rory hamilton usd 2661 quick quadratics ci 752a rory hamilton summer 2003

Rory Hamilton USD 266 1

Quick Quadratics

CI 752A

Rory Hamilton

Summer 2003


Quadratic Unit Algebra IIMaize High School

Graphing Quadratic FunctionsSolving Quadratic Equations by:

FactoringFinding Square RootsCompleting the SquareQuadratic Formula


Purpose

This action plan looks to further investigate graphing and solving quadratic equations. This action plan will look at integrating technology into our existing curriculum to improve the quality of instruction in our classroom and meet the standards set forth by the NCTM regarding technology in the classroom.


So What is a Quadratic?

Quadratics are functions whose graphs are in the form y= ax²+bx+c, where a0.

The graph of a quadratic function is a parabola.

The graph of the quadratic function y =ax²+bx+c represents all the values x and y that satisfy this equation..

The graph of y = x² is shown at the left.


Applications of Quadratics

“Vomit Comet” Stripped down KC-135 used to get

astronauts accustomed to weightlessness.

The weightlessness occurs when the jet climbs to roughly 33,000 feet, arcs over invisible peak, and then dives towards the ground.

For close to 23 seconds, the astronauts are weightless.

The path traveled by the KC-135 models a quadratic function similar to h(t) = 4800t - 160t²


Solving Quadratics by Graphing

When solving quadratic equations ax²+bx+c = 0, we are only interested in values of x that make the expression ax²+bx+c equal to 0.

These values termed zeros are represented by the points at which the graph of the function crosses the x-axis, since the y-values are 0.

The x-intercepts of the quadratic function are called the solutions or roots of the quadratic equation.

The graph at the left can be modeled by the equation y = x²-1, where the solutions or roots are at 1.


Technology for Graphing and Solving Quadratic Equations (1) Transformations of vertex form and standard form of quadratic

equations. www.exploremath.com/activ#2E578

Solving Quadratics by Graphing www.purplemath.com/module#2E566

Spreadsheet Analysis of quadratic functions and equations. http://mathforum.org/workshops/sum98/participants/sinclair/function.html

Green Globs Software tutorials allows students the opportunity to practice writing and graphing linear functions, quadratics, and conics.


Technology for Graphing and Solving Quadratic Equations (2) Families of Parabolas activity with graphing calculators. Students can pair up

and play battleship while learning basic function transformations. http://www.vh.cc.va.us/pc1spring99/Assignments/assignment_98126_function%20_families.htm#The%20family%20of%20quadratic%20functions

Investigating Parabolas with Graphing Calculators Activity. http://regentsprep.org/Regents/math/conics/Tparab.htm

Interactive Java Applet on Transformations of vertex form quadratics along with completing the square visual. http://www.rfbarrow.btinternet.co.uk/htmasa2/Quad2.htm

Excel Document with Graph help students visualize transformations of linear and quadratic functions.http://www.devon.gov.uk/dcs/maths/sec/res/alg/ma19.xls


Solving Quadratics by Completing the Square

When an equation does not contain a perfect square, you can create one by completing the square.

In a perfect square there is a relationship between the coefficient of the middle term and the constant term.

Take half of the coefficient of the middle term and square it and it completes the square.


Solving Quadratics by Completing the Square

Here is an example of completing the square.

Ex. x² + 6x Take half of 6 and square it (3)²= 9, then add to original We now have new expression

x² + 6x +9 Write as perfect square (x+3)²


Technology for Solving Quadratics by Completing the Square (1) Interactive Java Applet that shows transformations of quadratic

functions along with completing the square. http://www.rfbarrow.btinternet.co.uk/htmasa2/Quad2.htm

Java Applet Tutorial for Completing the Square. http://www.csm.astate.edu/algebra/csq.html

Online Tutorial with Additional Examples and Instruction. http://www.purplemath.com/modules/sqrquad.htm


Technology for Solving Quadratics by Completing the Square (2) Interactive Examples and Instruction. http://www.dessci.com

/en/reference/webmath/samples/javascript/completing_the_square/square.htm

Audio Lesson with interactive examples. GREAT VERBAL AND VISUAL!! http://www.math.hawaii.edu/~dale/flash/square.html


Solving Quadratics by Using Quardratic Formula

Last method taught in a quadratics unit.

Found by solving for x in the general form of a quadratic 0 = ax²+bx+c.

Works for any quadratic and is usually the “favorite method” by students since it always works.

Most difficult part of the lesson is getting the students to memorize the formula.


Solving Quadratics by Using Quardratic Formula

An example of solving using the quadratic formula is shown at the left.

The equation is 0 = 2x²-7x-5 It is also important for students to

know the nature of their answer by looking at the discriminant (b²-4ac).

If b²-4ac = 0 (1 real root) If b²-4ac > 0 (2 real, Q or I) If b²-4ac < 0 (2 imaginary) By evaluating the discriminant

students can visualize the nature of the graph.


Technology for Solving Quadratics by using Quadratic Formula (1) Sing Along to “Pop Goes the Weasel”. Helps students memorize the

quadratic formula. http://www.tcjc.cc.tx.us/campus_ne/mathfaculty/kbrown/Kim.html

Additional instructional video with practice exercises and assessment. http://spock.ulm.edu/~esmith/math093/unit5/5_7/5_7.htm

Additional Examples with instruction. http://www.purplemath.com/modules/quadform.htm


Technology for Solving Quadratics by using Quadratic Formula (2) Interactive Practice using Quadratic Formula. Allow students to plug

in any quadratic. http://www.bonita.k12.ca.us/schools/ramona/teachers/carlton/tutorialinteractives/inM-R/quadraticfolder/05quadsolver.html

This site has interactive quadratic generator and graph. It solves any quadratic and shows the graph as well. http://dev1.epsb.ca/math14_Jim/math9/strand2/factor2.htm

Quadratic Webquest that has numerous links to solving quadratics. http://www.scs.k12.tn.us/STT99_WQ/STT99/Collierville_HS/brashers/Webquest-Quadratics.htm


Ending with some HUMOR!

rory hamilton usd 2661 quick quadratics ci 752a rory hamilton summer 2003

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