Unit Title: Thinking with Mathematical Models
Suggested 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 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

Review linear equations

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*

Linear Equations

Solving Equations

Simplifying Expressions

Functions

Other online resources




Resources

Lab-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 Letters

Spanish 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