Polynomial Operations and Factoring


 Robert Morton
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1 Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring Overview Number of instructional days: 15 (1 day = minutes) Content to be learned Identify terms, coefficients, and degree of polynomials. Add and subtract polynomials. Multiply polynomials (monomials, binomials, trinomials including special cases) using the Distributive Property and FOIL method. Factor polynomials including trinomials of the form f (x) = ax 2 + bx + c, for a = 1, a 1, greatest common factors, perfect square trinomials, and difference of squares. Factor fourterm polynomials by grouping (e.g., 3x 3 12x 2 + 2x 8 ). Essential questions How are the operations and properties of real numbers related to polynomials? How can two algebraic expressions that appear to be different be equivalent? How is the factoring of polynomials related to the multiplication of polynomials? Mathematical practices to be integrated Attend to precision. Classify polynomials based on the number of terms. Make explicit use of degree of polynomials to add, subtract, multiply, and factor polynomials. Look for and make sense of structure. Factor trinomials by looking for and using the structure of the trinomial, considering parameters a, b, and c. Look for structure to factor four terms by grouping. Look for and express regularity in repeated reasoning. Use repeated reasoning to multiply binomials using the Distributive Property and the FOIL method. Use shortcuts for determining the square of a binomial and for multiplying to get a difference of squares. Use repeated reasoning to factor polynomials. Use shortcuts for factoring perfect square trinomials and differences of squares. What characteristics of a polynomial determine how to factor it completely? What are the special cases and patterns used to factor polynomials? 37
2 Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring (15 days) Written Curriculum Common Core State Standards for Mathematical Content Arithmetic with Polynomials and Rational Expressions AAPR Perform arithmetic operations on polynomials [Linear and quadratic] AAPR.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. Seeing Structure in Expressions ASSE Interpret the structure of expressions [Linear, exponential, quadratic] ASSE.1 Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r) n as the product of P and a factor not depending on P. ASSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 y 2 )(x 2 + y 2 ). Common Core Standards for Mathematical Practice 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. 38
3 Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring (15 days) 8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y 2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x 1)(x + 1), (x 1)(x 2 + x + 1), and (x 1)(x 3 + x 2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. Clarifying the Standards Prior Learning In grade 6, students applied the properties of operations to generate equivalent expressions. For example, students applied the Distributive Property to the expression 3(2 + x) to produce 6 + 3x. They also applied the Distributive Property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y). (6.EE.3) In grade 7, students applied properties of operations to add, subtract, factor, and expand linear expressions with rational coefficients. (7.EE.1) In grade 8, students applied the properties of integers (8.EE.1), and in Unit 2.3 of this course, they applied properties of exponents to rational exponents. Current Learning Students identify the degree, terms, and coefficients of a polynomial. They classify the polynomials based on the number of terms and degree. Students add and subtract polynomials. Students multiply polynomials using the distributive method, and they also square binomials. Students factor polynomials including trinomials of the form f (x) = ax 2 + bx + c, for a = 1, a 1, greatest common factor, perfectsquare trinomials, and difference of squares. They also factor fourterm polynomials by grouping. Future Learning Students will use factoring in the next unit when they solve quadratic equations. They will use operations of polynomials and factoring in this course and in later courses to factor higher degree polynomials and to express functions in various forms such as vertex form of a quadratic and standard form of a circle, parabola, and other conic sections. Operations of polynomials and factoring are necessary skills that will be needed for Algebra II, Geometry, Precalculus, and Calculus. Additional Findings According to Algebra of Polynomials (Lausch & Nöbauer, 1974), Polynomials are a classical subject of mathematics. The first steps towards the abstract concept of polynomials were the investigation of algebraic equations and the theory of real and complex functions f of the form f(x) = a n x n + +a 1 x + a 0. (p. ix) 39
4 Algebra 1, Quarter 4, Unit 4.1 Polynomial Operations and Factoring (15 days) 40
5 Algebra 1, Quarter 4, Unit 4.2 Quadratic Functions and Equations Overview Number of instructional days: 12 (1 day = minutes) Content to be learned Solve quadratic equations by factoring. Graph quadratic functions using x and yintercepts and the axis of symmetry. Transform the graph of f(x) = x 2, including translating, stretching, shrinking, and reflecting. Solve quadratic equations by completing the square. Graph quadratic functions using vertex form. Understand that the quadratic formula is derived from completing the square. Apply the quadratic formula and give solutions in simplified, radical form and as approximate values. Model with quadratic functions, interpret key features (intercepts, relative maximums and minimums, symmetries, end behavior) and sketch graphs given a verbal description of the relationship. Interpret the domain of a quadratic function in context of applications. Use graphing technology to explore and model quadratic relationships in realworld problem solving. Solve a simple system consisting of a linear equation and a quadratic equation algebraically and graphically. Mathematical practices to be integrated Model with mathematics. Identify key features of graphs and their relationship to the realworld situation they model. Choose which form of a quadratic equation to use when solving and interpreting different problems. Use appropriate tools strategically. Choose appropriate strategies according to task. Use graphing technology to explore and model quadratic relationships in realworld problem solving. Look for and make use of structure. Identify and examine algebraic expressions as single entities to evaluate characteristics of quadratic functions. Understand why the quadratic formula works based on completing the square. 41
6 Algebra 1, Quarter 4, Unit 4.2 Quadratic Functions and Equations (12 days) Essential questions What are the advantages of writing a quadratic equation in vertex form? What are the effects of a, h, and k on the graph of y = a(x h) 2 + k? What are the different methods to solve quadratic equations? When might one method be more beneficial to use than another? What are the key features of the graph of a quadratic function? How do you solve quadratic equations using different methods? What types of realworld situations can be modeled using quadratic equations? What are the characteristics of a quadratic function? Written Curriculum Common Core State Standards for Mathematical Content Seeing Structure in Expressions ASSE Write expressions in equivalent forms to solve problems [Quadratic and exponential] ASSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Reasoning with Equations and Inequalities AREI Solve equations and inequalities in one variable [Linear inequalities; literal that are linear in the variables being solved for; quadratics with real solutions] AREI.4 Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x p) 2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., for x 2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Solve systems of equations [Linearlinear and linearquadratic] AREI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = 3x and the circle x 2 + y 2 = 3. 42
7 Algebra 1, Quarter 4, Unit 4.2 Quadratic Functions and Equations (12 days) Interpreting Functions FIF Interpret functions that arise in applications in terms of the context [Linear, exponential, and quadratic] FIF.4 FIF.5 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Analyze functions using different representations [Linear, exponential, quadratic, absolute value, step, piecewisedefined] FIF.7 FIF.8 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. Building Functions FBF Build new functions from existing functions [Linear, exponential, quadratic, and absolute value; for F.BF.4a, linear only] FBF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. Common Core Standards for Mathematical Practice 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They 43
8 Algebra 1, Quarter 4, Unit 4.2 Quadratic Functions and Equations (12 days) are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, twoway tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 5 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. Clarifying the Standards Prior Learning Students worked with radicals and integer exponents in grade 8. They applied the properties of integer exponents to generate equivalent numerical expressions. (8.EE.1) Students evaluated square roots of small perfect squares and cube roots of small perfect cubes. (8.EE.2) In Unit 4.1, students used the Distributive Property and the FOIL method to multiply polynomials (monomials, binomials, and trinomials including special cases). They factored polynomials including trinomials of the form f(x) = ax 2 + bx + c, for a = 1, a 1, greatest common factors, perfect square trinomials, and difference of squares. 44
9 Algebra 1, Quarter 4, Unit 4.2 Quadratic Functions and Equations (12 days) Current Learning Students solve quadratic equations by factoring, completing the square, and the quadratic formula. They graph quadratic functions by determining the x and yintercepts and the axis of symmetry. By completing the square, students graph quadratic functions using vertex form. They also explore and model quadratic relations in realworld situations using graphing technology. Students transform the graph of f(x) = x 2 by translating, stretching, shrinking, and reflecting. They also solve simple systems consisting of a linear equation and a quadratic equation. Future Learning Quadratics have many applications to problems in physics and engineering that students will encounter in future math courses and careers. The study of quadratics prepares students for working with higher order polynomials, the Fundamental Theorem of Algebra, and the Rational Root Theorem. Additional Findings In relation to quadratics, John Allen Paulos wrote in Beyond Numeracy, Many situations in physics, engineering, and elsewhere lead to such equations. (p. 198) Relative to using graphical representations to solve equations, A Research Companion to Principals and Standards for School Mathematics states, One cannot simply expect students to be able to read these representations in the ways they are intended. The process of learning to read such representations is complex and requires teaching and learning. (p. 131) PARCC Model Content Frameworks for Mathematics notes that fluency in transforming expressions and chunking (seeing parts of an expression as a single object) is essential in factoring, completing the square and other mindful calculations. (p. 52) 45
10 Algebra 1, Quarter 4, Unit 4.2 Quadratic Functions and Equations (12 days) 46
11 Algebra 1, Quarter 4, Unit 4.3 Operations with Radicals Overview Number of instructional days: 12 (1 day = minutes) Content to be learned Write an expression with a rational exponent in radical form. Write a radical in exponential form. Simplify expressions involving rational exponents. Simplify radical expressions (square roots). (not in the CCSS) Add, subtract, and multiply radical monomials, expressing the solutions in simplified form (square roots). (not in the CCSS) Investigate the products and sums of two rational numbers, two irrational numbers, and a rational and irrational number (Closure Property). Essential questions What type of number(s) results from the sum of a rational number and an irrational number? What type of number(s) results from the product of a nonzero rational and an irrational number? What type of number(s) results from the sum of two irrational numbers? What type of number(s) results from the product of two irrational numbers? Mathematical practices to be integrated Attend to precision. State the meaning of the radical symbol and interpret it in terms of rational exponents. Look for and make use of structure. Review the properties of exponents to find structure in examples and apply the structure to simplifying rational expressions with exponents. Understand that the set of irrational numbers is closed under addition, but not under multiplication. How do you use rational exponents to represent radicals? How do you know when a radical expression is in simplest form? How do you know when an expression is in simplified rational exponent form? 47
12 Algebra 1, Quarter 4, Unit 4.3 Operations with Radicals (12 days) Written Curriculum Common Core State Standards for Mathematical Content The Real Number System NRN Extend the properties of exponents to rational exponents. NRN.1 NRN.2 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5 1/3 to be the cube root of 5 because we want (5 1/3 ) 3 = 5 (1/3)3 to hold, so (5 1/3 ) 3 must equal 5. Rewrite expressions involving radicals and rational exponents using the properties of exponents. Use properties of rational and irrational numbers. NRN.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. Common Core Standards for Mathematical Practice 6 Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 7 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered , in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older students can see the 14 as 2 7 and the 9 as They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 3(x y) 2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. 48
13 Algebra 1, Quarter 4, Unit 4.3 Operations with Radicals (12 days) Clarifying the Standards Prior Learning Students worked with radicals and integer exponents in grade 8. They applied the properties of integer exponents to generate equivalent numerical expressions. (8.EE.1) Students evaluated square roots of small perfect squares and cube roots of small perfect cubes. (8.EE.2) In Unit 4.1, students used the Distributive Property and the FOIL method to multiply polynomials (monomials, binomials, and trinomials including special cases). In Unit 4.2, students used the quadratic formula to simplify expressions with radicals. Current Learning Students extend their knowledge of exponents to include rational exponents. (NRN.2) They rewrite expressions involving radicals and rational exponents using the properties of exponents. Students add, subtract, and multiply radical expressions and realize that the set of irrational numbers is closed under addition but not multiplication. (NRN.3) Future Learning In Geometry, students will simplify radicals, work with trigonometric ratios, and solve special right triangles. (GSRT.8) In Algebra II and advanced algebra courses, students will connect the closure properties of irrational numbers to operations of complex number solutions for polynomials. (NCN.3, 7, 8) Additional Findings Principles and Standards for School Mathematics notes that high school algebra should provide students with insights into mathematical abstraction and structure. In grades 9 12, students should develop an understanding of algebraic properties that govern the manipulation of symbols in expressions, equations, and inequalities. It continues by adding that students should become fluent in performing such manipulations by appropriate means mentally, by hand, or by machine to solve equations and inequalities, to generate equivalent forms of expressions or functions, or to prove general results. (p. 297) 49
14 Algebra 1, Quarter 4, Unit 4.3 Operations with Radicals (12 days) 50
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