# Common Core High School Algebra Teaching and Learning Resources

## Domain: APR: Arithmetic with Polynomials and Rational Expressions

Select the cluster for resources on this page:

• APR-A: Perform arithmetic operations on polynomials

• APR-B: Understand the relationship between zeros and factors of polynomials

• APR-C: Use polynomial identities to solve problems

• APR-D: Rewrite rational functions

### APR-A: Perform arithmetic operations on polynomials

Standards:

• APR-A.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.

Technology-enhanced investigations:

Mathwords.com: Math Dictionary: Key vocabulary for this domain.  Use with APR-A, APR-B, APR-C and APR-D:

 Binomial Theorem Fundamental Theorem of Algebra Pythagorean triple Remainder Theorem factor of a polynomial Pascal's Triangle rational expression synthetic division Factor Theorem polynomial rational function zero of a function

McDougal Littell ClassZone, Algebra 2, 2011: Select resources in Ch. 5 Polynomials and Polynomial Functions for interactive practice: Animations (Virtual manipulatives, includes add, subtract, and multiply polynomials), Games/Vocabulary Flashcards, and PowerPoint lesson examples, and interactive problems within the eWorkbook. See multiple choice below for related sections to use within the eWorkbook.  Use with APR-A.1.

National Library of Virtual Manipulatives: Algebra Tiles: Visualize multiplying and factoring algebraic expressions using tiles.  Several activities are provided.  Use with APR-A.1.

Purple Math:

Study.com: Algebra: Exponents and Polynomials: 10 video lessons.  Each includes sample problems and a quiz.  View: How to Add, Subtract and Multiply Polynomials. Use with APR-A.1.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives: US Common Core State Standard Math HSA-APR.A.1: 4 manipulatives: Multiplying a monomial and a linear polynomial, squaring a binomial, multiplying binomials, polynomial long multiplication.

Zweigmedia.com:  Algebra Review Tutorials and examples using fill-in or multiple choice to test understanding.  Learners can also choose to do game versions of some topics.  Use with APR-A.1: Multiplying algebraic expressions.

Multiple Choice:

Khan Academy: see Algebra 1: Introduction to Polynomials: Practice questions with videos.

MathsIsFun.com: Tutorials on polynomials and operations with those.  Examples are shown.  Learners then complete 10 multiple choice problems in each section.  Aligns with APR-A.1:

McDougal Littell ClassZone, Algebra 2, 2011: Ch. 5.3 Quiz: Add, Subtract, and Multiply Polynomials

OpenEd:

Constructed-response:

MATHguide.com: Operations on Polynomials: Lessons on how to add, subtract, and multiply polynomials.  Multiplication with only two binomials could be via FOIL method; however, using a table can be done with any two polynomials--very interesting method.  Each lesson is followed by a quizmaster.  Fill-in and check answers.

OpenEd:

Illustrative Mathematics: Algebra: Powers of 11: Aligns with standards APR-A.1 and APR-C.5.

Mathematics Assessment Project: Standards: High School: Algebra: https://www.map.mathshell.org/stds.php?standardid=1190  Task 437: Manipulating Polynomials

Mathematics Vision Project, Secondary 3 Student Edition:

### APR-B: Understand the relationship between zeros and factors of polynomials

Standards:

• APR-B.2. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
• APR-B.3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

Technology-enhanced investigations:

Ohio Resource Center on YouTube: Tutorials for High School Mathematics: Generating Polynomial Equations for Given Roots.  Using factoring, multiplying binomials, and graphing to build equations.

Purple Math:

• Remainder Theorem: Use with APR-B.2.
• Factor Theorem: The Factor Theorem is the result of the Remainder Theorem.  Use with APR-B.2.
• Polynomial Graphs: End behavior, zeroes & multiplicity, "flexing", turning & "bumps", and graphing. Use with APR-B.3.

YouTube video lessons:  Use the following with APR-B.2:

YouTube video lessons:  Use the following with APR-B.3:

A. Dendane: Analyze Math:

McDougal Littell ClassZone, Algebra 2, 2011: Select resources in Ch. 5 Polynomials and Polynomial Functions for interactive practice: Animations (Virtual manipulatives, includes finding rational zeroes), Games/Vocabulary Flashcards, and PowerPoint lesson examples, and interactive problems within the eWorkbook. See multiple choice below for related sections to use within the eWorkbook.  Use with APR-B.

NCTM Illuminations: Function Matching: In this interactive demonstration, learners see the graph and determine parameters of the function.  The graph might be linear, a polynomial (up to degree 4), exponential, logarithmic, or periodic (sine, cosine, or tangent).  You can choose the type of graph to generate, or display graphs randomly.  When used with polynomials, learners can see the relationship of the zeroes to the graph of the polynomial.  Use with APR-B.3.

Study.com: Algebra:  Exponents and Polynomials: 10 video lessons.  Each includes sample problems and a quiz.  View: How to Graph Cubics, Quartics, Quintics and Beyond. Use with APR-B.3.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives.  Note: Within the Wolfram Demonstration Project are 5 manipulatives addressing APR-B.2 and 15 manipulatives addressing APR-B.3:  Among those are:

• Polynomial Roots: Learners manipulate the values of roots of a polynomial and choose the number of roots to see the affect on the graph of even or odd degree.  Aligns with APR-B.2 and APR-B.3.
• The number of distinct real roots of a real polynomial: Aligns with APR-B.2 and APR-B.3.
• Where are My Roots?: This manipulative shows a plot of the graph of the polynomial (x-a)(x-b)(x-c)(x-d).  Learners can change the parameters a, b, c, d to see the affect on the graph.  Aligns with APR-B.2, APR-B.3 and CN-C.9.
• Quadratic Equation with Factored Form: Learners see a graph of a quadratic.  By moving the sliders or selecting parameters a, b, c in the quadratic equation ax2 + bx + c, new quadratics are plotted together with the zeros and apex. If the roots are real, the factored form is shown.  Aligns with APR-B.3.

Zweigmedia.com:  Algebra Review Tutorials and examples using fill-in or multiple choice to test understanding.  Learners can also choose to do game versions of some topics.  Use with APR-B.3:

Multiple Choice:

A. Dendane: Analyze Math: Graphs of Polynomial Functions: Multiple choice self-test to identify the graph of the given function provided in its standard form.

MathsIsFun.com: Tutorials on polynomials.  Examples are shown.  Learners then complete 10 multiple choice problems in each section.  Aligns with APR-B.2 and APR-B.3:

McDougal Littell ClassZone, Algebra 2, 2011:

OpenEd:

• Zeroes of Polynomials: 5 questions (4 multiple choice, 1 free response).  Includes related review resources.  Aligns with SSE-B.3.

Constructed-response:

L. Spector: The Math Page: Topics in Precalculus.  Students see worked examples of concepts and following the presentation within each section, students are presented problems in constructed response form.  They would solve using paper-pencil and then by using their mouse to roll over a box, they see the solution to problems presented.

OpenEd:

• Remainder theorem: 4 questions that combine free response and multiple choice.  Includes related review resources.  Aligns with APR-B.2.
• Arithmetic with polynomials and rational expressions: 21 questions that combine multiple choice and free response.  Includes related review resources.  Aligns with SSE-A.1, SSE-B.2, SSE-B.3, SSE-C.4, SSE-C.5, SSE-D.6, SSE-D.7.

Illustrative Mathematics: Algebra: Following align with APR-B.2:

Mathematics Assessment Project: Standards: High School: Algebra: https://www.map.mathshell.org/stds.php?standardid=1190 Task 436: Representing Polynomials Graphically

Mathematics Vision Project, Secondary 3 Student Edition:

### APR-C: Use polynomial identities to solve problems

Standards:

• APR-C.4. 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.
• APR-C.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 for example by Pascal’s Triangle.

Technology-enhanced investigations:

Thinking Mathematics: Generating Pythagorean Triples: A YouTube video for generating Pythagorean triples using an ordinary multiplication table.  Use with APR-C.4.

Dr. Ron Knott of the University of Surrey in the UK: Pythagorean Triangles and Triples: Lesson and interactives to learn more about Pythagorean Triples with several online calculators.  Use with APR-C.4.

Art of Problem Solving: Counting and Probability Videos: These are the videos that go with each of the sections within the 14-chapter textbook, Introduction to Counting & Probability.  Ch. 12 is on Pascal's Triangle;  Ch. 14 videos are on the Binomial Theorem.  Use with APR-C.5.

Interactive Mathematics: Binomial Theorem also includes examples.  Use with APR-C.5.

McDougal Littell ClassZone, Algebra 2, 2011: Select resources in Ch. 10 Counting Methods and Probability for interactive practice: Animations (Virtual manipulatives, includes binomial distributions), Games/Vocabulary Flashcards, and PowerPoint lesson examples, and interactive problems within the eWorkbook. See multiple choice below for related sections to use within the eWorkbook.  Use with APR-C.5.

Purple Math: Binomial Theorem: Formulas on the binomial theorem with worked examples.  Pascal's triangle is included.  Use with APR-C.5.

Study.com: Algebra: Probability Mechanics: This chapter of four video lessons includes how to work with factorials and the binomial theorem.  Practice problems are included.  Each video includes a quiz and there is a multiple choice chapter test following the last video.  Use with APR-C.5.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives:

YouTube video lessons:  Use the following with APR-C.5:

Multiple Choice:

MathsIsFun.com: Tutorial on the Binomial Theorem.  Examples are shown.  Learners then complete 10 multiple choice problems.  Use with APR-C.5.

McDougal Littell ClassZone, Algebra 2, 2011: Ch. 10.2 Quiz: Use Combinations and the Binomial Theorem.  Use with APR-C.5.

Study.com: Algebra: Probability Mechanics multiple choice chapter test.  Learners should have completed the chapter Probability Mechanics with its four video lessons on factorials and the binomial theorem before taking the chapter test.  Use with APR-C.5.

Constructed-response:

Ohio Department of Education:

• Lesson Plan: Using the Binomial Theorem: This lesson includes practice problems with their answers.  "Students develop the Binomial Theorem by looking for patterns as they expand binomials with different exponents. Students find coefficients of specific terms within binomial expansions using combinatorics, i.e., choose notation involving factorials. Students apply their understanding of the use of the Binomial Theorem by finding solutions to practical applications."  Aligns with APR-C.5.

L. Spector: The Math Page: Topics in Precalculus:

• The Binomial Theorem.  This lesson helps learners to understand the binomial theorem and use of Pascal's triangle, and includes practice problems with answers provided.  Aligns with APR-C.5.

Math Warehouse: Algebra:

• Binomial Theorem: Lesson and practice problems.  Includes a binomial calculator to expand any expression.  Aligns with APR-C.5.

OpenEd:

Illustrative Mathematics: Algebra:

Mathematics Vision Project, Secondary 3 Student Edition:

### APR-D: Rewrite rational functions

Standards:

• APR-D.6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
• APR-D.7. (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

Technology-enhanced investigations:

Khan Academy video lessons.  Use the following with APR-D.6:

LearnZillion:

McDougal Littell ClassZone, Algebra 2, 2011: Select resources in  Ch. 8 Rational Functions for interactive practice: Animations (Virtual manipulatives), Games/Vocabulary Flashcards, and PowerPoint lesson examples, and interactive problems within the eWorkbook. See multiple choice below for related sections to use within the eWorkbook.  Use with APR-D.

Purple Math:  Use with APR-D.6 and APR-D.7.

Study.com: Algebra: Rational Expressions: This chapter includes six video lessons on rational expressions, each of which includes sample problems and a short quiz.  Videos provide the how-to's for addition, subtraction, multiplication and division of rational expressions, how to solve a rational equation, and practice problems.  There is a multiple choice chapter test following the last video in the chapter.  Use with APR-D.6 and APR-D.7.

Multiple Choice:

McDougal Littell ClassZone, Algebra 2, 2011:

Study.com: Algebra: Rational Expressions multiple choice chapter test.  Learners should have completed the Rational Expressions chapter with its six video lessons prior to taking the chapter test.  Use with APR-D.6 and APR-D.7.

Constructed-response:

Khan Academy: Algebra 1: Introduction to Polynomials: Add, subtract, multiply, and divide polynomial expressions; and Algebra 2: Rational Expressions, Equations, and Functions: Practice questions with videos.  Use the following with APR-D.6 and APR-D.7:

OpenEd:

West Texas A & M University: Virtual Math Lab: Intermediate Algebra: Tutorial 36: Practice Test on Tutorials 32-35.  This practice test includes addition, subtraction, multiplication, and division of rational expressions.  Tutorials 32-35 are available from the practice test page, if needed.  Use with APR-D.6 and APR-D.7.

West Texas A & M University: Virtual Math Lab: College Algebra: Lesson and practice problems.  Use with APR-D.6 and APR-D.7: