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1. Quick Quadratics CI 752A Rory Hamilton Summer 2003 Rory Hamilton USD 266

2. Quadratic Unit Algebra IIMaize High School • Graphing Quadratic Functions • Solving Quadratic Equations by: • Factoring • Finding Square Roots • Completing the Square • Quadratic Formula Rory Hamilton USD 266

3. 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. Rory Hamilton USD 266

4. So What is a Quadratic? • Quadratics are functions whose graphs are in the form y= ax²+bx+c, where a0. • 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. Rory Hamilton USD 266

5. 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 Rory Hamilton USD 266

6. 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. Rory Hamilton USD 266

7. 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. Rory Hamilton USD 266

8. 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 Rory Hamilton USD 266

9. 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. Rory Hamilton USD 266

10. 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 Rory Hamilton USD 266

11. 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 Rory Hamilton USD 266

12. 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 Rory Hamilton USD 266

13. 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. Rory Hamilton USD 266

14. 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. Rory Hamilton USD 266

15. 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 Rory Hamilton USD 266