Big Idea 1: A function’s rate of change and initial value determine its other properties and behaviors.

Common Core Standards: 
F-IF.AUnderstand The Concept Of A Function And Use Function Notation ( 3Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1. ) 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 ( 7aGraph linear and quadratic functions and show intercepts, maxima, and minima., 7bGraph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. ) 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., 1aProve that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals., 1bRecognize situations in which one quantity changes at a constant rate per unit interval relative to another. ) 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. )
Evidence of Understanding: 
  • analyze a function’s rate of change and initial value to uniquely define it and orient it in the coordinate plane
    • estimate the rate of change from a linear, exponential, piecewise, step, or absolute value graph and use it to describe patterns for different function families
    • determine the intercepts exactly (for integers) or approximately, from a table or graph
    • describe how altering the rate of change and/or initial value impacts the function’s graph and table

 

  • compare linear and exponential function families using rate of change
    • distinguish situations and number sequences (esp. arithmetic and geometric)  that are best modeled by linear, exponential or other functions
    • recognize situations in which one quantity changes at a constant rate per unit interval relative to another
    • analyze examples and nonexamples to define linear functions as growing by equal differences over equal intervals, and exponential functions growing by equal factors over equal intervals

 

  • use a function’s rate of change to predict future states/generate next steps
    • use the rate of change from a situation, graph, table, or number sequence to determine specific output or input values given the other
    • calculate and describe the meaning of the average rate of change in relation to context
      • recognize when the average rate of change is zero and describe its meaning
    • describe the rate of change patterns using words for recursive and explicit forms
    • accurately graph a linear or exponential function and label its intercepts given at least three coordinate pairs and justify reasoning
Core Resource: 
  1. Comparing Rates of Change
    Students start by creating definitions of linear and exponential functions by analyzing examples and non-examples of these types of functions. Students then use what they have learned about linear and exponential functions to solve word problems related to these function types. What is not covered in this resource is representing piecewise, step, or absolute functions as graphs, calculating average rate of change, and writing explicit and recursive descriptions of functions.
Instructional Routines: 
  1. Find the y-intercept #1
    Contemplate then Calculate
    Use the structure of a table to find a missing value in the table.
  2. Find the y-intercept #2
    Contemplate then Calculate
    Use the structure of line on a grid to be able to calculate the coordinates of a missing point (y-intercept).
  3. Find the y-intercept #3
    Contemplate then Calculate
    Use the structure of an exponential function to be able to calculate the coordinates of a point (called the y-intercept).
  4. Analyzing Linear Sequences
    Contemplate then Calculate
    Use the structure of a visual pattern to find the value of the 6th term of a linear sequence.
  5. Comparing Values on a Graph
    Connecting Representations
    Students will make connections between examples of verbal descriptions of the relationship between x and y and their corresponding graphs, using structural elements of linear relationships.
  6. How Many Birds
    Connecting Representations
    Students will chunk descriptions of growth/decay in order to connect these chunks to the same chunks of their associated graphs.
Additional Resources:
Develop conceptual understanding: 
rate of change, slope, initial value, y intercept, x intercept, arithmetic sequence, geometric sequence, linear, exponential, recursive
Supporting terms to communicate: 
function, average rate of change, interval, dependent, independent, input, output, domain, range, integer, continuous, discrete, function family