LITTLESTOWN AREA SCHOOL DISTRICT ALGEBRA II

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1 Subject Algebra II Grade Level 9th - 12th Title of Unit Unit 4: Quadratics Time Frame days Stage 1 - Desired Results Established Goals (Learning Outcomes) What content standards (designate focus standards with an F ) and program- or mission-related goal(s) will this unit address? What habits of mind and cross disciplinary goals will this unit address (example 21 st Century Skills)? Common Core Standards: A.SSE.1.a F.IF.9 A.CED.2 F.IF.4 A.SSE.2 F.IF.8.a N.CN.1 N.CN.2 N.CN.7 F.IF.8.a Interpret parts of an expression, such as terms, factors, and coefficients. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tabl verbal descriptions). Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate with labels and scales. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of quantities, and sketch graphs showing key features given a verbal description of the relationship. Use the structure of an expression to identify ways to rewrite it. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symme graph, and interpret these in terms of a context. 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. Use the relation i 2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply com numbers. Solve quadratic equations with real coefficients that have complex solutions. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symm the graph, and interpret these in terms of a context. A.SSE.1.b Interpret complicated expressions by viewing one or more of their parts as a single entity.

2 F.BF.3 A.CED.1 A.CED.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 Create equations and inequalities in one variable and use them to solve problems. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Transfer What kinds of long term independent accomplishments are desired? Students will be able to independently use their learning to.. Solve quadratic equations Transfer properties of quadratics equations to other families of equations Apply properties of quadratics to real life problems Enduring Understandings What understandings about the big ideas are desired? What misunderstandings are predictable? Students will understand that... Relations and functions are mathematical relationships that can be represented and analyzed using words, tables, graphs, and equations. Mathematical functions are relationships that assign each member of one set (domain) to a unique member of another set (range), and the relationship is recognizable across representations. Families of functions exhibit properties and behaviors that can be recognized across representations. Functions can be transformed, combined, and composed to create new functions in mathematical and real world situations. Numbers, measures, expressions, equations, and inequalities can represent mathematical situations and structures in many equivalent forms. Meaning Essential Questions What provocative questions will foster inquiry, meaning-making and transfer? Students will keep considering. What are some common characteristics of quadratic functions? How can you graph a quadratic function using the properties of parabolas? How can you use transformations to help graph quadratic functions? Why do we factor quadratic expressions? How can we solve quadratic equations? What are complex numbers?

3 There are some mathematical relationships that are always true and these relationships are used as the rules of arithmetic and algebra and are useful for writing equivalent forms of expressions and solving equations and inequalities. What are imaginary numbers and how are they used? How can we rewrite a quadratic function in an equivalent form? What facts and basic concepts should students know and be able to recall? This content knowledge may come from the established goals or address pre-requisite knowledge essential for this unit. Acquisition What discrete skills and processes should students be able to use? List the skills and/or behaviors students will be able to exhibit as a result of their work in this unit. Students will know... Vocabulary: Quadratic function Standard form of a quadratic function Parabola Axis of symmetry Vertex of a parabola Quadratic function Standard form of a quadratic function Parabola Axis of symmetry Vertex of a parabola Maximum Minimum Vertex form of a quadratic function Factoring Greatest common factor Perfect square trinomial Difference of squares Standard form of a quadratic equation Zero-product property Students will be skilled at Classify a function as linear, quadratic, or neither. Identify the vertex, the axis of symmetry, and the corresponding points of a parabola. Find a quadratic function given three points on the function. Find a quadratic function to model real-world data. Graph a quadratic function of the form Graph a quadratic function of the form y ax 2 c y ax 2 bx c Find the vertex, axis of symmetry, and y-intercept of a parabola. Find the minimum or maximum value of a quadratic function. Solve real-world max/min problems using a quadratic function that models the situation.

4 Zero of a function Root of an equation i Imaginary number Complex numbers Absolute value of a complex number Completing the square Quadratic formula Discriminant Students will understand: The graph of a quadratic function is a parabola. Quadratic functions have a vertex and an axis of symmetry. The standard form of a quadratic function is The graph of a quadratic function is a parabola. f (x) ax 2 bx c. Quadratic functions have a vertex and an axis of symmetry. The standard form of a quadratic function is Every parabola has either a maximum or minimum. f (x) ax 2 bx c. Many real-world problems can be modeled by quadratics. This allows you to maximize or minimize things like profit, cost, revenue, height, etc The vertex form of a quadratic function is f (x) a(x h) 2 k, where the vertex is h,k. The a value determines the opening direction of the parabola and the width. The values of h and k determine the horizontal and vertical translations respectively. Each member of a family of functions is a transformation of the parent function. Factoring is creating an equivalent expression by rewriting an expression as a product. Factoring quadratic expressions can help solve quadratic equations. The standard form of a quadratic equation is ax 2 bx c 0. Graph a quadratic function of the form y a(x h) 2 k. Write the equation of a parabola given the vertex and a point on the parabola. Convert a quadratic function from standard form to vertex form and vice versa. Identify the reflection, the stretches or shrinks, and the vertical translations and horizontal translations of a quadratic function and use this information to graph the function. Factor out a greatest common factor. Factor a quadratic trinomial of the form Factor a perfect square trinomial. Factor the difference of two squares ( Solve a quadratic equation by factoring. Solve a quadratic equation of the form ax 2 bx c. a 2 b 2 ). ax 2 c by taking square roots. Solve a quadratic equation by graphing on the graphing calculator. Solve quadratic equations that represent real-world situations. Simplify numbers using i. Find the absolute value of a complex number. Find the additive inverse of a complex number. Add, subtract, multiply, and divide complex numbers Find complex solutions of quadratic equations. Solve a perfect square trinomial equation. Find a missing value of a, b, or c in the expression perfect square trinomial. ax 2 bx c to build a Rewrite a quadratic function in vertex form by completing the square.

5 Some quadratic equations can be solved by factoring, taking square roots, or graphing. Each x-intercept of a quadratic function is a zero of the function and a root of the equation. A complex number can be written in the form a bi, where a and b are real numbers. Solve a quadratic equation using the quadratic formula. Find the discriminant of a quadratic equation. Use the discriminant to determine the type and number of solutions for a given quadratic equation. Imaginary numbers are numbers containing 1, which is denoted with an i. Imaginary numbers are used when solving equations that do not have real solutions such as x Completing the square can be used to rewrite a standard from quadratic equation, y ax 2 bx c in the vertex form f (x) a(x h) 2 k, where the vertex is h,k. The standard form of a quadratic equation is ax 2 bx c 0. Quadratic equations can be solved using the quadratic formula. The quadratic formula is x b b2 4ac 2a.

6 Stage 2 Evidence Performance Task Through what authentic performance task will students demonstrate the desired understandings, knowledge, and skills? By what criteria will performances of understanding be evaluated? Students will show their learning by Ace Problems Word Problems/Problem based learning Worksheets Exit Tickets Other Evidence Through what other evidence (work samples, observations, quizzes, tests, journals or other means) will students demonstrate achievement of the desired results? Pre-Assessments Warm-up problems Exit Tickets Quizzes Homework Unit Tests Student Self-Assessment How will students reflect upon or self-assess their learning? Homework checks Daily classwork Closure problems January 2015 Adapted from: Wiggins, G. and McTighe, J., (1998). Understanding by Design. Alexandria, VA: ASCD.