Insights Into Algebra 1: Teaching for Learning
Quadratic functions model a number of real-world situations. They describe the path of a ball in flight, represent a cross section of a headlight’s reflector, and enable business people to forecast sales.
A quadratic function is a second degree equation – that is, 2 is the highest power of the independent variable. Written in standard form, the equation y = ax2 + bx + c (a 0) represents quadratic functions.
When graphed in the coordinate plane, a quadratic function takes the shape of a parabola. To see a parabola in the real world, throw a ball. The path the ball traces as it travels through the air – in an arc to its highest point, then back to the ground in a similar arc – is a parabola. Naturally, the ball bounces after hitting the ground, and each time it does so, it traces another parabola.
In another example, suppose a builder wants to build a parking lot that is rectangular in shape and measures 250 feet around three of the four sides. Write an equation that models the area of the parking lot as a function of the width of the parking lot. Find the dimensions of a parking lot that will enclose the greatest area. If the width of the parking lot is x, then the length of the parking lot is 250 – 2x. So, the area of the parking lot can be modeled by A = (250-2x)x. A graph of the function shows that the maximum area occurs at the vertex of the parabola. This point is located at (62.5, 7812.5). This means that the width of the parking lot with the greatest area is 62.5 feet, the length is 125 feet, and the area is 7812.5 square feet.
- During freefall, the distance d (in feet) that an object falls is represented by the quadratic function d = 16t2, where t is the time (in seconds) that the object has fallen.
- Quadratic functions describe the relationship between height (from the ground) and time (in seconds) of a ball as it bounces.
- A rectangle with a border of 100 feet has perimeter 2l + 2w = 100, which can be rewritten as:
The quadratic function that represents the area:
A quadratic function is an equation of the form y = ax2 + bx + c (a 0). Its graph is a parabola.
Another widely accepted definition is: A quadratic polynomial is a polynomial of the second degree – that is, a polynomial of the form ax2 + bx + c. A quadratic function is a function f whose value f(x) at x is given by a quadratic polynomial. If f(x) = ax2 + bx + c, then the graph of f is the graph of the equation y = ax2 + bx + c and is a parabola with vertical axes.
(Source: James, Robert C. and Glenn James. Mathematics Dictionary (4th edition). New York: Chapman & Hall, 1976.)
Role in the Curriculum
In Algebra 1, students explore linear functions in great detail, and they learn to graph lines by using tables and plotting points. They can use the same skills to explore quadratic functions. Indeed, it is most appropriate for students to first graph quadratic functions by making a table of x and y values and plotting points. This way, in addition to discovering the parabolic shape of quadratic equations, they can practice using previously acquired skills.
But quadratic equations provide more than just an opportunity for embedded review of graphing. The exploration of functions – particularly quadratic functions and cubic functions – allows students to investigate important topics and explore mathematical patterns. The National Council of Teachers of Mathematics (NCTM) recommends in Principles and Standards for School Mathematics (PSSM):
Students should have substantial experience in exploring the properties of different classes of functions. For instance, they should learn that the function f(x) = x2 – 2x – 3 is quadratic, that its graph is a parabola, and that the graph opens “up” because the leading coefficient is positive. They should also learn that some quadratic equations do not have real roots and that this characteristic corresponds to the fact that their graphs do not cross the x-axis. And they should be able to identify the complex roots of such quadratics.
In addition to identifying properties of a parabola from the standard form (y = ax2 + bx + c), it is also important for students to recognize the properties of a parabola in the vertex form (y = a (x – h)2 + k). Of these two forms, the vertex form provides more valuable information to assist in graphing the function.
In vertex form, the point (h, k) is the vertex of the parabola, and the value of adetermines the vertical stretch or shrink of the parabola. A positive value of awill make the parabola open upward, while a negative value of a will make the parabola open downward.
Understanding why and how the transformations alter the graph is important. Students need to understand the effects of a, h, and k in the equation y = a (x – h)2 + k.
To understand the effects of k, students can compare the functions y = x2 and y = x2 +3. The table below shows the pattern of values.
By looking at both the table of values and the graph, students will begin to understand that k produces a vertical shift; in this case, k = 3 shifts the parabola up 3 units. In general, the graph is shifted k units up if k is positive, and k units down if k is negative.
To understand the effects of h, students can compare the functions y = x2 and y = (x + 3)2. The table below shows the patterns of values.
By looking at both the table of values and the graph, students will begin to understand that h produces a horizontal shift; in this case, h = -3 shifts the parabola 3 units to the left. In general, the graph is shifted h units to the right if h is positive, and h units to the left if h is negative. Expressing the equation in vertex form, y = (x + 3)2 = (x- (-3))2.
To understand the effects of a, students can compare the functions y = x2 and y = 3x2. The table below shows a pattern of values.
By looking at both the table of values and the graph, students will begin to understand that a produces a vertical stretch or shrink of the parabola; in this case, a = 3 stretches the parabola vertically by a factor of 3 and makes it appear “skinnier”. In general, the parabola is stretched by a factor of a. In addition, if the value of a is negative, the parabola will open downward instead of upward.
Because the standard and vertex forms of a quadratic function reveal different pieces of information, it is important for students to recognize both and be able to convert one to the other. While symbolic manipulation receives less attention today than in the past, it is a necessary skill for converting a quadratic function from the standard form to the vertex form.
As its name implies, the vertex form is more useful when it’s important to know the vertex. When a salesman wants to know the maximum profit, or when a rocket scientist needs to know the maximum height of a projectile, the vertex form is preferable. However, when it’s more important to know the x-intercepts – for instance, when a pilot wishes to determine the point at which a package dropped from a helicopter will land – the standard form is preferable, because the well-known quadratic formula can be used to determine the roots (or solutions) of the equation.
(The quadratic formula is
where a, b, and c represent the constants in y = ax2 + bx + c.)
The links below are to pages within stable sites and are current as of the date of publication of this workshop. Due to the ever-changing nature of the Web, it is possible that some links may change. Should you reach a non-working link, we recommend entering a couple words from its description into the site’s search function, or into a Web-based search engine.
NCTM Middle Grades Algebra Standard
This Web page describes what students should know and be able to do algebraically in grades 6-8, and offers suggestions for the type of classroom activities necessary to develop conceptual understanding.
NCTM High School Algebra Standard
This Web page describes what students should know and be able to do algebraically in grades 9-12, and offers suggestions for the type of classroom activities necessary to develop conceptual understanding.
Quadratic Functions Lessons and Resources:
The Quadratic Function – Descartes
Designed by the Ministry of Education of El Salvador, this site was intended to demonstrate the power of Descartes, a Web-based applet. However, the content used in the demonstration revolves around quadratic functions, and the examples provide interactive content for a secondary classroom.
Quadratic Equation Worksheets
EdHelper.com provides this collection of worksheets involving various aspects of quadratic equations.
Seymour, Dale and Margaret Shedd. Finite Differences: A Pattern-Discovery Approach to Problem-Solving. Palo Alto, CA: Dale Seymour Publications, 1997.
This book, though not exclusively focused on quadratic equations, teaches a method for identifying polynomial functions when given a pattern of coordinates.
Resources on Collaborative Learning:
“Collaborative Learning Enhances Critical Thinking”
In the Journal of Technology Education, scholar Anuradha A. Gokhale examines the effectiveness of individual learning versus collaborative learning in enhancing drill-and-practice skills and critical-thinking skills.
Andrini, Beth. Cooperative Learning & Mathematics (Grades K-8). Kagan Cooperative Publishing.
This book offers various cooperative learning activities that can be used for elementary and middle school mathematics.
Kagan, Spencer. Cooperative Learning. Kagan Cooperative Publishing.
Spencer Kagan provides information and tips on forming teams, classroom management, and lesson planning, and provides some of the research and theory that supports cooperative learning.
Kushnir, Dina. Cooperative Learning & Mathematics High School Activities (Grades 8-12). Kagan Cooperative Publishing.
This book offers various cooperative learning activities that can be used in a high school mathematics class.
Resources on Alternative Assessment:
“Inside the Black Box: Raising Standards Through Classroom Assessment”
Written by researchers Paul Black and Dylan William, this article from the Phi Delta Kappan (Oct. 1988) argues that assessment is an essential component of classroom work and that its development can raise standards of achievement.
Student Self Assessment
This site gives links to resources that will help involve students in the assessment process.
Introduction to Performance Assessment Scoring Rubrics (Chicago Public Schools)
A tutorial on creating rubrics for classroom tasks and several examples of math rubrics from various organizations.
“The Limits of Standardized Tests for Diagnosing and Assisting Student Learning”
This article, presented by the National Center for Fair & Open Testing, discusses the limits of standardized testing for student learning.
General Math Rubric
Consider this math rubric for use in the classroom.
Arter, Judith A. and Jay McTighe. Scoring Rubrics in the Classroom. Corwin Press, 2000.
The authors of this book instruct teachers on how to use rubrics to be more consistent in judging student performance, and how to help students become more effective at addressing their own learning.
Danielson, Charlotte and Elizabeth Marquez. A Collection of Performance Tasks and Rubrics: High School Mathematics. Larchmont, NY: Eye on Education, 1998.
This book contains information on creating and using performance tasks, helpful suggestions for using rubrics to grade tasks, and a collection of tasks ready for classroom.
Workshop 1 Variables and Patterns of Change
In Part I, Janel Green introduces a swimming pool problem as a context to help her students understand and make connections between words and symbols as used in algebraic situations. In Part II, Jenny Novak's students work with manipulatives and algebra to develop an understanding of the equivalence transformations used to solve linear equations.
Workshop 2 Linear Functions and Inequalities
In Part I, Tom Reardon uses a phone bill to help his students deepen their understanding of linear functions and how to apply them. In Part II, Janel Green's hot dog vending scheme is a vehicle to help her students learn how to solve linear equations and inequalities using three methods: tables, graphs, and algebra.
Workshop 3 Systems of Equations and Inequalities
In Part I, Jenny Novak's students compare the speed at which they write with their right hands with the speed at which they write with their left hands. This activity enables them to explore the different types of solutions possible in systems of linear equations, and the meaning of the solutions. In Part II, Patricia Valdez's students model a real-world business situation using systems of linear inequalities.
Workshop 5 Properties
In Part I, Tom Reardon's students come to understand the process of factoring quadratic expressions by using algebra tiles, graphing, and symbolic manipulation. In Part II, Sarah Wallick's students conduct coin-tossing and die-rolling experiments and use the data to write basic recursive equations and compare them to explicit equations.
Workshop 6 Exponential Functions
In Part I, Orlando Pajon uses a population growth simulation to introduce students to exponential growth and develop the conceptual understanding underlying the principles of exponential functions. In Part II, a scenario from Alice in Wonderland helps Mike Melville's students develop a definition of a negative exponent and understand the reasoning behind the division property of exponents with like bases.
Workshop 7 Direct and Inverse Variation
In Part I, Peggy Lynn's students simulate oil spills on land and investigate the relationship between the volume and the area of the spill to develop an understanding of direct variation. In Part II, they develop the concept of inverse variation by examining the relationship of the depth and surface area of a constant volume of water that is transferred to cylinders of different sizes.
Workshop 8 Mathematical Modeling
This workshop presents two capstone lessons that demonstrate mathematical modeling activities in Algebra 1. In both lessons, the students first build a physical model and use it to collect data and then generate a mathematical model of the situation they've explored. In Part I, Sarah Wallick's students use a pulley system to explore the effects of one rotating object on another and develop the concept of transmission factor. In Part II, Orlando Pajon's students conduct a series of experiments, determine the pattern by which each set of data changes over time, and model each set of data with a linear function or an exponential function.