Absolute Value

  • Building Understanding of Absolute Value

    HS Algebra

    Description

    Explore the meaning of absolute value using visual representations, including dynagraphs and graphing on the coordinate plane.


    Storyboard

    This ARC aims to create contexts for students to experience a variety of visual representations that reinforce the definition of absolute value. Explorations connecting back to elementary concepts and progressing in complexity provide intentional transitions for building deeper meaning. Representations considered in subsequent activities include the following topics:

    • Number line - as the (positive) distance from zero.
    • Double number lines - as functions in one-dimension.
    • Coordinate Plane - as a graph of a piecewise function.

    Audio

    High-school teachers Tina, Janet, and Deidra reflect on their own learning and describe some 'aha moments' they experienced while working as a team to write this series of lessons.


    Hook

    How do you break a function into pieces?

    Lessons

    Lesson 1 of 3
    Use dynagraphs to visualize and represent absolute value relationships.
    Lesson 2 of 3
    Students grapple with congruence through rigid transformations, then conjecture a "shortcut" set of conditions that also ensure congruence..
    Lesson 3 of 3
    Solving absolute value equations graphically and algebraically.

    ARC Assessement

    Technology Resources


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  • ARC Global Essential Question(s)

    What is the meaning of absolute value?

    Keywords

    Algebra, Functions, Reasoning/Sense Making/Proof, Representation, Functions and Relations, Technology, absolute value, reflections, absolute value equations, absolute value graphs, dynagraph, piecewise functions, system of equations

    Vocabulary

    • absolute value
    • function
    • domain
    • range
    • reflection
    • piecewise function
    • system of equations

    Standards

    CCSS, Content Standards to Domain Level

    • HSA.REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
    • HSA.REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
    • HSF-IF.C.7B Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.
    • HSF.IF.A.1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
    • HSF.BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
    • HSF.IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.
    • HSF.IF.C.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

    CCSS, Standards for Mathematical Practices

    • SMP 5 Use appropriate tools strategically.
    • SMP 7 Look for and make use of structure.
    • SMP 8 Look for and express regularity in repeated reasoning.

    Contributors

    Original Source/Author: n/a
    New Editors: Janet Oien, Deidra Baker, Tina Cardone, Jerel Welker, Brenda Gardunia, Marshall Lassak