Grade Levels: Algebra 1. Quadratic Functions Unit Plan Samantha Parks Millicent Brown Rachel White Time Frame This unit is designed to take 3-4 weeks, depending on the needs of students and the time allocated to mathematics class. Prerequisite Knowledge 1. Operate with whole numbers, fractions, and decimals. 2. Use graphs to organize, display, and interpret data. 3. Identify and organize information and look for patterns. 4. Identify patterns that grow at a constant rate and be able to generalize them. Example 1: 2, 4, 6, 8, are consecutive even numbers, just add 2 to the previous number to get the next one. Example 2: 3, 7, 11, 15, to get the next number, add 4 to the one before it. 5. Identify relationships that are modeled by linear functions. 6. Interpret slope as constant rate of change. 7. Interpret y-intercept as the starting value for y. 8. Recognize the relationships between slope and y-intercept in graphic, contextual, and algebraic representations of linear functions. 9. Solve linear equations and interpret the results in context. Learning Objectives 1. Identify relationships that are modeled with quadratic functions. (HSF-LE.A.1) 2. Interpret the slope as increasing or decreasing by a fixed amount with each unit change in x. (HSF-LE.A.1a) 3. Construct quadratic functions from a given graph, description of a relationship, or two input-output pairs (include reading these from a table). (HSF-LE.A.2) 4. In varying contexts, be able to interpret the range of x and y values and their representations in a quadratic function. (HSF-LE.B.5) 5. Identify the critical points of a quadratic function using factorization and apply them by constructing a rough graph. (HAS-APR.B.3) Common Core State Standards (Highlighted sections refer specifically to linear functions.): Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Construct viable arguments and critique the reasoning of others. 3. Model with mathematics. 4. Use appropriate tools strategically.
6. Look for and make use of structure. 7. Look for and express regularity in repeated reasoning. CCSS Algebra Standards: HSF-LE.A.1: Distinguish between situations that can be modeled with linear functions and with exponential (quadratic) functions. a. Prove that linear functions grow by equal differences over equal intervals, and that exponential (quadratic) functions grow by equal factors over equal intervals. HSF-LE.A.2: Construct linear and exponential (quadratic) functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). HSA-APR.B.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. Overview and Learning Activities This unit on quadratic functions is presented at the introductory level. It is assumed that the background students have needs significant review and relearning. 1. Introduction to Quadratic Graphs: To introduce students to the shape of a parabola, which is the graph of a quadratic function, we will begin by conducting an in-class experiment. The teacher will select three students to assist in the demonstration. The teacher will have two of the students stand in a line parallel to the white-board, facing each other. One student will toss a ball underhand to the other student, in an arcing fashion. The third student will record the time it takes for the ball to travel from the first student s hands to the second student s hands. As the ball is tossed, the teacher will draw the path of the ball on the board. Once the shape is drawn, the teacher then draws the x-axis, explaining that each of the base points lies on this axis because they are the starting and ending times of the ball. The x-axis is labeled Time. The teacher will then draw the y-axis, explaining that this represents the ball s height. The y-axis is labeled Height. The teacher will then ask the class to identify the highest point of the ball s path on the graph drawn. It is then explained that this is called an Absolute Maximum. Students will then observe how on the left side of this point, the height is increasing, and on the right side of this point the height is decreasing. The teacher will then draw a dotted line vertically through this point to show students the symmetry of the graph. Lesson 1 assumes students have little experience in interpreting quadratic graphs. It is designed to help build visual understanding of quadratic graphs by having them act out a situation that models a quadratic function. Takes about one 50- minute class periods to complete.
2. How Quadratic Functions Arise from Linear Functions: Students will use the program ParabolaX to observe how the product of two linear functions yields a quadratic function. Students should be allowed to play with the program and manipulate the linear functions to see how the quadratic function is affected. The teacher should emphasize that when both lines are increasing or both lines are decreasing, the parabola opens upward. When one line is increasing and the other is decreasing, the parabola opens downward. Eventually, students are introduced formally to these concepts through definition and concrete examples in Lesson 4. Takes approximately one 50-minute class period to complete. 3. Introduction to Writing Quadratic Equations: To begin, we will have students explore how changes in the coefficients a, b, and c affect quadratic functions of the form y = ax! + bx + c. This will be done by having the students graph and make value tables for three sets of equations. In each set, either the a, b, or c value will be varied in order to show the effect of these changes on the graph. When the coefficient a is greater than 0, the graph will open upward. When the value of a is greater than 1, as the value of a increases, y values increase at a faster rate with respect to x, which causes the parabola to appear more narrow. When the coefficient a is less than 0, the graph will open downward. When the value of a is less than -1, as the value of a decreases, y values decrease at a faster rate with respect to x, which causes the parabola to appear more narrow. When a is between -1 and 0, or 0 and 1, then y increases or decreases at a slower rate and causes the graph to appear wider. When a=0, we have a linear function. Every quadratic function has either a maximum or minimum that lies on the axis of symmetry. This x-value of the axis of symmetry can be found using the equation x =!. The changing coefficient b shifts the axis of symmetry and the!! maximum or minimum value of the parabola. When b is negative, it causes the graph to shift negatively on both the x and y-axis. When b is positive, it causes the graph to shift positively on both the x and y-axis. The coefficient c determines the y-intercept of the parabola. When x = 0, y = c. This will take two 50-minute class periods, one to introduce the coefficients a and c, and one to introduce the coefficient b. 4. Introduction to Solving Quadratic Equations: Students learn to solve quadratic functions using algebra tiles starting with simple examples and working into more difficult ones. They learn to multiply two linear functions together and how to factor quadratic functions. We will do this using Multiplying and Factoring with Algebra Tiles, from Teaching and Learning High School Mathematics by Charlene Beckmann, Denisse Thompson, and Rheta Rubinstein. The first part of Lesson 4 builds student intuition for symbolic manipulation for solving quadratic equations. By the end of the lesson, students
are able to solve equations of the form y = (ax + b)(cx + d) and y = ax! + bx + c. Then we will upon their foundational intuition skills from the Algebra Tiles and teach the students how to directly solve equations of the form y = (ax + b)(cx + d) and y = ax! + bx + c. Students will learn how to expand quadratic equations using the foiling method. Then students will learn how to factor by completing the square solving for x when y=0. They will also learn to use the Quadratic Formula to solve for x when the equation is not easily factored. Takes approximately two 50-minute class periods. 5. How to Interpret a Quadratic Function from a Table of Values: Students will learn how to determine that a table represents a quadratic function by examining the first and second differences of y. Students will discover that the first differences in the y values are increasing at a constant rate with respect to x. They will also find that the second differences in the y values are constant and thus define a constant function. The goal is to have the students be able to identify a quadratic function from a given table of values by finding that the second difference in the y values is constant. This takes one 50-minute class period. 6. Recognize a Quadratic Function: The students will be given 10 cards, 5 of which model quadratic function, and 5 that could be easily mistaken as quadratic functions. Each of the two sets of cards have one graph, one equation, one table, one children s story, and one real life context. The students should be able to apply their previous knowledge to determine which of the cards represent a quadratic function and which do not. Takes approximately one 50-minute class periods. 7. Applying Quadratic Functions to Real Life Situations: Students are provided with problems that represent real life situations of quadratic functions. Students will be given a Problems in Context worksheet which has been created for further understanding of quadratic functions. They will apply their previous findings about quadratic functions to solve these problems. Students are asked to create graph, table, and regression line representations of the functions. Additionally the students will use the formulas they generated to answer specific questions pertaining to the situation. This allows students to see how quadratic functions can be used in every-day life. Takes two 50-minute class periods in which students work small in collective groups. 8. Ball Bounce Activity: This experiment is meant to act as a summary of all of the previous lessons in this unit. It touches on creating a graphic representation of a quadratic function from a real life context. This includes creating a regression equation, interpreting the graph relative to the motion of the ball, and comparing how the coefficients a, b, and c change relative to the ball used. This will take one 50 minute class period to complete the experiment and one 50 minute class period to complete the experimental analysis.
Unit Project: Actions Modeling Quadratic Functions For the end of unit project, the students will choose a activity in their own life that can be modeled by a quadratic function. They will be required to document the activity with pictures of them performing the activity. Then they will be asked to explain how their activity represents a quadratic function as well as creating a formula that models the action. Students must then explain their findings to the class in an interactive and creative way in a 5 minute presentation. This will take 5 minutes per student to present, so about three 50 minute class periods. Materials Needed 1. Ball (Basketball, Bouncy Ball, Tennis Ball) for Introduction to Quadratic Graphs activity, On a Roll, and Ball Bounce. 2. Stop Watch for Introduction to Quadratic Graphs activity. 3. Device compatible with ParabolaX application. 4. Algebra tiles, one set per pair of students. 5. Computer with application Geogebra. 6. Function Cards (Examples and Non-examples). 7. Children s counting books, including One Thousand Monsters (non-example) and Alaska s Twelve Days of Summer, and other children s stories with quadratic patterns in them. 8. CBR, one per group of 4 students. 9. Graphing calculators with statistical analysis and plotting capabilities. 10. Metric tape measure, one per group, for Ball Bounce. 11. Plastic rain gutter for On a Roll activity. Assessments 1. Daily warm-up problems 2. Daily observations of students interacting in small groups 3. Daily homework 4. Quizzes or short writing assignments 5. Project described in Unit Project 6. Test over the unit