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Exploring Quadratic Equations

Grade Focus
Grade 9, Grade 10, Grade 11
Age Level
14, 15, 16
Subject
Math
Technology Integration Activity
PowerPoint Presentations, Video Productions, Webpage Creation
Estimated Time of Completion
5 hours
Author: Sara AuthenticatedPublicUser Last modified: 05/01/2010

Exploring Quadratic Equations

Downloads for this lesson plan

Student Sketch (EXE, 180K)

Quadratic Forms (EXE, 143K)

Presentation Rubric (EXE, 11K)

Introduction

This lesson covers several days and introduces the Quadratic Fquation.  Students use laptops with Geometer's Sketchpad (GSP) software to explore properties of quadratics, make connections with polynomials and linear equations.  The Quadratic equation is not introduced until the end. The students should have multiple graphs and will be able to measure coordinates, identify x- intercepts, y intercepts, the vertex and line of symmetry.

Prerequisite Experience

·        This lesson builds and continues from the  Polynomials Unit
·        Students should understand degree of polynomial
·        Linear equations, how to solve for x-intercept and y intercept
·        The hand out is broken into sections so that in the end they slowly solve/build the Quadratic.
·        Students have prior experience with Geometer’s Sketchpad, coordinates and graphing lines

·       Students should have experience with PowerPoints

·     They can view Video Tutorials found at http://edu.learnit-teachit.org/technology/PowerPoint_Presentations/ for additional help

Teacher Prep Time

It requires very little preparation as long as the software is installed on each student's computer.  Students do better with this lesson if they have had prior experience with GSP software and laptops.

Project

Student copy graphs from Geometer's Sketchpad to create presentations that explain vocabulary associated with

Quadratic equations as the process of solving the Quadratic Equation.

Pressentation of findings can be in the form of a Movie, Glogster, Cartoon, Video Thread

Assessment/Grading

Use the attached rubric for student presentation

Lesson Plan Details

Engage

Teacher Questions

·   What is the Quadratic Equation?
·    How are quadratic functions different from linear equations?
·    How does the formula look the same?
·    How does the formula look different?

·    What website helped to answer your questions best?Why?

 Student Instructions

·         Students work in pairs or individually and search the Internet to find the answers to the questions.   Teacher moves around classroom.

 ·         Students take turns sharing answers to their questions 

Explore

Teacher Directions


With given examples and open ended questioning.  Students guide discussion, Teacher is more of a facilitator 

Students should observe and make a connection with the degree of the polynomial and ask to continue with more examples.  If not keep guiding 

Students should observe a lineStudents should observe a parabola
Students should observe a parabola on top of existing curve and make the connection between multiplying polynomials and the standard form 

Continue with examples and discussion 

Students should identify x intercepts – roots
Students should identify y intercept
Students should identify vertex and line of symmetry

Student Instructions Using Geometer's Sketchpad

Write: ax2 + bx +c = 0; Write: ax +  b = 0
Write equation & In Geometer’s Sketchpad  

Graph, plot function x - 4 = y
Graph, plot function x2 + x - 4 = y 

In a new document
Graph, plot function 3x - 4 = y
Graph, plot function x2 + 3x - 4 = y
Graph, plot function (x + 4) (x – 1) = y  

Graph, plot function (x + 1) (x + 1)
Graph, plot function (x - 1) (x - 1)
Graph, plot function (x + 1) (x + 1)
Graph, plot function x2 + x - 4 = y
Graph, plot function -x2 + x - 4 = y   

Explain

Teacher Observations

Students should be able to explain to a partner what the x intercepts are… the root or solution. 


 

Student should explain the axis of symmetry by reading the graph


Student will be able to pick values left and right of the vertex and make a table of x and y values   

Student Instructions 

Students fold their paper at the vertex, name coordinates of the point

Standard equation is written on the board

Practice writing in standard form
     2x2  - 4 + 3x = 02
     x2  + 3x – 4 = 0  

Practice identifying a =, b = , c =

Students are introduced to the equation and solve for the vertex

Finally making tables and graphing on paper

Elaborate

Quadratic  Equation is introduced and students use a calculator to solve multiple problems

Students can use GSP software to graph and solve

Students should demonstrate understanding of what it means to solve a Quadratic equation through either graphing of formulas

Students should understand that a Quadratic Equation must be written in standard form before the quadratic formula can be used

 

Evaluate

 

 

Students solve various quadratics

 

Create a PowerPoint to explain the process and vocabulary associated with Quadratic Equations

 

Students are also assessed through standard Unit Test 

 

Extend

Find real life applications of Quadratic functions 

The following was found at www.purplemath.com 

An object in launched directly upward at 64 feet per second (ft/s) from a platform 80 feet high. What will be the object's maximum height? When will it attain this height?

Hmm... They didn't give me the equation this time. But that's okay, because I can create the equation from the information that they did give me. The initial height is 80 feet above ground and the initial speed is 64 ft/s. Since my units are "feet", then the number for gravity will be 16, and my equation is:

    s(t) = –16t2 + 64t + 80

They want me to find the maximum height. For a negative quadratic like this, the maximum will be at the vertex of the upside-down parabola. So they really want me to find the vertex. From graphing, I know how to find the vertex; in this case, the vertex is at (2, 144):

    h = –b/2a = –(64)/2(–16) = –64/–32 = 2
    k = s(2) = –16(2)2 + 64(2) + 80 = –16(4) + 128 + 80 = 208 – 64 = 144

But what does this vertex tell me? According to my equation, I'm plugging in time values and extracting height values, so the input "2" must be the time and the output "144" must be the height.   Copyright © Elizabeth Stapel 1999-2009 All Rights Reserved

    It takes two seconds to reach the maximum height of 144 feet.