Big Idea 3: Rate of change describes how one quantity changes with respect to another.

Common Core Standards: 
F-IF.BInterpret Functions That Arise In Applications In Terms Of The Context ( 4For 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., 6Calculate 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-IF.CAnalyze Functions Using Different Representations ( 9Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. ) F-LE.AConstruct And Compare Linear, Quadratic, And Exponential Models And Solve Problems ( 1Distinguish between situations that can be modeled with linear functions and with exponential functions., 1bRecognize situations in which one quantity changes at a constant rate per unit interval relative to another., 3Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. ) F-LE.BInterpret Expressions For Functions In Terms Of The Situation They Model ( 5Interpret the parameters in a linear or exponential function in terms of a context. ) S-ID.CInterpret Linear Models ( 7Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. )
Evidence of Understanding: 

 

  • use the rate of change to extend a pattern, sequence, table, or graph
    • identify and justify the rate at which the independent and dependent quantities/variables are changing in relation to each other
      • describe rate of change with its attached units (e.g., miles per hour)
      • calculate the rate of change between pairs of values
         
  • interpret a mathematical representation using the rate of change
    • calculate and articulate the usefulness of the average rate of change in relation to context
      • recognize when the average rate of change is zero and describe its meaning
    • convert one or both units for a given relationship, including the use of proportions and unit rate
      • Ex: convert a situation relating distance to time from miles per hour to feet per second
      • understand unit conversion is a unit rate that is proportional and is linear when graphed
    • recognize a constant rate of change in a table of values, graph, or situation
      • explain why a line results when graphing a constant rate of change

 

  • compare linear, quadratic, and exponential functions using the rate of change
    • recognize the second difference (“rate of change of the rate of change”) for quadratic functions is constant
    • recognize exponential functions grow by equal factors over equal intervals
      • rate of change for exponential growth increases over the domain
      • rate of change for exponential decay decreases over the domain
    • match a situation with a graph or table using the function’s rate of change
Instructional Routines: 
  1. Looking at Rate of Change
    Contemplate then Calculate
    Supports understanding rate of change on a graph
  2. Driving Down the Highway
    Contemplate then Calculate
    Supports understanding rate of change from a situation
  3. Quadratic Tables
    Contemplate then Calculate
    Supports understanding rate of change from a table
  4. Sequences and Tables
    Connecting Representations
    Students will chunk a pattern or a table to focus on two items and the rate of change between these items.
  5. Interpreting Rate of Change
    Connecting Representations
    Students will connect visual graphs with descriptions of rate of change.
Additional Resources:
Develop conceptual understanding: 
rate, rate of change, average rate of change, proportion, constant rate of change, line, exponential function, exponential growth, exponential decay
Supporting terms to communicate: 
function, f(x), axes, units, scale, coordinate point, ordered pair, table of values, interval, increasing, decreasing, positive, negative, turning point, maximum, minimum, intercept, dependent, independent, domain, range