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Gr. HS Unit: Polynomial, Rational and Radical Relationships

Unit Overview

Algebra II Unit 1develops the structural similarities between the system of polynomials and integers within the system of real numbers. Students draw on analogies between polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property. Students identify zeros of polynomials, including complex zeros of quadratic polynomials, and make connections between zeros of polynomials and solutions of polynomial equations. Rational numbers extend the arithmetic of integers by allowing division by all numbers except 0. Similarly, rational expressions extend the arithmetic of polynomials by allowing division by all polynomials except the zero polynomial. A central theme of this unit is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. Students continue their study of quadratics by connecting the geometric and algebraic definitions of the parabola.

Essential Questions:

  • When and how is mathematics used in solving real world problems?
  • What characteristics of problems would determine how to model the situation and develop a problem solving strategy?
    • What is the role of complex numbers in the equation solving process?
  • When and why is it necessary to follow set rules/procedures/properties when manipulating numeric or algebraic expressions?
    • How do the ordered pairs on the graph of an equation relate to the equation itself and then to a system which contains the given equation?
    • What are some similarities and differences between the algorithms used for performing operations on rational numbers and the algorithms used for performing operations on rational expressions?
    • Why does the equation solving process sometimes produce extraneous solutions?

A question is essential when it stimulates multi-layered inquiry, provokes deep thought and lively discussion, requires students to consider alternatives and justify their reasoning, encourges re-thinking of big ideas, makes meaningful connections with prior learning, and provides students with opportunities to apply problem-solving skills to authentic situations.

Unit Lesson

Additional information such as Teachers Notes, Enduring Understandings,Content Emphasis by Cluster, Focus Standards, Possible Student Outcomes, Essential Skills and Knowledge Statements and Clarifications, and Interdisciplinary Connections can be found in this Lesson Unit.

Available Model Lesson Plans

The lesson plan(s) have been written with specific standards in mind. Each model lesson plan is only a MODEL - one way the lesson could be developed. We have NOT included any references to the timing associated with delivering this model. Each teacher will need to make decisions related ot the timing of the lesson plan based on the learning needs of students in the class. The model lesson plans are designed to generate evidence of student understanding.

This chart indicates one or more lesson plans which have been developed for this unit. Lesson plans are being written and posted on the Curriculum Management System as they are completed. Please check back periodically for additional postings.

CCSC Alignment: N.CN.1, N.CN.2

The student will know that the definition of the imaginary unit is and is used to express the square root of a negative number write complex numbers in form: identify the real and imaginary parts of the number perform operations on complex numbers and simplify algebraic expressions that contain complex numbers.

CCSC Alignment: A.APR.3

The student will use the factors of a given polynomial to identify its zeros and make a rough sketch of the graph of a polynomial using zeros and end behavior. They will also solve polynomial equations.

Available Model Lesson Seeds

The lesson seed(s) have been written with specific standards in mind. These suggested activity/activities are not intended to be prescriptive, exhaustive, or sequential; they simply demonstrate how specific content can be used to help students learn the skills described in the standards. Seeds are designed to give teachers ideas for developing their own activities in order to generate evidence of student understanding.

This chart indicates one or more lesson seeds which have been developed for this unit. Lesson seeds are being written and posted on the Curriculum Management System as they are completed. Please check back periodically for additional postings.

CCSC Alignnment: N.CN.1

This lesson seed requires students associate numbers with the appropriate number sets to which they belong.
Practice: This could be used as a warm up or guided practice. This activity provides an interactive way of reviewing various number sets and how the newly introduced set of Complex Numbers fits in with prior knowledge of number sets.

CCSC Alignnment: F.IF.7, A.APR.3

This activity provides a means for having students to work collaboratively and provides a legitimate reason for movement while providing practice that should help to strengthen student’s proficiency with graphing quadratics and identifying key characteristics.
Practice: This activity is best used as a review or for skill maintenance. Guiding questions will show how this activity can be used to introduce transformation of graphs. This activity is called “Pass the Problem.” Pass the Problem works well as a means of reviewing multiple concepts and can be used to provide distributed practice over time.

CCSC Alignnment: A.REI.11

The activity in this lesson seed requires students to solve systems of equations which are comprised of one linear equation and one quadratic equation. It could easily be adapted to solve systems comprised of one linear and any other type of function covered in this unit or other units. Students also explore systems which have two, one or no solutions.
Practice: This lesson seed describes a cooperative learning activity where students work collaboratively where students will build their proficiency with solving systems of equations graphically.

CCSC Alignnment: A.APR.4

This is a cooperative learning activity that will help build understanding of how is used to describe a numerical relationship.
Motivation: This lesson seed could be used as motivation for a lesson that targets standard A.APR.4

CCSC Alignnment: A.REI.2, A.REI.11

This lesson seed describes an investigation that students could use to help them recognize extraneous solutions that are often produced when solving a radical equation algebraically.
Investigation: This lesson seed describes an investigation that could be used after teaching the algebraic methods for solving radical equations. This activity will provide additional opportunities for students to solve radical equations algebraically and graphically. This activity will also help students to understand the extraneous solutions.

CCSC Alignnment: A.SSE.2, A.SSE.3, A.SSE.3a, A.SSE.3b, A.APR.6, F.IF.1, F.IF.7d

This lesson seed begins with an activity to be completed in groups that is designed to build conceptual understanding about the characteristics of rational functions that lead to vertical asymptotes or holes in the graph. This leads into a discussion of the domain of a rational function and the restrictions caused by these characteristics.
Enrichment: The activities in this lesson seed could be used to extend the study of rational functions. The lesson seed helps students to discover the relationship between the function rule and the graph of a rational function and the restrictions on the domain and how they can be determined from each.

CCSC Alignnment: A.REI.2

This lesson seed provides practice in solving radical equations.
Practice: This lesson seed describes a novel activity. Students solve rational equations and then match problems and solutions on provided puzzle pieces that are then cut out a glued together to form an octahedron.