FANTASTIC FACTORING!!! Greatest Common Factor Difference of Squares Perfect Square Trinomial Leading Coefficient of One Leading Coefficient Not One All Five Types Appendices Teacher’s Guide Don’t let factoring blow you away! This presentation is to be used for educational purposes only. EXIT

TEACHER’S GUIDE Introduction Aim Rationale Learners Prerequisites Subject Matter Goals and Objectives Instructional Plan Materials Assessment

Introduction  Factoring…students often find it hard to learn and teachers often find it hard to teach! Students wonder how they can factor polynomials when each one seems to be a different style than the last. Teachers wonder how they can find enough time to teach it thoroughly when they have so many other topics to cover.

Aim  The purpose of this curriculum web is to provide a resource for students and teachers where students can learn and practice factoring the most common types of binomials and trinomials.

Rationale Factoring is often one of the mathematical skills that students find the most difficult to learn, along with graphing and proofs. Factoring can also be a source of frustration for teachers who think students should be learning the concepts faster or who find themselves re-teaching factoring in courses they feel they should not have to spend time doing. Factoring is often one of the mathematical skills that students find the most difficult to learn, along with graphing and proofs. Factoring can also be a source of frustration for teachers who think students should be learning the concepts faster or who find themselves re-teaching factoring in courses they feel they should not have to spend time doing.

Rationale Continued… Sometimes students wonder why they have to learn factoring. There are at least four reasons learning to factor polynomials is important. Sometimes students wonder why they have to learn factoring. There are at least four reasons learning to factor polynomials is important. First, factoring is a skill that is needed to simplify rational expressions, locate holes in graphs, and decompose fractions. It can be used to solve some quadratic equations. First, factoring is a skill that is needed to simplify rational expressions, locate holes in graphs, and decompose fractions. It can be used to solve some quadratic equations.

Rationale Continued… Second, factoring is found throughout the high school math curriculum. Factoring occurs in Geometry and Precalculus but factoring is found predominantly in Algebra I and Algebra II. Second, factoring is found throughout the high school math curriculum. Factoring occurs in Geometry and Precalculus but factoring is found predominantly in Algebra I and Algebra II. Third, as students practice these factoring skills, they will also practice the very important algebraic skill of multiplying polynomials which involves other fundamental skills such as working with exponents, combining like terms, and multiplying positive and negative numbers. Third, as students practice these factoring skills, they will also practice the very important algebraic skill of multiplying polynomials which involves other fundamental skills such as working with exponents, combining like terms, and multiplying positive and negative numbers.

Rationale Continued… Fourth, factoring and skills used in factoring are among the academic standards for state testing. For example, the Ohio Academic Content Standards for Mathematics lists three educational objectives covered by factoring: “Solve quadratic equations…by factoring,” “Add, subtract, multiply and divide monomials and polynomials…” and “Simplify rational expressions by eliminating common factors…” Fourth, factoring and skills used in factoring are among the academic standards for state testing. For example, the Ohio Academic Content Standards for Mathematics lists three educational objectives covered by factoring: “Solve quadratic equations…by factoring,” “Add, subtract, multiply and divide monomials and polynomials…” and “Simplify rational expressions by eliminating common factors…”

Rationale Continued… This curriculum web can be a resource for teachers to provide for their students when they do not have the personal time to review how to factor the five different types of polynomials presented. It can also be a resource parents can provide their children who are struggling with factoring. It can even be a resource for parents themselves to learn factoring and explain it to their children. This curriculum web can be a resource for teachers to provide for their students when they do not have the personal time to review how to factor the five different types of polynomials presented. It can also be a resource parents can provide their children who are struggling with factoring. It can even be a resource for parents themselves to learn factoring and explain it to their children.

Rationale Continued… This curriculum web can enable students to realize that most factoring problems are not unique but fall into general categories. Students can realize that factoring polynomials does employ algorithms (step-by-step procedures to solve them). Lastly, students can overcome their fear of failure, start to succeed at factoring, and gain confidence in approaching factoring problems. This curriculum web can enable students to realize that most factoring problems are not unique but fall into general categories. Students can realize that factoring polynomials does employ algorithms (step-by-step procedures to solve them). Lastly, students can overcome their fear of failure, start to succeed at factoring, and gain confidence in approaching factoring problems.

Learners  This curriculum web is designed to be used by high school students who have previously been taught how to factor binomials and trinomials. Although it could be used to initially learn how to factor, it is designed to provide explanations and practice for those who have been exposed to factoring.

Prerequisites  Currently taking Algebra I or higher  Previous exposure to factoring binomials and trinomials  Interest in learning more details about factoring and practicing different problem styles

Subject Matter  Algebra I and II  Factoring binomials and trinomials  Multiplying polynomials to check factoring

Goals and Objectives  Students will learn to factor five different styles of polynomials:  Greatest Common Factor  Difference of Squares  Perfect Square Trinomial  Trinomials with a leading coefficient of one  Trinomials with a leading coefficient of an integer other than one

Instructional Plan This curriculum web is designed to be used by students without the help of a teacher. Students can access the curriculum web and follow the instructions. Students will find: This curriculum web is designed to be used by students without the help of a teacher. Students can access the curriculum web and follow the instructions. Students will find: Detailed explanations of problems Detailed explanations of problems Practice quizzes on each factoring style Practice quizzes on each factoring style Helpful resources for further exploration Helpful resources for further exploration

Materials  An Internet-linked computer and web browser  Paper and pencil to perform calculations and check work before submitting answers  Hand-held calculator is optional and online calculator links are provided

Assessment  Learning that occurs through this curriculum web can be assessed by the online quizzes that can be taken and the results ed to the teacher, parent, or student.  Students can do a self-assessment by using a rubric to examine their improving skill level.  EVALUATION: The curriculum web itself can be evaluated by using a link to a feedback form in the Appendices.

Appendices Resources Resources Ask Dr. Math Ask Dr. MathAsk Dr. MathAsk Dr. Math Does the CAREER I want use MATH?! Does the CAREER I want use MATH?!Does the CAREER I want use MATH?!Does the CAREER I want use MATH?! More Factoring Practice More Factoring Practice Leading Coefficient of One Leading Coefficient of OneLeading Coefficient of OneLeading Coefficient of One Leading Coefficient Not One Leading Coefficient Not One Leading Coefficient Not One Leading Coefficient Not One Online Calculator Online Calculator Online Calculator Online Calculator

Appendices Glossary (in your language!) Glossary (in your language!) Binomial – two terms Binomial – two terms Coefficient – number in front of the letter Coefficient – number in front of the letter Degree - highest exponent Degree - highest exponent Factoring – factors that can be multiplied to get the original problem Factoring – factors that can be multiplied to get the original problem Greatest Common Factor – what all the terms have in common Greatest Common Factor – what all the terms have in common Perfect Square – has the same two factors Perfect Square – has the same two factors Trinomial – three terms Trinomial – three terms

Appendices Have you ever used a product and had an idea you KNOW would make it better? Well here’s your chance to make your voice heard! (Fill out this short and simple four-question evaluation.) EVALUATION

Fantastic Factoring Greatest Common Factor Explanation Online Practice Quizzes Difference Of Squares Explanation Online Practice Quizzes Perfect Square Trinomial Explanation Online Practice Quizzes Leading Coefficient of One Explanation Online Practice Quizzes Leading Coefficient Not One Explanation Online Practice Quizzes All Five Types Online Practice Website Appendices Resources Glossary Evaluation Teacher’s Guide

Teacher’s Guide Introduction Aim Rationale Learners Prerequisites Subject Matter Goals and Objectives Instructional Plan Materials Assessment

Appendices If you would like to see if your factoring skills are improving you can click on the link below to rank yourself on a scale from 0 to 5. SELF-ASSESSMENT

GREATEST COMMON FACTOR Detailed Example Practice Quizzes Easier than facing a 300 lb GORILLA!

DIFFERENCE OF SQUARES Detailed Example Practice Quizzes Go for the GOLD!!!

PERFECT SQUARE TRINOMIAL Detailed Example Practice Quizzes I love trinomials!!!

LEADING COEFFICIENT OF ONE Detailed Example Practice Quizzes Your future’s so bright you have to wear shades!!!

LEADING COEFFICIENT NOT ONE Detailed Example Practices Quizzes Come on sleepy head!!!

ALL FIVE TYPES Practice Quizzes Don’t be a chicken!!!

Greatest Common Factor Factor 3ty ty 2 – –Largest number that goes into 3 and 36? Answer: 3 – –Highest power of t in both? Answer: t – –Highest power of y in both? Answer: y 2 Overall Answer: 3ty 2 (y + 12)

Difference of Squares Factor x 2 – 36 Factor x 2 – 36 –What times what would equal x 2 ? Answer: x –What times what would equal 36? Answer: 6 –Put x and 6 together, one with a plus and one with a minus. Final answer: (x + 6)(x - 6) Final answer: (x + 6)(x - 6)

Perfect Square Trinomial Factor x 2 – 10x + 25 Factor x 2 – 10x + 25 –What times what would equal x 2 ? Answer: x –What times what would equal 25? Answer: 5 –Put x and 5 together using the same signs as the negative 10x. Final Answer: (x - 5)(x - 5) Final Answer: (x - 5)(x - 5)

Leading Coefficient of One Factor x 2 - 2x - 48 Factor x 2 - 2x - 48 –List factors of 48. Answer: 1&48, 2&24, 4&12, 6&8 –Choose pair that gets 2. Answer: 6&8 –Get signs from Answer: one + and one – –Put bigger number with the same sign as middle term of - 2x Final Answer: (x – 8)(x + 6) Final Answer: (x – 8)(x + 6)

Leading Coefficient Not One Factor 8x x + 30 Factor 8x x + 30 –Find 8 times 30. Answer: 240 –List factors of 240. Answer: 1&240, 2&120, 3&80, 4&60, 5&48, 6&40, 8&30, 12&20 –Choose the pair that gets 46. Answer: 6&40 –Write - 46x as - 6x and - 40x.  Answer: 8x x - 6x + 30 –Continued on next page…

Leading Coefficient Not One Problem Continued… Problem Continued… > 8x x - 6x + 30 > Factor first two terms. Answer: 8x (x – 5) > Factor last two terms. Answer: - 6 (x – 5) > Put 8x and - 6 together. Answer (8x – 6) > Write (x – 5) only once. Overall Answer: (8x – 6)(x – 5) Overall Answer: (8x – 6)(x – 5)

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