Math 3 Unit 3b: Polynomial and Rational Modeling
Approximate Time Frame: 2 – 3 Weeks
Connections to Previous Learning:
In Math 1 and 2, students worked with one-variable equations. They studied linear, exponential and quadratic functions and compared them in a modeling context. Students looked at their multiple representations and how to translate them on a graph or in an equation. Students also looked at the structure of square root, cube root, piecewise, absolute value and step functions and represented these on graphs. Students manipulate expression and equations to reveal new information about these functions and how to solve systems with these types of functions.
Focus of this Unit:
This unit has students now comparing the functions from Math 1, 2 and Polynomial and Rational functions. Students will see connections between these and the equations, functions, tables and graphs they have used in the past courses. Students will specifically compare polynomial and rational functions and polynomial and root functions. Students will use all of these functions to create and compare models of real-life situations.
Visualizing solutions graphically:
Just as the algebraic work with equations can be reduced to a series of algebraic moves unsupported by reasoning, so can the graphical visualizations of solutions. The simple idea that an equation f(x) – g(x) can be solved (approximately) by graphing y = f(x) and y = g (x) and finding the intersection points involves a number of pieces of conceptual understanding. This seemingly simple method, often treated as obvious, involves the rather sophisticated move of reversing the reduction of an equation in two variables to an equation in one variable. Rather, in seeks to convert an equation in once variable, f(x) = g(x), to a system of equations in two variables, y = f(x), and y = g (x), by introducing a second variable y and setting it to equal to each side of the equation. If x is a solution to the original equation the f(x) and g(x) are equal, and thus (x,y) is a solution to the new system. This reasoning is often tremendously compressed and presented as obvious graphically; in fact following it graphically in a specific example can be instruction.
Connections to Subsequent Learning:
Students that continue their study of mathematics will continue to learn about new functions and more complex versions of the functions they have already studied. In college or a career situation, students will frequently be asked to model situations as they arise in real-life. Frequently, students will need to take data and represent it appropriately and make decisions from their representations.
Desired Outcomes
Standard(s):Create equations that describe numbers or relationships
- A.CED.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.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- N.Q.2 Define appropriate quantities for the purpose of descriptive modeling.
- F.IF.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; relate maximums and minimums, symmetries, end behavior, and periodicity.
- F.IF.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.
- A.REI.11 Explain 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 equations 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.
- S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
WIDA Standard: (English Language Learners)
English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.
English language learners benefit from:
- Guided discussions regarding the connection between a context and the graphs, tables, equations and inequalities used to represent it.
- Explicit vocabulary instruction regarding the key features of graphs and tables.
Understandings: Students will understand…
- Rational equations can be created from given information and solved algebraically or graphically.
- Polynomial functions have key features that can be represented on a graph and can be interpreted to provide information to describe relationships of two quantities. These functions can be compared to each other or other functions to model a situation.
- The average rate of change can be estimated, calculated or analyzed from a polynomial or rationalfunction or graph.
- Data can be represented on and interpreted from a scatter plot and modeled with a best-fit function.
- Systems can be solved graphically, algebraically or from a table.
Essential Questions:
- How can you create and solve equations involving rational equations?
- What do the key features of a graph represent in a modeling situation?
- What new information will be revealed if this equation is written in a different but equivalent form?
- How do you create an appropriate function to model data or situations given within context?
- When changes are made to an equation, what changes are made to the graph?
- How do you create or interpret a scatter plot from data and fit a function to this data?
- What are the different methods that can be used to find the solutions of a system of equations?
- How do logarithms relate to exponential models and how can they be used to solve exponential equations?
Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.)
*1. Make sense of problems and persevere in solving them. Students will consider a real-life situation, determine if it is best modeled by a linear, exponential, quadratic, polynomial or rational function, and answer questions and solve problems based on this model.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
*4. Model with mathematics. Students will use graphs, tables, and equations to model polynomial and rational equations. Students will interpret the appropriateness of linear, exponential, quadratic, polynomial or rational models for a given data set or situation.
5. Use appropriate tools strategically.
6. Attend to precision.
*7. Look for and make use of structure. Students will see connections between polynomial and root functions and between polynomial and rational functions and use this structure to write equations and create graphs.
8. Look for and express regularity in repeated reasoning.
Prerequisite Skills/Concepts:
Students should already be able to:
- Represent and solve one variable equations and know the steps for solving equations and looking at equivalent forms limited to linear, exponential and quadratic equations.
- Choose and interpret appropriate models limited to linear, exponential and quadratic functions.
- Define a function and the different parts in expressions, equations and functions.
- Recognize how functions can be represented on a graph and in a table and how scale and labels can change the look of a representation.
- Interpret these representations with regards to linear, exponential and quadratic functions.
Some students may be ready to:
- Use advanced skills of the complex numbers before, could find the conjugate of a complex number and use conjugates to find moduli and quotients of complex number when working with rational equations and functions. (N.CN.3+)
- Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication and division by a nonzero rational expression. Students can also add, subtract, multiply and divide rational expressions. (A.APR.7+)
- Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. (F.IF.7d+)
- Verify that one function is the inverse of another. Read values of an inverse function from a graph or a table, given that the function has an inverse. Produce an invertible function from a non-invertible function by restricting the domain. (F.BF.4b,c,d+)
- Fit a function to the data; use polynomial functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context.
Knowledge: Students will know…
All standards in this unit go beyond the knowledge level. / Skills: Students will be able to …
- Write a polynomial equation and/or function to model a real-life situation.
- Use a model of a polynomial function to interpret information about a real-life situation.
- Define appropriate quantities for the purpose of descriptive modeling.
- Translate among representations of polynomial functions including tables, graphs, equations and real-life situations.
- Distinguish between linear, exponential, quadratic, polynomial and rational functions from multiple representations.
- Rewrite polynomial and rationalequations to reveal new information.
- Estimate, calculate and interpret average rate of change over a specified interval.
- Compare two functions represented in different ways.
- Create a scatterplot from data and interpret the relationship of the quantities represented.
- Appropriately fit a model to data.
Academic Vocabulary:
Critical Terms:
Rational function / Supplemental Terms:
Tables
Graphs
Real-life situations
Equations
2/22/2013 8:53:59 AM Adapted from UbD frameworkPage 1