Big Idea 1: Linear functions are characterized by a constant rate of change.

Common Core Standards: 
A-SSE.AInterpret The Structure Of Expressions ( 1Interpret expressions that represent a quantity in terms of its context., 1aInterpret parts of an expression, such as terms, factors, and coefficients., 1bInterpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P. ) A-CED.ACreate Equations That Describe Numbers Or Relationships ( 1Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions., 3Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. ) A-REI.DRepresent And Solve Equations And Inequalities Graphically ( 11Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. )
Evidence of Understanding: 

 

  • create a linear equation or inequality in 2 variables from a situation, table of values, or graph
    • explain the difference between an expression, an equation, and an inequality
    • understand the difference between variables and fixed quantities in a function rule
    • given f(x) = mx + b, compare how the constant (b) and the lead coefficient (m) relate to the graph, table, or situation
      • Ex: explain how y = 2x + 5 and y = 5x + 2 are different
         
  • write linear equations or inequalities in 1 variable from a situation
    • in the form px + q = r where p, q, and r are rational numbers, describe how r is one specific output value of the general function f(x) = px + q
      • describe the units of a variable or quantity and explain what each term/value in the equation represents
    • represent an equation or inequality in 1 variable using a visual diagram or model
    • recognize equations have a single valid input for a given output and inequalities have several possible inputs for a given output
    • articulate (p, r), (x, r), (q, r) as possible function relationships and understand why (p, q), (x, q), and (p, x) do not make sense as a function relationship (spiral back to Unit 1, Big Idea 1)
    • in the form px + q = r, describe how increasing or decreasing p, q, or r impacts x
Core Resource: 
  1. Generating Equations and Inequalities
    During these three activities, students focus on the relationships between quantities describe in situations and represent those relationships using tables of values, graphs, and function rules. Students then explore what happens to those representations when the parameters embedded within the situations change.
Instructional Routines: 
  1. Which is bigger? #1
    Contemplate then Calculate
    Use the structure of two visuals representing sequences to determine which is greater for a specific term.
  2. Which is bigger? #2
    Contemplate then Calculate
    Use the structure of two tables representing sequences to determine which is greater for a specific term.
  3. Sequences and Functions #1
    Connecting Representations
    Pay attention to the structure of an visual representation to identify the rate of change and initial value to connect it to a function.
  4. Linear Functions
    Connecting Representations
    Use the language structure of situations to connect these to linear functions.
  5. Sequences and Functions #2
    Connecting Representations
    Use the structure of a visual to connect it to a function by chunking the visual by color and by how it is changing/staying the same.
  6. Expressions and Sentences
    Connecting Representations
    Use the structure of a sentence and an expression to make connections between the language and the expression.
Additional Resources:
Develop conceptual understanding: 
expression, equation, inequality, variable, fixed quantity, coefficient, constant, rational, equality, equal, reasonable
Supporting terms to communicate: 
function, function rule, initial value, rate of change, domain, range, dependent, independent, linear equation, greater than, greater than or equal to, less than, less than or equal to, at least, at most