Families of Functions

11 Components
Initial Task
Sorting Functions
Big Ideas
Big Idea 1 Big Idea 2 Big Idea 3 Big Idea 4 Big Idea 5
Formative Assessment Lesson
Functions & Situations
Re-engagement
End of Unit Assessments
End of Unit Assessment for Unit 1 Performance Task for Unit 1 End of Unit Assessment Teacher Resources ALG II Unit 1 Find aligned Regents questions with Quiz Banker
A2 U1: Families of Functions
Browse Components
Initial Task
Sorting Functions
Big Ideas
Big Idea 1 Big Idea 2 Big Idea 3 Big Idea 4 Big Idea 5
Formative Assessment Lesson
Functions & Situations
Re-engagement
End of Unit Assessments
End of Unit Assessment for Unit 1 Performance Task for Unit 1 End of Unit Assessment Teacher Resources ALG II Unit 1 Find aligned Regents questions with Quiz Banker
Big Idea:

Big Idea 2

Mathematical models illustrate the behavior of real-world situations.

1 week
See Standards
Common Core: Major Standards:
F-IF.B.4

Interpret functions that arise in applications in terms of the context. 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. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

F-IF.B.6

Interpret functions that arise in applications in terms of the context. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

F-BF.A.1a

Build a function that models a relationship between two quantities. Determine an explicit expression, a recursive process, or steps for calculation from a context.

F-BF.A.1b

Build a function that models a relationship between two quantities.  Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

Common Core: Supporting Standards:
F-LE.A.2

Linear, Quadratic, & Exponential Models: Construct and compare linear, quadratic, and exponential models and solve problems. Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Common Core: Additional Standards:
F-LE.B.5

Interpret expressions for functions in terms of the situation they model. Interpret the parameters in a linear or exponential function in terms of a context.

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.

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