

Essential Terms Developed in This UnitUseful Terms Referenced in This UnitTerms Developed in Previous UnitLinear model

Mathematical

Breaking weight

Demand

Direct variation

Multiplicative Inverse

Supply

average speed proportion

coefficient rate

constant term rate of change

coordinate graph ratio

coordinate pair rise

dependent variable run

fact families scale

independent variables scatter plot

inequality slope

length surface area

linear equation table

linear relationship variables

patterns of change width

point of intersection y-intercept

prism

## Thinking With Mathematical Models

Review linear equationsThinking With Mathematical Modelsis a review of the seventh grade book titledMoving Straight Ahead. Students are exposed to situations that can be represented as a linear model. Students perform three experiments and record results in a table. When they graph the results, they discover a pattern that can be expressed as a linear equation. The second focus of this unit is a week-long review of solving linear equations.## Example of a linear function:

y= 2x+ 7## For an in-depth explanation of unit goals, specific questions to ask your student and examples of core concepts from the unit, go to Thinking With Mathematical Models.

Online resources for*Thinking With Mathematical Models## Linear Equations

## Virtual bridge experiment (for Investigation I)

## What is a linear equation?

## Graphing lines using slope/intercept form

## Visualizing slope/intercept form

## Solving Equations

## Intro to solving equations

## Solving equations

## Balancing scales

## Simplifying Expressions

## Combining like terms (addition)

## Combining like terms (subtraction)

## Combining like terms (multiplication)

## Functions

## Function Machine

## Graphing functions

Other online resources## General Homework Tips

## Homework examples from Thinking With Mathematical Models

ResourcesLab-Sheet

Additional Practice/Skills Worksheets

CMP2 Website –online & technology resources

Formal Assessment

- Check-Ups
- Partner Quiz
- Unit Test

Assessment Options- Notebook check
- Multiple-Choice
- Question Bank
- ExamView CD-ROM

Parent Guide-Unit LettersSpanish Assessment Resources

PHSchool.com

TeacherExpress CD-ROM

LessonLab Online Courses Unit Technology Tips

Goals of the Unit• Recognize linear and nonlinear patterns from verbal descriptions, tables, and graphs and describe those patterns using words and equations

• Write equations to express linear patterns appearing in tables, graphs, and verbal contexts

• Write a linear equation when given specific information, such as two points or a point and the slope • Approximate linear data patterns with graph and equation models • Solve linear equations • Develop an informal understanding of inequalities • Write equations describing inverse variation • Use linear and inverse variation equations to solve problems and to make predictions and decisions