Modeling – Equations & Functions
Unit Length and Description:
In this unit, students learn to anticipate the graph of a quadratic function by interpreting various forms of quadratic expressions. Students expand their experience with functions to include more specialized functions—absolute value, step, and those that are piecewise-defined. (Mathematics Appendix A, p.25)
· Construct and compare linear, quadratic, and exponential models and solve problems.
· Build new functions from existing functions.
· Build a function that models a relationship between two quantities.
· Interpret functions that arise in applications in terms of a context.
· Create equations that describe numbers or relationships.
N-Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.
N-Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
A-CED.A.1 Create 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.★
A-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.★
F-IF.B.4 For 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.★
F-IF.B.5 Relate 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-IF.B.6 Calculate 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-BF.A.1 Write a function that describes a relationship between two quantities. ★
F-LE.A.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.★
F-LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).★
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 define descriptive modeling
· I can determine appropriate quantities for the purpose of descriptive modeling
· I can identify appropriate units of measurement to report quantities
· I can determine the limitations of different measurement tools
· I can choose and justify a level of accuracy and/or precision appropriate to limitations on measurement when reporting quantities
· I can identify important quantities in a problem or real-world context
· I can solve linear and exponential equations in one variable
· I can solve inequalities in one variable
· I can describe the relationships between the quantities in the problem (for example, how the quantities are changing or growing with respect to each other); express these relationships using mathematical operations to create an appropriate equation or inequality to solve
· I can create equations (linear and exponential) and inequalities in one variable and use them to solve problems
· I can create equations and inequalities in one variable to model real-world situations
· I can compare and contrast problems that can be solved by different types of equations (linear and exponential)
· I can identify the quantities in a mathematical problem or real-world situation that should be represented by distinct variables and describe what quantities the variables represent
· I can create at least two equations in two or more variables to represent relationships between quantities
· I can Justify which quantities in a mathematical problem or real-world situation are dependent and independent of one another and which operations represent those relationships
· I can determine appropriate units for the labels and scale of a graph depicting the relationship between equations created in two or more variables
· I can graph one or more created equation on a coordinate axes with appropriate labels and scales
· I can define and recognize the key features in tables and graphs of linear and exponential functions: intercepts; intervals where the function is increasing, decreasing, positive, or negative, and end behavior
· I can identify whether the function is linear or exponential, given its table or graph
· I can interpret key features of graphs and tables of function in the terms of the contextual quantities the function represents
· I can sketch graphs showing key features of a function that models a relationship between two quantities from a given verbal description of the relationship
· I can, given the graph or a verbal/written description of a function, identify and describe the domain of the function
· I can identify an appropriate domain based on the unit, quantity , and type of the function it describes
· I can relate the domain of the function to its graph and, where applicable, to the quantitative relationship it describes
· I can explain why a domain is appropriate for a given situation
· I can recognize slope as an average rate of change
· I can calculate the average rate of change of a function (presented symbolically or as a table) over a specified interval
· I can estimate the rate of change from a linear or exponential graph
· I can interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval
· I can define “explicit function” and “recursive process”
· I can write a function that describes a relationship between two quantities by determining an explicit expression, a recursive process, or steps for calculation from a context
· I can recognize that linear functions grow by equal differences over equal intervals
· I can recognize that exponential functions grow by equal factors over equal intervals
· I can distinguish between situations that can be modeled with linear functions and with exponential functions to solve mathematical and real-world problems
· I can prove that linear functions grow by equal differences over equal intervals
· I can prove that exponential functions grow by equal factors over equal intervals
· I can recognize situations in which one quantity changes at a constant rate per unit (equal differences) interval relative to another to solve mathematical and real-world problems
· I can recognize arithmetic sequences can be expressed as linear functions
· I can recognize geometric sequences can be expresses as exponential functions
· I can construct linear functions, including arithmetic sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table)
· I can construct exponential functions, including geometric sequences, given a graph, a description of relationship, or two input-output pairs (include reading these from a table)
· I can determine when a graph, a description of a relationship, or two input-output pairs (include reading these from a table) represents a linear or exponential function in order to solve problems
· Algebraic representations can be used to generalize patterns in mathematics.
· Relationships between quantities can be represented symbolically, numerically, graphically, and verbally in the exploration of real world situations.
· Mathematical rules that reflect recurring patterns facilitate efficiency in problem solving.
· When analyzing linear and exponential functions, different representations may be used based on the situation presented
· There are multiple algorithms for finding a mathematical solution and those algorithms are frequently associated with different contexts.
· Functions can be used to model real-life situations.
· There is an important distinction between solving an equation and solving an applied problem modeled by an equation.
· What characteristics of problems would determine how to model the situation and develop a problem solving strategy?
· How can patterns, relations, and functions be used as tools to best describe and help explain relationships between quantities?
· How can I use patterns to establish relationships that will help make decisions in real-life situations?
· How can a new function be created from an existing function?
· How do parameters introduced in the context of the problem affect the symbolic, numeric and graphical representations of a quadratic function?
· How are patterns of change related to the behavior of functions?