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Understanding the Least-Squares Regression Line with a Visual Model

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Understanding the Least-Squares Regression Line with a Visual Model

Grade Levels

10th Grade, 11th Grade, 12th Grade, 9th Grade

Course, Subject

Related Academic Standards
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  • Big Ideas
    Bivariate data can be modeled with mathematical functions that approximate the data well and help us make predictions based on the data.
    Families of functions exhibit properties and behaviors that can be recognized across representations. Functions can be transformed, combined, and composed to create new functions in mathematical and real world situations.
    Mathematical functions are relationships that assign each member of one set (domain) to a unique member of another set (range), and the relationship is recognizable across representations.
    Mathematical statements can be justified through deductive and inductive reasoning and proof.
    Numbers, measures, expressions, equations, and inequalities can represent mathematical situations and structures in many equivalent forms.
    Objects can be transformed in an infinite number of ways. Transformations can be described and analyzed mathematically.
    Patterns exhibit relationships that can be extended, described, and generalized.
    Relations and functions are mathematical relationships that can be represented and analyzed using words, tables, graphs, and equations.
    Similarity relationships between objects are a form of proportional relationships. Congruence describes a special similarity relationship between objects and is a form of equivalence.
    Some geometric relationships can be described and explored as functional relationships.
    Spatial reasoning and visualization are ways to orient thinking about the physical world.
    There are some mathematical relationships that are always true and these relationships are used as the rules of arithmetic and algebra and are useful for writing equivalent forms of expressions and solving equations and inequalities.
  • Concepts
    2- and 3-dimensional figures
    Algebraic properties and processes
    Algebraic properties, processes and representations
    Analysis of one and two variable (univariate and bivariate) data
    Analytic Geometry
    Exponential functions and equations
    Functions and multiple representations
    Geometric Relations: Congruence and Similarity
    Geometric Representations
    Linear relationships: Equation and inequalities in one and two variables
    Linear system of equations and inequalities
    Polynomial functions and equations
    Quadratic functions and equations
    Trigonometric Ratios
  • Competencies
    Define, describe, and analyze 2- and 3-dimensional figures, their properties and relationships, including how a change in one measurement will affect other measurements of that figure.
    Represent a polynomial function in multiple ways, including tab les , graphs, equations, and contextual situations, and make connections among representations; relate the solution of the associated polynomial equation to each representation.
    Represent a quadratic function in multiple ways, including tab les , graphs, equations, and contextual situations, and make connections among representations; relate the solution of the associated quadratic equation to each representation.
    Represent exponential functions in multiple ways, including tab les , graphs, equations, and contextual situations, and make connections among representations; relate the growth/decay rate of the associated exponential equation to each representation.
    Represent functions (linear and non-linear) in multiple ways, including tables, algebraic rules, graphs, and contextual situations and make connections among these representations. Choose the appropriate functional representation to model a real world situation and solve problems relating to that situation.
    Use algebraic properties and processes in mathematical situations and apply them to solve real world problems.
    Use concepts of congruence and similarity to relate and compare 2- and 3-dimensional figures, including trigonometric ratios.
    Use coordinates and algebraic techniques to interpret, represent, and verify geometric relationships.
    Write, solve, and interpret systems of two linear equations and inequalities using graphing and algebraic techniques.
    Write, solve, graph, and interpret linear equations and inequalities to model relationships between quantities.

Description

This example allows students to explore three methods for measuring how well a linear model fits a set of data points. The Data Analysis and Probability Standard calls for students to explore how residuals (the difference between a predicted and observed value) may be used to measure the "goodness of fit" of a linear model. In this example, two of the methods use residuals and the third uses the shortest distance between a data point and the line given by the model. 

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