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Readiness Standards for Mathematics Exemplar Lesson Plan College- and Career- Readiness Standards for Mathematics Exemplar Lesson Plan “Polynomials, Maximum or Minimum?” A-SSE.2 A-APR.3 F-IF.4 Carey M. Wright, Ed.D., State Superintendent of Education Office of the Chief Academic Officer Kim S. Benton, Ed.D., Chief Academic Officer Office of Professional Development Trecina Green, Executive Director Carol Ladner, Professional Development Coordinator Elizabeth Fulmer, Professional Development Coordinator Office of Secondary Education Jean Massey, Executive Director Marla Davis, Ph.D., NBCT, Bureau Director Office of Elementary Education and Reading Nathan Oakley, Executive Director Mississippi College- and Career-Readiness Standards Mathematics Exemplar Lesson Plan COURSE: Algebra II Title: Polynomials, Maximum or Minimum? Real World Purpose: Estimated Duration: 3 to 4 Days Polynomials can be found in many different and diverse industries such as aerospace, architecture, financial planning, legal, and construction to name a few. But what about polynomial division? Ever think you might do that again after a high school or college math course? You will if you become a pharmacist and need to determine the correct amount of a drug to administer to a patient. The same goes for a chemist. They use polynomial division to derive chemical formulas. I Can: A-SSE.2: Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2 thus recognizing it as a 2 2 2 2 difference of squares that can be factored as (x – y ) (x + y ). 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. 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.* Page 1 of 25 Mississippi College- and Career-Readiness Standards Mathematics Exemplar Lesson Plan Prerequisite Skills: A-SSE.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. 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. 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.6: Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. A-REI.4: Solve quadratic equations in one variable. b. Solve quadratic equations by inspection (e.g., for x2 = 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. A-REI.10: Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Page 2 of 25 Mississippi College- and Career-Readiness Standards Mathematics Exemplar Lesson Plan Materials/Resources: Key Vocabulary: • www.classzone.com • Algebra tiles • Quadratic • T-chart • www.msemac.redwoods.ed • Graphing calculator • Linear factor • Punnett square u/~kyokoyama/math106/M • Graph paper • Long division • Sum of Cubes ath380pdf • Mini-white board • Synthetic division • Difference of Cubes • www.illustrativemathemati • Expo marker • Polynomial • Difference of Two Square c.org • Eraser • Zero • Complex solutions • www.engageny.com • Algebra II Pearson • Solution • u-substitution • • https://mathstackexchange. Prentice Hall • Root X-intercept • Decreasing interval com • Attachments (Total: 8) • Relative maximum • Union • https://wvde.state.wv.us/st • Relative minimum • • Multiplicity rategybank/FrayerModel.ht Increasing interval • Zero Product Property ml Lesson Introduction Student Exploration Activity: What does the graph model? Student Independent Work (15 minutes estimated) Directions: The teacher will distribute the activity Real-World Context – Graph – Function (Attachment #1). The teacher will review the directions at the top of the activity. The teacher will give students 10 minutes to match the real-world situation to the graph and to match the graph to the function without the use of a calculator. The teacher will spend 5 minutes as closure reviewing students’ work sample and introducing the unit Polynomials, Maximum or Minimum? Page 3 of 25 Mississippi College- and Career-Readiness Standards Mathematics Exemplar Lesson Plan Activity Objective: Using prior knowledge, students will work independently to determine which real-world situation models the polynomial represented by each graph. Students will have an opportunity to match the graph of the polynomial to the function. The teacher will explain the general characteristics of a polynomial function and how to relate the graph to its function. Lesson Activities Guidelines for Group Work • The teacher will remind students during the Pair and Share activities that collaboration is required. Students will share work collaboratively in solving the task given. Once one student finds the solution, the other student needs to check their work. Students will take turns problem solving until the task is complete. Students will use the “Flag It” technique to note any areas of difficulty during the activity on sticky notes. • The teacher will remind students during the Group activities that collaboration is required. Students will need to take turns completing parts of the task until the task is complete. When a student is not identifying or determining solutions, he/she should be checking the work of another student or asking questions of the group. Students will use the “Flag It” technique to note any areas of difficulty during the activity on sticky notes. • The teacher will circulate throughout the classroom to monitor student work and to guide them through the activity using any form of questioning technique they deem appropriate. The teacher is encouraged to ask guiding questions that will encourage productive struggle and critical thinking skills. The teacher will also monitor individual student understanding and general misconceptions. Day 1 1. Activity #1: A-SSE.3a Mini-Review Factoring Quadratic Expressions - All students using Mini-white Boards and Algebra Tiles (15 minutes estimated) The teacher will review factoring quadratics using mini-white boards which allow the teacher to assess students’ prior knowledge. The teacher will allow students to use Algebra Tiles if needed for struggling learners. Quadratics may range from an easier type x2 + 4x (where the leading coefficient is a = 1) to a harder type 4x2 - 4x -15 (where the leading coefficient is a ≠ 1). The teacher will select 5 to 10 quadratics to review during this activity. The teacher will remind students that they can also use T-charts and Punnett Squares if Page 4 of 25 Mississippi College- and Career-Readiness Standards Mathematics Exemplar Lesson Plan needed to factor the quadratics. The teacher will guide students through the activity using effective questioning techniques. Two questions a teacher might ask include, “Do you remember any factoring steps from Algebra I that could assist you?” and “What are the factors of 4 and 15?” The teacher will allow for opportunities at the beginning of the review for students to have productive struggle with factoring techniques for quadratics. Students will also have any opportunity to assess their own level of understanding by noting issues that arise during instruction. Activity Objective: Students were introduced to factoring quadratics in Algebra I and furthered their skills in factoring quadratics and finding zeros in this course. Students will review their prior knowledge and fluency of factoring quadratics to better prepare for Day 2 Activity #2. 2. Activity #2: A-APR.6 Mini-Review Long Division of Polynomials - Students using white board and paper/pencil at desk (20 minutes estimated) The teacher will call two students to the board to divide a 3rd degree or higher polynomial by a given linear factor. For example, x3 - 13x +12 divided by x + 4. One student will perform long division and another student may perform synthetic division. While at their desk, the remainder of the class may use paper/pencil and choose any method they prefer. The two students at the board will compare their solutions. Students at their desks will also check their work. The teacher will remind students that division is a technique to find factors of polynomials. The teacher will guide students to factor the quadratic into linear factors as well. The teacher will choose several different types of long division problems to showcase during the review activity. The teacher will call upon different pairs of students for each problem as time permits. The teacher will ask guided questions during the review activity to assess students’ conceptual understanding and circulate throughout the classroom to assess students’ procedural skill and determine fluency. Activity Objective: Students will review long division of polynomials as a procedural skill while improving their fluency with factoring. Students will have an opportunity to assess their understanding of polynomial long division during the review.
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