Algebra 2 Unit 3 Polynomials

BY THE END OF THIS UNIT:

CORE CONTENT

Cluster Title: Polynomial Operations
Standard A-APR.1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Concepts and Skills to Master
• Add, subtract, and multiply polynomials
• Synthetic Division
• Remainder Theorem

SUPPORTS FOR TEACHERS

Critical Background Knowledge
·  Rules of exponents
·  Combining like terms
·  Formulas for perimeter, area and volume of basic geometric shapes
Academic Vocabulary
Distributive, Closure, Synthetic Division, Quotient, Divisor, Dividend
Suggested Instructional Strategies
·  Teach various methods of adding/subtracting polynomials, including vertically and horizontally, through real-world and geometric examples and situations
·  Teach various methods of multiplying polynomials, including the “box method” and distributive property (FOIL), through real-world and geometric examples
·  Introduce synthetic division of a polynomial by a binomial (HONORS include when a coefficient is given for the variable in the binomial and long division)
·  Ensure that students understand that the remainder in synthetic division equals the function evaluated at that point (Remainder Theorem) / Resources
Textbook Correlation: 4-4 Beginning of Chapter, 5-4
Word Problem Applications Discovery Education Video
Add and Subtract Polynomials Discovery Education Video
Multiply Polynomials for Volume Discovery Education Video
Synthetic Division Brightstorm Video
Sample Formative Assessment Tasks
Skill-based task
1. (3x3 – 4x2 + 1) – (2x + 4)
2. (2x2 + 3) + 4(x – 2)2 / Problem Task
1) The area of a rectangle can be represented by the expression 2x3 – 17x2 + 31x – 6. If the width is represented by x – 6, what expression represents the length?

CORE CONTENT

Cluster Title: Interpreting Functions
Standard A-APR.5. Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined by Pascal’s Triangle.
Concepts and Skills to Master
• Binomial Expansion

SUPPORTS FOR TEACHERS

Critical Background Knowledge
• Multiplying Polynomials
• Identifying Patterns
Academic Vocabulary
Pascal’s Triangle
Suggested Instructional Strategies
·  Have students fill out Pascal’s Triangle for the first 12 rows. Then, they should multiply (x + y) to the 6th power manually until they discover the pattern. (Start the standard with the argumentation task so they discover the pattern.) / Resources
·  Textbook Correlation: 5 – 7
·  5-7 Puzzle: Pyramid Power (The Binomial Theorem)
Sample Formative Assessment Tasks
Skill-based task
1) Expand (x + y)5.
2) What is the fourth term in (2x – y)6? / Problem Task
Textbook pg. 329 #24

CORE CONTENT

Cluster Title: Interpreting Functions
Standard A-SSE.2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4as (x2)2 –(y2)2, thus recognizing it as a difference of squares.
Concepts and Skills to Master
• Factoring Polynomials by Greatest Common Factor
• Factoring Trinomials with and without leading coefficients
• Factoring a Sum and Difference of Cubes

SUPPORTS FOR TEACHERS

Critical Background Knowledge
• Rules for Exponents
•Multiplying Polynomials
• Basic Factoring Concepts (GCF, Difference of Squares, Trinomials)
Academic Vocabulary
Factoring, Trinomials, Greatest Common Factor, Coefficient
Suggested Instructional Strategies
·  Create factoring foldables/graphic organizers
·  For students needing additional help, use Algebra Tiles
·  Be sure to relate factoring to multiplying polynomials (factoring determines what multiplies to equal the polynomial)
·  Include the sum and difference of cubes
·  For HONORS students include problems like
, / Resources
·  Textbook Correlation: 4-4
·  4-4 Enrichment (Teacher Resources)
·  Factoring Notes Organizer
Sample Formative Assessment Tasks
Skill-based task
1) Factor: 3x2 -48
2) Factor: x3 - 27 / Problem Task
A box with no top is to be made from an 8 inch by 6 inch piece of metal by cutting identical squares from each corner and turning up the sides. The volume of the box is modeled by the polynomial 4x3 – 28x2 + 48x. Factor the polynomial completely. Then use the dimensions given on the box and show that its volume is equivalent to the factorization that you obtain.

CORE CONTENT

Cluster Title: Interpreting Functions
Standard A-APR.3.Identify zeroes of polynomials when suitable factorizations are available, and use the zeroes to construct a rough graph of the function defined by the polynomial.
Concepts and Skills to Master
• Finding zeroes of quadratics graphically
• Factor trinomials and differences of squares to solve quadratics algebraically
• Use the quadratic formula to solve quadratic equations
• Use the roots, end behaviors, and minimum/maximum point to sketch graphs of quadratics

SUPPORTS FOR TEACHERS

Critical Background Knowledge
• Factoring
• End Behavior
• Simplifying Radicals
• Finding the vertex of a parabola
Academic Vocabulary
Zeroes, Roots, Intercepts, Vertex, Quadratic Formula
Suggested Instructional Strategies
·  Teach factoring and quadratic formula first, and then have students discover the relationship between the solutions (especially those from factoring) and the x-intercepts of the graph using the calculator.
·  Have students discuss how to determine the best method for solving various quadratic equations in a Paideia.
·  Discuss the real-world implications of the roots and vertices of quadratics (connect back to Unit 1).
·  For HONORS, extend to include completing the square and introduce parabolas / Resources
·  Textbook Correlation: 4-5, 4-6, 4-7
·  Chapter 4 Performance Tasks 1 and 2: Quadratic Graphs
Sample Formative Assessment Tasks
Skill-based task
Solve the following equations:
1) x2 + 9x = 36
2) 3x2 + 8x + 2 = 0
3) 4x2 – 25 = 0
Graph the following functions by hand or technology, labeling key points:
4) f(x) = 2x2 – 5x – 12
5) f(x) = x2 – 5x – 2 / Problem Task
1) Textbook pg. 245 #39

CORE CONTENT

Cluster Title: Functions
Standard A-APR.4. Prove polynomial identities and use them to describe numerical relationships.
Concepts and Skills to Master
• Rational Root Theorem
• Given the roots of a polynomial, write the equation of the polynomial
• Find all zeroes of a function

SUPPORTS FOR TEACHERS

Critical Background Knowledge
• Factoring
• Synthetic Division
• Using the calculator to find roots (graphically, using a table, and evaluating a function)
Academic Vocabulary
Rational Root Theorem, Root, Zeroes
Suggested Instructional Strategies
·  Focus on technology to find zeroes (HONORS, introduce the Rational Root Theorem and Descartes Rules of Signs).
·  Discuss the rationale for synthetically dividing to create lower degree polynomials. / Resources
·  Textbook Correlation: 5-5
Sample Formative Assessment Tasks
Skill-based task
1) Find the roots of x3 – 3x2 + x – 3 = 0.
2) Write an equation in the least degree of a polynomial with the roots 2, 3, and -4. / Problem Task
Textbook pg. 309 #43

CORE CONTENT

Cluster Title: Functions
Standard F.IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (Specifically c – Graph polynomial functions, identifying zeroes when suitable factorizations are available and showing end behavior.)
Concepts and Skills to Master
• Graph higher degree polynomial functions.
• Identify relative extremes and concavity.

SUPPORTS FOR TEACHERS

Critical Background Knowledge
• Graphs of functions (including intercepts, maximum and minimum values, and end behaviors)
• Transformations of functions
Academic Vocabulary
Relative extremes, concavity
Suggested Instructional Strategies
• Relate everything we’ve done this unit (quadratics, solving by graphing/factoring, end behaviors, etc.) to graphing higher-order polynomials.
• Show how graphs of some polynomials can have minimum and maximum values for different intervals (use graphing technology to set left bounds and right bounds). / Resources
·  Textbook Correlation: 5-1, 5-2
Sample Formative Assessment Tasks
Skill-based task
Find the relative maximum, relative minimum, and zeroes of:
y = 2x3 – 23x2 + 78x – 72 / Problem Task
Textbook pg. 294 #46

Successive pages contain an unpacking of the standards contained in the unit. Standards are listed in alphabetical and numerical order

not suggested teaching order. Teachers must order the standards to form a reasonable unit for instructional purposes.