Big Idea 1: The relationship between the variables of a function can be represented visually by a graph.

Common Core Standards: 
N-Q.AReason Quantitatively And Use Units To Solve Problems. ( 1Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays., 2Define appropriate quantities for the purpose of descriptive modeling., 3Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. ) F-IF.AUnderstand The Concept Of A Function And Use Function Notation ( 1Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). ) 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., 5Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. ) 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. )
Evidence of Understanding: 
  • identify variables within a situation and describe how they relate to each other
    • identify dependent and independent variables given a situation/context
      • grapple with several possible relationships and distinguish the difference between a relation and a function
    • analyze quantities and their units in a situation to justify which ones are the most important
      • use units to describe quantities with precision
    • determine appropriate domain and range values of a function based on a given situation
      • state restrictions on a function’s domain and range within a given situation
      • select appropriate units and round as indicated by the context or direction
    • think flexibly about the relationship between quantities by converting among different-sized standard measurement units within a given measurement system (ie. inches to feet)

 

  • recognize the graph of a function as one way to represent all of the solutions between two quantities in a context/situation (NOTE: no equations!)
    • explain how two axes of a graph are related to each other and describe specific coordinate points that define that relationship
    • interpret the graph of a function by identifying its most important features and describing what those features mean within the context of the situation
      • key features include: intervals where the function is increasing or decreasing, positive or negative, intercepts, turning points/maximums and minimums, symmetries
    • analyze patterns within a function’s graph and use it to make predictions about outputs of a function in a situation (including piecewise and step functions)
      • recognize how restrictions on a function’s domain or range can cause a graph to be continuous or discrete
Core Resource: 
  1. Studying Situations
    A multiple-day resource that supports students in grappling with quantities, relationships, functions, and graphs.
Instructional Routines: 
  1. Staircase
    Contemplate then Calculate
    Find the number of squares - supporting looking at patterns.
  2. Counting Squares
    Contemplate then Calculate
    Find the number of squares - supporting looking at patterns.
  3. Counting Circles
    Contemplate then Calculate
    Find the number of circles - supporting looking at patterns.
  4. Discrete versus Continuous
    Connecting Representations
    Students will make connections between examples of linear and exponential verbal situations and graphs using discrete and continuous properties.
Additional Resources:
Develop conceptual understanding: 
dependent, independent, function, relation, units, domain, range, interval, continuous, discrete
Supporting terms to communicate: 
convert, ordered pair, axes, x-axis, y-axis, coordinate point, increasing, decreasing, positive, negative, turning point, maximum, minimum, intercept, piecewise function, step function