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2 2 TExES Standard II.006 Patterns and algebra. The teacher understands linear and quadratic functions, analyzes their algebraic and graphical properties, and uses them to model and solve problems. The beginning teacher: (G) Models and solves problems involving linear and quadratic equations and inequalities using a variety of methods, including technology. TExES Standard V.018 Mathematical processes and perspectives. The teacher understands mathematical reasoning and problem solving. The beginning teacher: (E) Understands the problem-solving process (i.e., recognizing that a mathematical problem can be solved in a variety of ways, selecting an appropriate strategy, evaluating the reasonableness of a solution). TExES Standard V.019 Mathematical processes and perspectives. The teacher understands mathematical connections both within and outside of mathematics and how to communicate mathematical ideas and concepts. (A) Recognizes and uses multiple representations of a mathematical concept (e.g., a point and its coordinates, the area of a circle as a quadratic function of the radius, probability as the ratio of two areas, area of a plane region as a definite integral). (B) Understands how mathematics is used to model and solve problems in other disciplines (e.g., art, music, science, social science, business). (C) Translates mathematical ideas between verbal and symbolic forms. (D) Communicates mathematical ideas using a variety of representations (e.g., numeric, verbal, graphical, pictorial, symbolic, concrete). (E) Understands the use of visual media, such as graphs, tables, diagrams, and animations, to communicate mathematical information. TEKS/TAKS focus TEKS 2A.1 Foundations for functions. The student uses properties and attributes of functions and applies functions to problem situations. The student is expected to: (B) collect and organize data, make and interpret scatter plots, fit the graph of a function to the data, interpret the results, and proceed to model, predict, and make decisions and critical judgments. TEKS 2A.2 Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. The student is expected to: (A) use tools including factoring and properties of exponents to simplify expressions and to transform and solve equations; and (B) use complex numbers to describe the solutions of quadratic equations. TEKS 2A.3 Foundations for functions. The student formulates systems of equations and inequalities from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situations. The student is expected to:

4 4 Summary The big ideas here are to examine various methods for defining a quadratic function given a pattern of a set of data. Participants use matrices, regression equations and finite differences to find quadratic equations. Assessment What follows is an assessment that participants should be able to complete upon finishing task

5 5 Reflect and Apply Consider the data below. x y Use the first three points to find a quadratic function that fits the data. Check your function using the 5 th data point. 2. Find the formula for a linear function that passes through the second point and the third point in the table. 3. Find the value of the linear functions at x = 3 and compare that value to the value of the quadratic at x = Compare the values of the linear and quadratic functions at x = Use a graphing calculator to graph both functions on the same axes to estimate your answers. Approximately what x-values would you have to use so that the resulting function values for the quadratic function and the linear function would not differ from each other by more than 0.5?

6 6 Reflect and Apply Solutions: 1. y = 0.01x 2 + 2x + 1, 0.01( 50) 2 + 2( 50) + 1 = y = 2.03x Linear function: x = 3, y = 7.07 Quadratic function: x = 3, y = Linear function: x = 50, y = Quadratic function: x = 50, y = Participants may examine the table or graph the functions to determine that this difference occurs for 0! x < 8.6. Task is best suited to the teacher journal, whereas Tasks and would work well with the student journal. Teacher use only Modify for students Ready for students Task * Task * Task *

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