Common Core High School Algebra Teaching and Learning Resources

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

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:

• Polynomials: Definition/Evaluations
• Polynomials: Adding and Subtracting: Horizontal and vertical
• Polynomials: Multiplying: Simple multiplying, FOIL, and general multiplying
• Polynomials: Dividing: Simplification, Reduction, and Long Division methods and worked examples.  There is also a widget in which learners can enter a problem and see the answer.

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.

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:

• Polynomials addresses the definition and associated terms used with polynomials.
• Adding and Subtracting Polynomials
• Multiplying Polynomials
• Polynomials: Long Multiplication involving multiple terms
• Dividing Polynomials
• Polynomials: Long Division involving multiple terms

Khan Academy: HSA-A-APR includes HSA-APR.A.1 Common Core aligned problems.

Also see Algebra 1: Introduction to Polynomial Expressions and Algebra 2: Unit 1 Polynomial Arithmetic: Practice questions with videos.

• Add Polynomials and Subtract Polynomials with one variable: Aligns with APR-A.1.
• Add and subtract polynomials with two variables: Aligns with APR-A.1.
• Multiplying binomials by polynomials: Aligns with APR-A.1.
• Multiply binomials of the form (ax +b): Aligns with APR-A.1.
• Special products of binomials: Aligns with APR-A.1.

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.

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

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

Mathematics Assessment Project: Standards: High School: Algebra: Task 437: Generating Polynomials from Patterns

Mathematics Vision Project, Secondary 3 Student Edition:

• Module 3: Polynomial Functions: This module contains 8 classroom tasks.  Task 4 includes algebra standard APR-A.1.

Interactive Mathematics: Polynomial Equations of Higher Degree delves into polynomial functions and equations, the remainder and factor theorems, how to factor polynomials, roots of a polynomial equation, and roots of polynomial equations using graphs.  Use with APR-B.

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:

• The Remainder Theorem - Example 1
• The Remainder Theorem - Example 2
• Dividing Polynomials and the Remainder Theorem Part 1
• Dividing Polynomials and the Remainder Theorem Part 2
• Dividing Polynomials and the Remainder Theorem Part 3
• The Factor Theorem
• The Factor Theorem and the Remainder Theorem
• Using the Remainder and Factor Theorem

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

• Real Zeroes, Factors, and Graphs of Polynomial Functions
• Polynomial Graphs Part 1: ProfRobBob introduces polynomial functions and give examples of what their graphs may look like.  He then goes over how to determine the End Behavior of these graphs.
• Polynomial Graphs Part 2: ProfRobBob includes finding y-intercepts, x-intercepts and their multiplicity, determining even or odd, and sketching the function.
• End Behavior of Polynomial Functions
• Sketch the graph of polynomials using zeros, end behavior, and y intercept

A. Dendane: Analyze Math:

• Polynomial Functions, Zeroes, Factors, and Intercepts: Tutorial and problems with detailed solutions on finding functions given their zeros and/or graphs and other information.  Aligns with APR-B.3.
• Multiplicity of Zeros and Graphs of Polynomials. A large screen applet helps you explore the effects of multiplicities of zeros on the graphs of polynomials the form f(x) = a(x-z1)(x-z2)(x-z3)(x-z4)(x-z5).  Aligns with APR-B.3.
• Polynomial Functions: An applet to explore polynomial functions up to degree 5, which are presented in standard form.

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.

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:

• Solving Polynomials addresses solving using algebraic methods and by graphing.  A Function Grapher is included.  Roots may be real or complex.
• Fundamental Theorem of Algebra
• Remainder Theorem and Factor Theorem

Khan Academy: HSA-A-APR includes HSA-APR.B.2 and HSA-APR.B.3 Common Core aligned problems.

Also see:

• Polynomial Remainder Theorem
• Zeroes of polynomials (with factoring)

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.

• Topic 7: The Roots, or Zeroes, of a Polynomial
• Topic 12: Synthetic Division by x-a: Includes the Remainder Theorem and Factor Theorem relationship.
• Topic 13: Roots of Polynomials of Degree Greater than 2: Includes the factor theorem and its proof, fundamental theorem of algebra and its proof, a strategy for finding roots, the integer root theorem, conjugate pairs.  Students also sketch a graph from its zeroes.

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:

• Factoring algebraic expressions, including solving quadratics
• Solving polynomial equations: two parts on equations and solving polynomials with methods reviewed

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

• The Missing Coefficient: This is on the Remainder Theorem
• Zeroes and factorization of a quadratic polynomial I
• Zeroes and factorization of a quadratic polynomial II
• Zeroes and factorization of a general polynomial
• Zeroes and factorization of a non polynomial function
• Graphing from factors Align with APR-B.2 and also APR-B.3.

Mathematics Assessment Project: Standards: High School: Algebra: Task 436: Representing Polynomials Graphically

Mathematics Vision Project, Secondary 3 Student Edition:

• Module 3: Polynomial Functions: This module contains 8 classroom tasks.  Algebra standards APR-B.2, APR-B.3 can be found in tasks 5-8.

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 in the UK: Pythagorean Right-Angled 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.

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:

• US Common Core State Standard Math HSA-APR.C.4: 1 manipulative: Square of the sum of two numbers
• US Common Core State Standard Math HSA-APR.C.5: 12 manipulatives.  Among those: Binomial cube, Binomial theorem, binomial distribution, Pascal's Triangle and the Binomial theorem, Binomial theorem step-by-step, Pascal's Triangle, Expansion of a polynomial, and more.

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

• Binomial Expansion Using Pascal's Triangle:  Great way to introduce the Binomial Theorem.
• Ex 1: The Binomial Theorem Using Pascal's Triangle
• Ex 1: The Binomial Theorem Using Combinations

MathsIsFun.com: Tutorial on the Binomial Theorem.  Examples are shown.  Learners then complete 10 multiple choice problems.  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.

Khan Academy: HSA-A-APR includes HSA-APR.C.4 and HSA-APR.C.5 Common Core aligned problems.

Also see:

• Binomial Theorem: Practice questions and videos. 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.

Illustrative Mathematics: Algebra:

• Trina’s Triangles: Aligns with standard APR-C.4.
• Powers of 11: Aligns with standards APR-A.1 and APR-C.5.

Mathematics Vision Project, Secondary 3 Student Edition:

• Module 3: Polynomial Functions: This module contains 8 classroom tasks.  Algebra standard APR-C.5 is found in task 5.

Imagine Learning Classroom (formerly LearnZillion) subscription needed:

• Lesson set: Rewrite simple rational expressions in different forms: This is a series three video lessons that also include guided practice on this topic.  Videos demonstrate rewriting expressions using inspection, division of the numerator by the denominator, and using long division.  Aligns with APR-D.6.
• Lesson set: Add, subtract, multiply, and divide rational expressions: This is a series five video lessons that also include guided practice on this topic.  Aligns with APR-D.7.

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

• Synthetic Division
• Synthetic Division Example 2

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

• Rational Expressions: Finding the domain and Simplifying
• Adding and Subtracting Rational Expressions
• Multiplying Rational Expressions
• Polynomial Division: Simplification and reduction; polynomial long division.  Long division includes examples with no remainders and examples with remainders as expressed in APR-D.6.

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.

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.

Khan Academy: HSA-A-APR includes HSA-APR.D.6 and HSA-APR.D.7 Common Core aligned problems.

Also see:

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

• Dividing polynomials by monomials with remainders
• Dividing polynomials by linear expressions with remainders
• Dividing polynomials by polynomials with remainders
• Simplifying rational expressions 1 by canceling monomial factors
• Simplifying rational expressions 2 by canceling binomial factors
• Simplifying rational expressions 3 using advanced factorization methods
• Adding and subtracting rational expressions with same denominators
• Adding and subtracting rational expressions 2 with unlike denominators
• Adding and subtracting rational expressions 3 with factored denominators
• Adding and subtracting rational expressions 5 by finding the least common multiple
• Multiplying and dividing rational expressions 1 with monomial numerators and denominators
• Multiplying and dividing rational expressions 2 with polynomial numerators and denominators
• Multiplying and dividing rational expressions 3 with advanced rational expressions

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 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:

• Tutorial 9: Multiplying and Dividing Rational Expressions
• Tutorial 10: Adding and Subtracting Rational Expressions

Illustrative Mathematics: Algebra: The following align with standard APR-D.6:

• Combined Fuel Efficiency
• Egyptian Fractions II

Mathematics Assessment Project: Standards: High School: Algebra: Task 221: Interpreting Algebraic Expressions

Mathematics Vision Project, Secondary 3 Student Edition:

• Module 4: Rational Functions: This module contains 7 classroom tasks.  Multiple standards are addressed. Standards APR-D.6, APR-D.7 are among those found in tasks 3-4.