Big Idea:

Big Idea 2

Mathematical models illustrate the behavior of real-world situations.

1 week

Evidence of Understanding

  • interpret a situation, graph, table of values, or equation using the rate of change
    • calculate and interpret average rate of change in relation to a context
      • explain the meaning of an average rate of change equal to zero, and when appropriate, make connections between the zero change and the graph’s symmetry and critical points  
    • recognize a constant rate of change in a table of values, graph, or situation
    • describe rate of change in relation to critical points or increasing/decreasing intervals
      • Example: when the rate of change is positive the function is increasing, negative when decreasing and approaches zero as it gets closer to a min or max
      • estimate the instantaneous rate of change through a point on the function using the qualities of the tangent line
    • interpret real world behavior using a function's average or instantaneous rate of change
       
  • determine which function families best model given sets of real world data
    • compare linear, quadratic, and exponential functions using the rate of change
      • Example: exponential functions grow by equal factors over equal intervals and the second difference ('rate of change of the rate of change') for quadratic functions is constant
    • use the rate of change to determine which function family fits best
    • relate key quantities in a situation to key characteristics of a function family and their graphs (especially linear, quadratic, or exponential functions)
    • create a graph, table of values, or an equation that best models a situation and use the model to evaluate the situation (find a missing value, make a prediction, etc.)

Develop conceptual understanding:

average rate of change, constant rate of change, instantaneous rate of change, tangent line, exponential function

Supporting terms to communicate:

end behavior, symmetry, zero, root, intercept, critical point, relative or absolute min/max interval, linear, quadratic, logarithmic, square root, cube root, polynomial, rational, periodic, step function
Core Resource
A core resource supports multiple days of instruction.
Instructional Routine: Contemplate then Calculate

These tasks are embedded within the instructional routine called Contemplate then Calculate.

  • Slope of the Tangent Line
    Zoom in on the tangent line in a picture and ignore the graph of the polynomial function around it and revisit strategies for finding the slope of that tangent line.
    Resource:
    Slope of the Tangent Line

    Zoom in on the tangent line in a picture and ignore the graph of the polynomial function around it and revisit strategies for finding the slope of that tangent line.

    All Resources From:
Instructional Routine: Connecting Representations
These tasks are embedded within the instructional routine called Connecting Representations.