Download Algebra 1 Unit 3: Systems of Equations

Algebra 2 Unit 3 Polynomials
Enduring understanding (Big Idea):
Students will understand how to factor polynomials in multiple forms, analyze polynomial functions and their graphs
by identifying end behavior and roots, graph polynomial functions and inverse.
Essential Questions:
1. How are factors and roots of a polynomial equation related?
2. How are a function and its inverse function related?
BY THE END OF THIS UNIT:
Students will know…
• Behavior of Polynomial Functions
• Real-World Applications of Higher-Degree Polynomials
Vocabulary:
• Zeroes
• Binomial Expansion
• Multiplicity
• Relative Extrema
• Concavity
Unit Resources – See attached standard guides
Suggested pacing time: 10 days
Students will be able to…
• Factor Polynomials
• Describe the end behavior of polynomials
• Find the inverse of functions
• Know and apply the Binomial Theorem
• Recognize a polynomial function in real-world situation
Mathematical Practices in Focus:
1.
2.
3.
4.
5.
6.
Make sense of problems and persevere in solving.
Reason abstractly.
Model with Mathematics.
Use appropriate tools strategically.
Look for and make use of structure.
Look for and express regularity in repeated
reasoning.
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.
Algebra 2 Unit 3 Polynomials
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 realworld 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
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.
Algebra 2 Unit 3 Polynomials
Sample Formative Assessment Tasks
Skill-based task
1. (3x3 – 4x2 + 1) – (2x + 4)
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?
2. (2x2 + 3) + 4(x – 2)2
Teacher Created Argumentation Tasks (W1-MP3&6)
Divide 3x3 + 5x2 – 7x – 4 by x + 3.
a) For f(x) = 3x3 + 5x2 – 7x – 4, evaluate f(-3).
b) Explain the relationship between the synthetic division and f(-3)?
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.
Algebra 2 Unit 3 Polynomials
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.)
Sample Formative Assessment Tasks
Skill-based task
Resources
 Textbook Correlation: 5 – 7
 5-7 Puzzle: Pyramid Power (The Binomial Theorem)
1) Expand (x + y)5.
2) What is the fourth term in (2x – y)6?
Textbook pg. 329 #24
Problem Task
Teacher Created Argumentation Tasks (W1-MP3&6)
1) Fill out the first 12 rows of Pascal’s Triangle.
2) Distribute and write out the solutions to: (x + y)0, (x + y)1, (x + y)2, (x + y)3, (x + y)4, (x + y)5, and (x + y)6.
3) Explain the relationship between Pascal’s Triangle and your solutions, and apply this relationship to compute (x + y)10.
4) How do you think the answer would change if you computed (x + 5)10? Justify your answer.
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.
Algebra 2 Unit 3 Polynomials
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
,
x 4  3x 2  4  x 2  4 x 2  1  x  2x  2 x 2  1
x 9 


x 3

x 3
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.
Algebra 2 Unit 3 Polynomials
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.
Teacher Created Argumentation Tasks (W1-MP3&6)
Task #1: 1) Factor x2 – 16 using the difference of squares rule.
2) Factor x2- 0x – 16 by factoring trinomials.
3) What similarities and differences do you see in the two original expressions? Explain why you got the same answer using two different methods.
Task #2: Explain why you can NOT just find two numbers that add to 8 and multiply to -20 to factor 3x2 + 8x – 20.
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.
Algebra 2 Unit 3 Polynomials
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
x  y2
Resources

Textbook Correlation: 4-5, 4-6, 4-7

Chapter 4 Performance Tasks 1 and 2: Quadratic Graphs
parabolas
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.
Algebra 2 Unit 3 Polynomials
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
Teacher Created Argumentation Tasks (W1-MP3&6)
Paideia Seminar on Solving Quadratic Equations
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.
Algebra 2 Unit 3 Polynomials
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.
Sample Formative Assessment Tasks
Skill-based task
1) Find the roots of x – 3x + x – 3 = 0.
2) Write an equation in the least degree of a polynomial with the roots 2, 3,
and -4.
3
2
Resources
 Textbook Correlation: 5-5
Problem Task
Textbook pg. 309 #43
Teacher Created Argumentation Tasks (W1-MP3&6)
Solving Higher-Order Equations Performance Task (use after solving 3rd degree equations)
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.
Algebra 2 Unit 3 Polynomials
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).
Sample Formative Assessment Tasks
Skill-based task
Find the relative maximum, relative minimum, and zeroes of:
y = 2x3 – 23x2 + 78x – 72
Resources
 Textbook Correlation: 5-1, 5-2
Problem Task
Textbook pg. 294 #46
Teacher Created Argumentation Tasks (W1-MP3&6)
Find the zeroes and relative minimum and maximum values for y = (x + 1)4, y = (x + 3)4, and y = (x + 1)4 + 2. What are the similarities and
differences between the values? Explain why this occurs based on what you have learned about transformations.
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.