Operations and Characteristics of Polynomials

Transcription

1 Algebra 1, Quarter 4, Unit 4.1 Operations and Characteristics of Polynomials Overview Number of instructional days: 10 (1 day = minutes) Content to be learned Understand and use characteristics of polynomials (standard form, degree, leading coefficient, and classifying). Simplify polynomial expressions through adding and subtracting. Evaluate polynomial expressions, given values of the variable. Multiply polynomials including 1) monomial times binomial, 2) binomial times binomial, 3) binomial times trinomial, and 4) squaring a binomial. Factor out the greatest common factor of a polynomial expression. Factor quadratic trinomials including special cases such as difference of squares and perfect square trinomials. Essential questions How do you classify a polynomial by term and degree? How do you identify like terms in order to add and subtract polynomials? How do you know when a polynomial expression is in its simplest form? How do you evaluate polynomials? Mathematical practices to be integrated Attend to precision. Use precise mathematical vocabulary, clear and accurate definitions, and symbols to communicate efficiently and effectively. Calculate and compute accurately (including technology). Look and make use of structure. Recognize the importance of a mathematical procedure and extend its use to other problems. Evaluate work and make modifications or try a new approach, if necessary. How are adding and subtracting polynomial expressions similar to each other, yet different from multiplying polynomial expressions? How is the factoring of a polynomial expression related to the multiplication of two polynomial expressions? 39

2 Algebra 1, Quarter 4, Unit 4.1 Operations and Characteristics of Polynomials (10 days) Written Curriculum Common Core State Standards for Mathematical Content Arithmetic with Polynomials and Rational Expressions A-APR Perform arithmetic operations on polynomials [Linear and quadratic] 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. Seeing Structure in Expressions A-SSE Write expressions in equivalent forms to solve problems A-SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 y 2 )(x 2 + y 2 ) A-SSE.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. Common Core Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 40

3 Algebra 1, Quarter 4, Unit 4.1 Operations and Characteristics of Polynomials (10 days) 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. Clarifying the Standards Prior Learning In grade 4, students began using symbols to represent unknown quantities to write simple linear algebraic expressions. In grade 5, students began evaluating linear algebraic expressions using whole numbers. In grade 6, students began working with four operations to write and evaluate linear algebraic expressions with more than one variable. By grade 8, students had written and evaluated algebraic expressions with rational numbers and exponents. Current Learning Students understand and use characteristics of polynomials, and they simplify polynomial expressions through adding and subtracting. Students evaluate polynomial expressions, given values of the variable. Students multiply polynomials and factor out the greatest common factor of a polynomial expression. Students factor quadratic trinomials. Future Learning In grades 11, 12, and advanced mathematics, students will manipulate, evaluate, and simplify algebraic and numerical expressions; add, subtract, multiply, and divide polynomials; add, subtract, multiply, and divide rational expressions; simplify complex fractions; factor quadratic and higher-degree polynomials, including difference of squares; and apply properties of logarithms. Students will solve polynomial equations graphically and algebraically. Additional Findings According to Principles and Standards for School Mathematics, students will understand meanings of operations and how they relate to one another. (p. 292) Benchmarks for Science Literacy states: Students should know that there is no one right way to solve a math problem; different methods have different advantages and disadvantages (p. 28). 41

4 Algebra 1, Quarter 4, Unit 4.1 Operations and Characteristics of Polynomials (10 days) 42

5 Algebra 1, Quarter 4, Unit 4.2 Graphing and Transformations of Quadratic Functions Overview Number of instructional days: 5 (1 day = minutes) Content to be learned Calculate the vertex of a quadratic function from the equation. Identify the vertex (maximum/minimum) and intercepts (solutions) from the graph of a quadratic function. Accurately sketch a quadratic function from the vertex and table of values. Identify the axis of symmetry of the graph of a quadratic function. Identify transformations of quadratic functions that change the position or the size of the parent function. Essential questions How do the values of a, b, and c in the standard form of a quadratic function help in graphing the parabola? How do you determine the vertex and x-intercepts of a parabola? Mathematical practices to be integrated Attend to precision. Use precise mathematical vocabulary, clear and accurate definitions, and symbols to communicate efficiently and effectively. Calculate and compute accurately (including technology). Look and make use of structure. Recognize the importance of a mathematical procedure and extend its use to other problems. Evaluate work and make modifications or try a new approach, if necessary. What is 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), and how do you find the value of k given the graphs. 43

6 Algebra 1, Quarter 4, Unit 4.2 Graphing and Transformations of Quadratic Functions (5 days) Written Curriculum Common Core State Standards for Mathematical Content Building Functions F-BF Build new functions from existing functions [Linear, exponential, quadratic, and absolute value; for F.BF.4a, linear only] F-BF.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.\ Interpreting Functions F-IF Analyze functions using different representations [Linear, exponential, quadratic, absolute value, step, and piecewise-defined] 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. a. Graph linear and quadratic functions and show intercepts, maxima, and minima. Common Core Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 44

7 Algebra 1, Quarter 4, Unit 4.2 Graphing and Transformations of Quadratic Functions (5 days) 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. Clarifying the Standards Prior Learning In grade 4, students began using symbols to represent unknown quantities to write simple linear algebraic expressions. In grade 5, students began evaluating linear algebraic expressions using whole numbers. In grade 6, students began working with four operations to write and evaluate linear algebraic expressions with more than one variable. By grade 8, students had written and evaluated algebraic expressions with rational numbers and exponents. Current Learning Students graph quadratic functions and apply transformations to them. Students identify the different characteristics of quadratic functions. Future Learning In grades 11, 12, and advanced math, students will manipulate, evaluate, and simplify algebraic and numerical expressions; add, subtract, multiply, and divide polynomials; add, subtract, multiply, and divide rational expressions; simplify complex fractions; factor quadratic and higher-degree polynomials, including difference of squares; and apply properties of logarithms. Students will solve polynomial equations graphically and algebraically. Additional Findings A Research Companion to Principles and Standards for School Mathematics (NCTM), pp , indicates that graphs, diagrams, charts, number sentences, formulas, and other representations play an increasingly important role in mathematical activities. Benchmarks for Science Literacy states: Students should know that there is no one right way to solve a math problem; different methods have different advantages and disadvantages (p. 28). 45

8 Algebra 1, Quarter 4, Unit 4.2 Graphing and Transformations of Quadratic Functions (5 days) 46

9 Algebra 1, Quarter 4, Unit 4.3 Solving and Applying Quadratics Overview Number of instructional days: 15 (1 day = minutes) Content to be learned Solve quadratic equations by finding square roots, factoring, using the quadratic formula, and completing the square. Calculate the discriminant and use it to determine the number and types of solutions of a quadratic equation. Write quadratic functions in vertex form. Solve real-world problems that are modeled by quadratic functions. Essential questions What kinds of solutions to a quadratic equation are possible? How can you tell when a quadratic function has complex solutions? What do the solutions to a quadratic function tell you about the graph? Mathematical practices to be integrated Attend to precision. Use precise mathematical vocabulary, clear and accurate definitions, and symbols to communicate efficiently and effectively. Calculate and compute accurately (including technology). Look and make use of structure. Recognize the importance of a mathematical procedure and extend its use to other problems. Evaluate work and make modifications or try a new approach, if necessary. How do you choose the best method for solving a quadratic equation? How do you use the different methods to solve a quadratic equation? What type of real-world situations can be modeled by quadratic equations? 47

10 Algebra 1, Quarter 4, Unit 4.3 Solving and Applying Quadratics (15 days) Written Curriculum Common Core State Standards for Mathematical Content Seeing Structure in Expressions Write expressions in equivalent forms to solve problems A-SSE.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. A-SSE 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 Solve equations and inequalities in one variable A-REI.4 Solve quadratics in one variable. A-REI 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. Interpreting Functions Analyze functions using different representations F-IF-8 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. The Real Number System Use properties of rational and irrational numbers. N-RN-3 F-IF N-RN 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. 48

11 Algebra 1, Quarter 4, Unit 4.3 Solving and Applying Quadratics (15 days) Quantities N-Q Reason quantitatively and use units to solve problems. N-Q.2 Define appropriate quantities for the purpose of descriptive modeling Common Core Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. Clarifying the Standards Prior Learning In grade 4, students began using symbols to represent unknown quantities to write simple linear algebraic expressions. In grade 5, students began evaluating linear algebraic expressions using whole numbers. In grade 6, students began working with four operations to write and evaluate linear algebraic expressions with more than one variable. By grade 8, students had written and evaluated algebraic expressions with rational numbers and exponents. 49

12 Algebra 1, Quarter 4, Unit 4.3 Solving and Applying Quadratics (15 days) Current Learning Students use various methods to solve quadratic equations. Students use these methods to solve realworld problems that can be modeled by quadratic functions. Future Learning In grades 11, 12, and advanced math, students will manipulate, evaluate, and simplify algebraic and numerical expressions; add, subtract, multiply, and divide polynomials; add, subtract, multiply, and divide rational expressions; simplify complex fractions; factor quadratic and higher-degree polynomials, including difference of squares; and apply properties of logarithms. Students will solve polynomial equations graphically and algebraically. Additional Findings A Research Companion to Principles and Standards for School Mathematics (NCTM), pp , indicates that graphs, diagrams, charts, number sentences, formulas, and other representations play an increasingly important role in mathematical activities. Benchmarks for Science Literacy states: Students should know that there is no one right way to solve a math problem; different methods have different advantages and disadvantages (p. 28). 50

13 Algebra 1, Quarter 4, Unit 4.4 Special Functions Overview Number of instructional days: 8 (1 day = minutes) Content to be learned Given an equation, graph the following functions by hand: absolute value, piece-wise, step, square root, and cube root. Identify the characteristics (domain, range, shape, intercepts, end behavior) of each of the functions above. Mathematical practices to be integrated Attend to precision. Use precise mathematical vocabulary, clear and accurate definitions, and symbols to communicate efficiently and effectively. Calculate and compute accurately (including technology). Look and make use of structure. Recognize the importance of a mathematical procedure and extend its use to other problems. Evaluate work and make modifications or try a new approach, if necessary. Essential questions What are the different identifying characteristics of each special function? How can you determine the type of special function from its graph? How can you determine the type of special function from its equation? 51

14 Algebra 1, Quarter 4, Unit 4.4 Special Functions (8 days) Written Curriculum Common Core State Standards for Mathematical Content Interpreting Functions Analyze functions using different representations 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. F-IF-9 a. Graph square root, cube root, and piecewise-defined functions, including step function and absolute value functions. Compare properties of two functions each represented in a different way (graphically, numerically in tables, or by verbal description). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Common Core Standards for Mathematical Practice 1 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 2 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. F-IF 52

15 Algebra 1, Quarter 4, Unit 4.4 Special Functions (8 days) Clarifying the Standards Prior Learning In grade 4, students began using symbols to represent unknown quantities to write simple linear algebraic expressions. In grade 5, students began evaluating linear algebraic expressions using whole numbers. In grade 6, students began working with four operations to write and evaluate linear algebraic expressions with more than one variable. By grade 8, students had written and evaluated algebraic expressions with rational numbers and exponents. Current Learning Students graph and analyze functions including absolute value, piece-wise, step, square root, and cube root. Future Learning In grades 11, 12, and advanced math, students will manipulate, evaluate, and simplify algebraic and numerical expressions; add, subtract, multiply, and divide polynomials; add, subtract, multiply, and divide rational expressions; simplify complex fractions; factor quadratic and higher-degree polynomials, including difference of squares; and apply properties of logarithms. Students will solve polynomial equations graphically and algebraically. Additional Findings Principles and Standards for School Mathematics notes that high school students should be able to interpret functions in a variety of formats. Students should solve problems in which they use tables, graphs, words and symbolic expressions to represent and examine functions and patterns of change. Students should learn algebra both as a set of concepts and competencies tied to the representation of quantitative relationships and as a style of mathematical thinking for formalizing patterns, functions, and generalizations (p. 287). By the completion of algebra 1, high school students should have substantial experience in exploring the properties of different classes of functions. 53

16 Algebra 1, Quarter 4, Unit 4.4 Special Functions (8 days) 54