Thinking With Mathematical Models Teachers Guide
Unit Title: 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).
Mathematics Teaching Practices that Support Mathematics Learning for All Students. Math Content by Unit. Thinking With Mathematical Models.
8.F.4- Construct a function to model a linear relationship between two quantities. Fluke 99 scopemeter manual. 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.
Mathematical Models
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. 225 KB 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 = 2 x + 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.