SFUSD Mathematics Core Curriculum Development Project
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1 1 SFUSD Mathematics Core Curriculum Development Project Creating meaningful transformation in mathematics education Developing learners who are independent, assertive constructors of their own understanding
2 Algebra 2 A.10 Polynomials and Rational Expressions 2 Number of Days Lesson Reproducibles Number of Copies Materials 2 Entry Task Sequences and Equations Sequences and Equations (revisited) 2 Lesson Series 1 CPM CCA (3 pages) CPM CCA (6 pages) CPM CCA (6 pages) CPM CCA (4 pages) 2 Apprentice Task Cubic Graphs and their Equations Cubic Graphs and their Equations (revisited) 8 Lesson Series 2 CPM CCA (8 pages) CPM CCA (5 pages) CPM CCA (5 pages) 1 per student 1 per student 1 per student 1 per student Sequences of Dots card set (1 per group) Adding Polynomials card set (1 per group) Poster paper and glue sticks Resource Page (optional) Resource Page (3 pages) for teacher Card sets (1 per group): Cubic Graphs, Cubic Functions, Statements to Discuss Poster paper and glue sticks Poster graph paper and markers 2 Expert Task The Hill and the Riverbed 1 per student 4 Lesson Series 3 CPM CCA (5 pages) CPM CCA (6 pages) CPM CCA (7 pages) Resource Page CPM CCA (7 pages) CPM CCA (11 pages) CPM CCA (5 pages) Poster paper and markers Scissors Tape Graph paper 1 Milestone Task Consecutive Integers and Cookie Boxes (2 pages) 1 per student
3 Unit Overview Big Idea 3 Polynomial expressions can be factored to reveal properties of the graph of the associated function; the degree of the polynomial gives the number of real and complex roots. Like the integers, polynomials are closed under addition, subtraction, and multiplication; dividing polynomials leads to rational expressions in the same way that dividing integers lead to fractions in earlier grades. Unit Objectives Students will be able to solve quadratic equations with both real and complex solutions. Students will be able to express complex numbers in standard form (a + bi). Students will be able to add, subtract, and multiply complex numbers. Students will be able to factor polynomial expressions. Students will be able to perform the four basic operations on polynomials. Students will prove polynomial identities and use them to describe numerical relationships. Students will be able to graph polynomial functions with and without graphing calculators. Students will be able to solve higher order polynomials equations. Students will model real-world and mathematical problems using polynomials. Unit Description In this unit, students will continue their study of polynomial functions, which includes: graphing, factoring, finding roots, writing equations given graphs. They apply previous knowledge of volume and quadratics to solve real-world problems involving polynomials. N.CN The Complex Number System Perform arithmetic operations with complex numbers. CCSS-M Content Standards N.CN.1 Know there is a complex number i such that i 2 = 1, and every complex number has the form a + bi with a and b real. N.CN.2 Use the relation i 2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. Use complex numbers in polynomial identities and equations. N.CN.7 Solve quadratic equations with real coefficients that have complex solutions. [The following (+) standards are not in the Scope & Sequence but are covered informally in this unit.] N.CN.8 (+) Extend polynomial identities to the complex numbers. For example, rewrite x as (x + 2i)(x - 2i) N.CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
4 4 A-SSE Seeing Structure in Expressions Interpret the structure of expressions A.SSE.1 Interpret expressions that represent a quantity in terms of its context.* A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients. A.SSE.1b 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. 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 ) 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 ). A-APR Arithmetic with Polynomials and Rational Expressions Perform arithmetic operations on polynomials 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. Understand the relationship between zeros and factors of polynomials A.APR.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). A.APR.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. Use polynomial identities to solve problems A.APR.4 Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x 2 + y 2 ) 2 = (x 2 y 2 ) 2 + (2xy) 2 can be used to generate Pythagorean triples. F-IF Interpreting Functions Interpret functions that arise in applications in terms of the context F.IF.4 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.* Analyze functions using different representations F.IF.7c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
5 Progression of Mathematical Ideas 5 Prior Supporting Mathematics Current Essential Mathematics Future Mathematics In Algebra 1, in Unit A.4 Working With Expressions, students explain why a particular form of an expression is desirable for a given context. Students perform operations on polynomials and expressions. In Units A.5 Quadratic Functions and A.6 Quadratic Equations, students graph quadratic functions, identify features such as the vertex and line of symmetry, and find the roots by both graphical and analytic methods. In Algebra 2 Unit A.9 Functions, students graph, interpret, and analyze a variety of functions, including polynomial functions. In this unit, students understand the relationship between factors and zeros through their study of polynomial functions. They solve polynomial equations with both real and complex roots, express complex numbers in standard form (a + bi), and add, subtract and multiply complex numbers. Students use the structure of expressions to rewrite polynomial expressions in different forms: standard form, factored form and quotient/remainder form. They use factored form to find the zeros of polynomials. Additionally, students perform operations, including division, on polynomials, and understand that polynomials form a system closed under addition, subtraction, and multiplication, but not division. In future mathematics courses (Precalculus), students will address related (+) standards, extending their understanding of numbers to understand that rational expressions form a system analogous to the rational numbers. They will also graph rational functions, identifying zeros and asymptotes. In the realm of complex numbers, they will find conjugate of complex numbers and use it to find quotients. Students will use the Binomial Theorem to expand expressions of the form (x + y) n.
6 Entry Task Apprentice Task Unit Design All SFUSD Mathematics Core Curriculum Units are developed with a combination of rich tasks and lessons series. The tasks are formative assessments of student learning. The tasks are designed to address four central questions: Expert Task Lesson Series 1 Lesson Series 2 Lesson Series 3 Milestone Task 6 Entry Task: What do you already know? Apprentice Task: What sense are you making of what you are learning? Expert Task: How can you apply what you have learned so far to a new situation? Milestone Task: Did you learn what was expected of you from this unit? 2 days 2 Days 2 Days 8 Days 2 Days 4 Days 1 Day Total: 21 days
7 7 Entry Task Manipulating Polynomials Apprentice Task Representing Polynomials Expert Task The Hill and the Riverbed Milestone Task Consecutive Integers and Cookie Boxes CCSS-M Standards A.APR.3 F.IF.4, F.IF.7c A.SSE.2 A.APR.1, A.APR.2, A.APR.3, A.APR.4 A.APR.3 F.IF.7c A.SSE.2 A.APR.1, A.APR.2, A.APR.3, A.APR.4 Brief Description of Task In the context of dot diagrams representing patterns, students switch between visual and algebraic representations of polynomial expressions and perform operations on polynomial expressions. Students make connections between graphs and equations of polynomial functions. Students find a cubic polynomial to represent the curve of a hill and riverbed. Given additional information, they assess whether that is the appropriate model. Students model a numeric relationship using a polynomial expression, compare equivalent polynomial expressions, and prove the relationship algebraically. Then students model a real-world situation and explore a problem that is structurally the same. Source Manipulating Polynomials FAL, MARS Representing Polynomials FAL, MARS EngageNY Algebra 2, Mid-Module Assessment Task Problem 1 EngageNY Algebra 2, Mid-Module Assessment Task Problems 2 & 3 Lesson Series 1 Lesson Series 2 Lesson Series 3 CCSS-M Standards A.APR.1, A.APR.4 A.SSE.1a, A.SSE.1b, A.SSE.2 A.APR.3 F.IF.4, F.IF.7c N.CN.1, N.CN.2, N.CN.7, N.CN.8 (+), N.CN.9 (+) A.APR.6 Brief Description of Lessons Students rewrite polynomial expressions with the goal of putting polynomial expressions into useful equivalent forms. They determine whether polynomial expressions are equivalent by testing values, using graphs and tables, or rewriting the expressions. Students perform operations on polynomial functions, leading to the conclusion that polynomials are a system closed under addition, subtraction, and multiplication, but not division. Students learn that the factored form of polynomial equations is convenient for sketching graphs by using the relationship between the factors and the x-intercepts. Students write exact polynomial equations given the x-intercepts and using another point to determine the stretch factor. Students are introduced to complex numbers, learn properties of imaginary and complex numbers, and understand that polynomial functions can have complex roots. They learn how to divide polynomials by a factor to find other factors. In a culminating problem on Day 7, students apply their knowledge to maximize the volume of a tank. On Day 8, students reflect upon their own learning and assess areas where they may need assistance. Sources CPM CCA 2 - Unit 3 CPM CCA 2 - Unit 8 CPM CCA 2 - Unit 8
8 Entry Task Manipulating Polynomials What will students do? 8 Mathematics Objectives and Standards Math Objectives: Students will be able to switch between visual and algebraic representations of polynomial expressions. Students will be able to perform arithmetic operations on algebraic representations of polynomials, factoring, and expanding appropriately when it helps to make the operations easier CCSS-M Standards Addressed: A.APR.1, A.SSE.1a, A.SSE.1b, A.SSE.2 Potential Misconceptions: Students draw dot diagram incorrectly. Students rely on counting. Students factorization of algebraic expressions is incorrect or omitted. Framing Student Experience Launch: Before the lesson students work individually on the assessment task Sequences and Manipulations. Review the responses and create questions for students to consider to help improve their work. During: Introduce the lesson by showing students how the card matching activity works. Students then work collaboratively in small groups to match diagrammatic and algebraic representations of polynomials. They display their work on posters. Students share their finished posters, comparing reasoning and checking that explanations are clear and complete. In a whole-class discussion, reflect on the strategies students have used to complete the task. Closure/Extension: Students work individually to review their solutions to the initial assessment task before completing a second task, Sequences and Equations (revisited).
9 9 Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 7. Look for and make use of structure. Structures for Student Learning: Academic Language Support: Vocabulary: polynomial Manipulating Polynomials How will students do this? Differentiation Strategies: Participation Structures (group, partners, individual, other): Individual - Group - Individual
10 Lesson Series #1 10 Lesson Series Overview: Students rewrite polynomial expressions with the goal of putting polynomial expressions into useful equivalent forms. They determine whether polynomial expressions are equivalent by testing values, using graphs and tables, or rewriting the expressions. On Day 4, students perform operations on polynomial functions, leading to the conclusion that polynomials are a system closed under addition, subtraction, and multiplication, but not division. Seeing structure in expressions is an important underlying theme of this lesson series. CCSS-M Standards Addressed: A.APR.1, A.APR.4, A.SSE.1a, A.SSE.1b, A.SSE.2 Time: 4 days Lesson Overview Day 1 Students focus on equivalence and strategies for rewriting equations and expressions. For detailed Teacher Notes, including strategies for working with groups, focus questions, mathematical overviews, and suggestions for closure, please reference the CPM Teacher Guide pages for these lessons. Lesson Overview Day 2 Students work in teams to find multiple ways to show that (x + y) 2 x 2 + y 2. This two-day lesson is being cut to one day. (If time permits, you can use Day 2: 3-18 to 3-22.) Lesson Overview Day 3 Students will solve equations by rewriting them in equivalent forms. This two day lesson is being cut to one day. (If time permits, you can use Day 2: 3-39 to 3-44) Are they equivalent? Lesson (3-1 to 3-4) HW (3-5 to 3-10) How can I rewrite it? Lesson (3-13 to 3-17) HW (3-23 to 3-25, 3-27 to 3-30) How can I solve it? Lesson (3-37 to 3-38) HW (3-45, 46, 47, 50, 51) Lesson Resource Page
11 11 Lesson Overview Day 4 Students will add, subtract, multiply and divide polynomials, in order to see that dividing polynomials will NOT result in a polynomial. They explore the graphs of the functions created by these operations. The investigation problem 3-57 is open-ended. If your students need more guidance they can refer to 3-58 to Where does the graph go? Lesson (3-57 to 3-62) HW (3-63 to 3-69) Lesson Resource Page
12 Apprentice Task Representing Polynomials What will students do? 12 Mathematics Objectives and Standards Math Objectives: Recognize the connection between the zeros of polynomials when suitable factorizations are available, and graphs of the functions defined by polynomials. Identify the connection between transformations of the graphs and transformations of the functions obtained by replacing f(x) by f(x + k), f(x) + k, -f(x), f(-x). CCSS-M Standards Addressed: A-SSE: Interpret the structure of expressions A-APR: Understand the relationship between zeros and factors of polynomials F-IF: Analyze functions using different representations F-BF: Build new functions from existing functions Potential Misconceptions Framing Student Experience Launch: Before the lesson students work individually on the assessment task Cubic Graphs and Their Equations. You review the responses and create questions for students to consider, to help improve their work. During: Students work collaboratively in pairs or threes, matching functions to their graphs and creating new examples. Throughout their work students justify and explain their decisions to peers. During a wholeclass discussion, students explain their reasoning. Closure/Extension: Students work individually again to improve their solutions to the assessment task.
13 13 Representing Polynomials How will students do this? Focus Standards for Mathematical Practice: 2. Reason abstractly and quantitatively. 7. Look for and make use of structure. Structures for Student Learning: Academic Language Support: Vocabulary: cubic Differentiation Strategies: Participation Structures (group, partners, individual, other): Individual - Group - Individual
14 Lesson Series #2 14 Lesson Series Overview: Students move between polynomial equations and graphs, learning that the factored form is convenient for sketching graphs by using the relationship between the factors and the x-intercepts. Students will also write exact polynomial equations given the x-intercepts and using another point to determine the stretch factor. CCSS-M Standards Addressed: A.APR.3, F.IF.4, F.IF.7c Time: 3 days Lesson Overview Day 1 Students will match polynomial functions in factored form to their graphs. This two-day lesson is cut to one day. Students can create a stand-alone poster instead of presentations on a second day. Take out any problems if time is an issue How can I describe a graph? Lesson (8-1 to 8-6) HW (8-8 to 8-13, 8-17) (If you have CPM Algebra 2 Connections textbooks, Lesson is a similar lesson.) Lesson Overview Day 2 Students will graph polynomial functions written in factored form. Lesson Overview Day 3 Students will write exact polynomial equations given the x-intercepts and one additional point How can I predict the graph? Lesson (8-26 to 8-35) HW (8-36 to 8-44) (If you have CPM Algebra 2 Connections textbooks, Lesson is a similar lesson.) How can I find the equation? Lesson (8-45 to 8-53) HW (8-54 to 8-62) (If you have CPM Algebra 2 Connections textbooks, Lesson is a similar lesson.)
15 Expert Task The Hill and the Riverbed What will students do? 15 Mathematics Objectives and Standards Math Objectives: Students will write exact polynomial equations given the x-intercepts and one additional point. Students model a real-world situation with a polynomial function. CCSS-M Standards Addressed: A.APR.3, F.IF.7c Potential Misconceptions: Framing Student Experience Launch: Before starting the Expert Task, begin the class with a warm-up problem on writing exact equations. You might use homework problem During: Students work in groups of 2 to 4 on the Expert Task. Closure/Extension: See rubric and solution in general folder.
16 16 Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 4. Model with mathematics. Structures for Student Learning: Academic Language Support: Vocabulary: geographer, depth, peak, shallow The Hill and the Riverbed How will students do this? Differentiation Strategies: Participation Structures (group, partners, individual, other): Groups of 2 to 4
17 Lesson Series #3 17 Lesson Series Overview: Students are introduced to complex numbers, learn properties of imaginary and complex numbers, and understand that polynomial functions can have complex roots. They learn how to divide polynomials by a factor to find other factors. CCSS-M Standards Addressed: N.CN.1, N.CN.2, N.CN.7, N.CN.8 (+), N.CN.9 (+), A.APR.6 Time: 7 days Lesson Overview Day 1 Students will solve equations using imaginary and complex numbers. Make sure to refer to the CPM Teacher Guide for Chapter 8. Lesson Overview Day 2 Students will practice operations with complex numbers. Lesson Overview Day 3 Students will investigate the number of factors a polynomial can have and how the roots and factors are related. (Optional: Students will graph complex numbers.) This lesson addresses N.CN.4, which is a (+) standard not explicitly covered in this unit. You may omit problems 8-97, 8-98, What are imaginary numbers? Lesson (8-63 to 8-69) HW (8-70 to 8-78) (If you have CPM Algebra 2 Connections textbooks, Lesson is a similar lesson.) What are complex roots? Lesson (8-78 to 8-86) HW (8-87 to 8-96) (If you have CPM Algebra 2 Connections textbooks, Lesson is a similar lesson.) What are the complex numbers? Lesson (8-97 to 8-99 (optional), to 8-103) HW (8-104 to 8-112) (If you have CPM Algebra 2 Connections textbooks, Lesson is a similar lesson.) Lesson Resource Page
18 18 Lesson Overview Day 4 Students will factor polynomials using polynomial division. This lesson uses the area model to organize factors and treats the polynomial division as a puzzle. You should work through the problems yourself before the lesson to familiarize yourself with this approach. Lesson Overview Days 5 and 6 Day 5: Students will work on problems to ending with a poster. Day 6: Gallery walk for the first 20 minutes. Students then complete problems to Lesson Overview Day 7 Students will work on the application problem County Fair Game Tank. Omit problems if time is an issue. Problem is open-ended; problems 8-166, 8-167, and provide more guidance if students need it. Lesson Overview Day 8 Students reflect upon their own learning and synthesize concepts. You may choose to do one or more of the review activities: Team Brainstorm, Making Connections, or Portfolio How can I divide polynomials? Lesson (8-112 to 8-119) HW (8-120 to 8-125) (If you have CPM Algebra 2 Connections textbooks, Lesson is a similar lesson.) How can I solve it? Lesson (8-129 to 8-137) HW (8-138 to 8-149) (If you have CPM Algebra 2 Connections textbooks, Lesson is a similar lesson.) How can I use it? Lesson (8-165 to 8-168) HW (8-169 to 8-173, to 8-177) (If you have CPM Algebra 2 Connections textbooks, Lesson is a similar lesson.) What have I learned? Lesson (CL to CL 8-187) HW (Complete lesson for homework)
19 Milestone Task Consecutive Integers and Cookie Boxes What will students do? 19 Mathematics Objectives and Standards Math Objectives: Students will prove polynomial identities and use them to describe numerical relationships. Students will model real-world situations with polynomials and solve graphically or algebraically. CCSS-M Standards Addressed: A.SSE.2, A.APR.1, A.APR.2, A.APR.3, A.APR.4 Potential Misconceptions: Launch: Framing Student Experience During: Students complete the Milestone Task individually. These problems are taken from the EngageNY Algebra 2 Mid-Module Assessment Task - Problems 2 and 3. Closure/Extension: If students finish early, ask them to explain the relationship between the original problem and Juan s variation.
20 20 Consecutive Integers and Cookie Boxes How will students do this? Focus Standards for Mathematical Practice: 4. Model with mathematics. 7. Look for and make use of structure. Structures for Student Learning: Academic Language Support: Vocabulary: consecutive integers, cubed Differentiation Strategies: Participation Structures (group, partners, individual, other): Individual
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