Rational and Polynomial Functions

9 Components
Initial Task
Big Ideas
Big Idea 1 Big Idea 2 Big Idea 3 Big Idea 4 Big Idea 5
Formative Assessment Lesson
Representing Polynomials
Re-engagement
End of Unit Assessments
End of Unit Assessment for Unit 4 End of Unit Assessment Teacher Resources ALG II Unit 4 Find aligned Regents questions with Quiz Banker
A2 U4: Rational and Polynomial Functions
Browse Components
Initial Task
Big Ideas
Big Idea 1 Big Idea 2 Big Idea 3 Big Idea 4 Big Idea 5
Formative Assessment Lesson
Representing Polynomials
Re-engagement
End of Unit Assessments
End of Unit Assessment for Unit 4 End of Unit Assessment Teacher Resources ALG II Unit 4 Find aligned Regents questions with Quiz Banker
Big Idea:

Big Idea 4

Mathematical models illustrate the behavior of real-world situations.

1 week
See Standards
Common Core: Major Standards:
A-CED.A.1

Create equations that describe numbers or relationships. 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-REI.A.1

Understand solving equations as a process of reasoning and explain the reasoning. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

A-REI.D.11 Algebra II

Represent and solve equations and inequalities graphically.  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 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.

A-SSE.B.4

Write expressions in equivalent forms to solve problems.  Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.

Common Core: Additional Standards:
A-APR.C.4

Use polynomial identities to solve problems.  Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 - y2)2 + (2xy)2 can be used to generate Pythagorean triples.

A-REI.C.6

Solve Systems of equations. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

A-REI.C.7

Solve Systems of equations. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x2 + y2 = 3.

G-GPE.A.2

Translate between the geometric description and the equation for a conic section. Derive the equation of a parabola given a focus and directrix.

Evidence of Understanding

  • solve polynomial functions in the context of a situation
    • build equations that model the relationship between two variables in a function
    • connect equivalent mathematical representations (an equation to a situation, etc.) and justify the connection
    • use multiple mathematical representations (equation, table, graph) to uncover solutions to a problem or answer questions for a given situation
  • solve systems with two or three variables in the context of a situation
    • algebraically solve systems containing a linear equation and either a polynomial equation or the equation of a circle
      • possible extension: solve a system containing a quadratic and the equation of a circle
    • graphically solve systems containing any two function types with the use of technology
    • solve 3x3 systems algebraically
    • build and combine functions by using the arithmetic operations of addition, subtraction, and multiplication
    • identify values in the solution set and describe solution(s) in the context of a situation  
  • apply geometric ideas to create and solve polynomial equations
    • recognize and apply polynomial identities describing numerical relationships
      • connect Pythagorean Triples to (x2 + y2)2 = (x2 - y2)2 + (2xy)2
    • derive the equation of a parabola, the focus, or the directrix using information
    • derive the formula for the sum of a finite geometric series and use it to solve problems

Develop conceptual understanding:

polynomial identity, Pythagorean Triple, focus, directrix, geometric series

Supporting terms to communicate:

polynomial, linear, quadratic, cubic, zeros, roots, simultaneous solution, vertex, maximum/minimum point
Core Resource

A core resource supports multiple days of instruction.

Instructional Routine: Connecting Representations

These tasks are embedded within the instructional routine called Connecting Representations.