Unit Length and Description:
The concept of a function started being developed during 6th grade when students took their first look at the relationship between two quantities and attempted to describe the way the quantities changed in relation to one another. In the 7th grade, students studied direct proportions algebraically and geometrically including identifying the constant of proportionality. That work culminates in this unit with the concept of a function with a strong emphasis on linear functions. Unit 5 repositions previous work with tables and graphs by putting them in a new context of input/output rules. Students should leave this unit with a strong conceptual understanding of functions, the ability to compare functions represented in different forms (algebraically, graphically, numerically in tables, or by verbal descriptions), the understanding that the equation y=mx+b defines a line, the ability to identify functions that are linear and ones that are not, and the ability to construct a function to model a linear relationship between two quantities. Sufficient coverage of nonlinear functions should be addressed to avoid giving students the misleading impression that all functional relationships are linear.
8.F.A.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
8.F.A.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
8.F.A.3 Interpret the equation 𝑦=𝑚𝑥+𝑏 as defining a linear function whose graph is a straight line; give examples of functions that are not linear. For example, the function 𝐴=𝑠2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9) which are not on a straight line.
8.F.B.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models and in terms of its graph or a table of values.
8.F.B.5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
Focus Standards of Mathematical Practice:
MP.1 Make sense of problems and persevere in solving them.
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and critique the reasoning of others.
MP.4 Model with mathematics.
MP.5 Use appropriate tools strategically.
MP.6 Attend to precision.
MP.7 Look for and make use of structure.
MP.8 Look for and express regularity in repeated reasoning.
Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:
· I can determine if an equation represents a function.
· I can apply a function rule for any input that produces exactly one output.
· I can generate a set of ordered pairs from a function and graph the function.
· I can identify functions algebraically including slope and y-intercept.
· I can identify functions using graphs.
· I can identify functions using tables.
· I can identify functions using verbal descriptions.
· I can compare and contrast two functions with different representations.
· I can draw conclusions based on different representations of functions.
· I can recognize that a linear function is graphed as a straight line represented as an equation in the form y = mx + b.
· I can recognize the equation y=mx+b is the equation of a function whose graph is a straight line where m is the slope and b is the y-intercept
· I can provide examples of nonlinear functions using multiple representations (tables, graphs, and equations).
· I can compare the characteristics of linear and nonlinear functions using various representations.
· I can recognize that slope is determined by the constant rate of change.
· I can recognize that the y-intercept is the initial value where x=0.
· I can determine the rate of change (slope) from two (x,y) values, a verbal description, values in a table, or graph.
· I can determine the initial value (y-intercept) from two (x,y) values, a verbal description, values in a table, or graph.
· I can construct a function to model a linear relationship between two quantities.
· I can relate the rate of change and initial value to real world quantities in a linear function in terms of the situation modeled and in terms of its graph or a table of values.
· I can sketch a graph given a verbal description of its qualitative features.
· I can interpret the relationship between x and y values by analyzing a graph.
· I can analyze a graph and describe the functional relationship between two quantities using the qualities of the graph.
· Our world is filled with functions. By learning how to represent, construct, and analyze functions we gain a better understanding of how our world works.
· Verbal descriptions, tables, equations, and graphs can be used to represent linear and nonlinear functions.
· What is a function?
· How can you determine if a relation is a function?
· How can you represent a function?
· How does a change in the independent variable affect the dependent variable?
· What types of relationships can be represented as functions?
· How can you model relationships between quantities?
· How can you use words, tables, equations, and graphs to represent linear and nonlinear functions?
· How do you define, evaluate, and compare functions?