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Course 1, Unit 1 - Patterns of Change

Overview
The intent of this unit, which begins Core-Plus Mathematics Course 1, is to focus student attention on the variety of types of change inherent in problem situations. This unit will provide students with a broad picture of patterns of change. Students will explore linear, quadratic, inverse variation, and exponential patterns of change throughout the unit. Within this unit there is an effort to make a distinction between cause-and-effect change relationships and change-over-time relationships. In the third unit of this course, linear functions will be analyzed as a class of functions with a specific pattern of change. The unit should be completed in under 4 weeks of classes that meet approximately 50 minutes each day.

Key Ideas from Course 1, Unit 1

  • Linear: Linear functions have graphs that are straight lines, rules that can be written in the form y = a + bx, and tables of (x, y) values in which the ratio of change in y to change in x is constant. These ideas are formally developed in Unit 3. (See student book pages 150-167.)

  • Exponential: Exponential functions have curved graphs showing the dependent variable increasing at an increasing rate (for exponential growth) and decreasing at a decreasing rate (for exponential decay) and rules that can be written in the form y = a(b)x, where b is the constant growth or decay factor. In tables of (xy) values for exponential functions, if successive x values differ by 1, then the ratio of corresponding y values is b. Ideas about exponential growth and decay will be developed more formally in Course 1, Unit 5. (See student book pages 289-303, 322-331.)

  • Quadratic: Quadratic functions have graphs that are parabolas, rules that can be written in the form y = ax2 + bx + c, and tables of (xy) values in which y values change in a symmetric pattern centered at a maximum or minimum value. For example, y = x2 - 4.

  • x y
    -3 5
    -2 0
    -1 -3
    0 -4
    1 -3
    2 0
    3 5
  • NOW-NEXT rules (pages 26-33): In many problem situations it is important to study the pattern of change in a single variable that changes with passage of time. Observing values of that variable at regular time intervals, it is natural to look for a pattern relating each value of the variable to the next value. The NOW-NEXT language is an informal way of capturing this perspective on patterns of change. Writing linear and exponential patterns of change in NOW-NEXT form highlights the constant additive and constant multiplicative patterns of change that characterize those two fundamental quantitative relationships. These ideas are developed further in Course 1, Units 3 and 5. (See student book pages 157-161.) Examples:

    x y
    0 2
    1 5
    2 8
    3 11
    4 14
    Linear Relationship

    To get NEXT y, add 3 to the current y-value.
    Two symbolic ways to represent this pattern are
    NEXT = NOW + 3, starting at 2, and = 3x + 2.
    x y
    0 2
    1 6
    2 18
    3 54
    4 162
    Exponential Relationship

    To get NEXT y, multiply the current y-value by 3.
    Two symbolic ways to represent this pattern are
    NEXT = 3NOW, starting at 2, and y = 2(3x).

    Examples of other patterns introduced:

    y = 3/x

    y = x3

    This work with NOW-NEXT patterns of change is also a precursor to work with sequences and series in future units (see Course 3, Unit 7, Recursion and Iteration).

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