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Unit 2 - Patterns of Change

Summary
In the "Patterns of Change" unit, students extend their understanding and skill in algebra in three ways. They learn how to recognize relationships between independent and dependent variables in problems and experiments, and describe patterns in quantitative variables that change over time. They learn how to read and construct data tables and graphs that display relationships between variables. They begin developing symbol sense - the ability to express and reason about linear, exponential, and quadratic functions using symbolic formulas and equations. All of this is an essential precursor to more formal work with algebraic symbols.

Key Ideas from Course 1, Unit 2

  • 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.
  • 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 (x, y) 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 6.
  • Quadratic: Quadratic functions have graphs that are parabolas, rules that can be written in the form y = ax2 + bx + c, and tables of (x, y) 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 equations: 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 6. 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 y = 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
    y = x0.5

    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).

 

 

 

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