Real+World+Quadratic+Equations

= = toc =Pre-Assignment= Have the students take a digital picture of a parabolic curve in the real world for homework the night before or as part of the beginning portion of the class. The students may use a digital camera or a smartphone to take the picture and then upload the picture to their computer. (Option: have students work in pairs or groups of three in order to take pictures if this is a part of the start of the lesson) //Suggestions to offer students: McDonald's sign, bridge, water fountain, hose//

=Content Knowledge Required:= Previous discussion of vertex form of a quadratic equation and identifying the components of a parabola in the graph. =Fitting a Quadratic Equation to a Real-World Picture= (**to do: create a screencast guiding students through this process for them to refer back to**)

Using GSP to obtain the vertex and a point on a parabola
1. Open Geometer's Sketchpad and your picture on your computer. 2. Use Control-C to copy the picture to your clipboard. 3. Use Control-V to paste the picture into your GSP sketch. 4. Go to **Graph - Define a Coordinate System** and you will see a coordinate grid appear on top of your picture like below: (Option: have students move the coordinate system to place the origin at a more convenient location, such as the bottom left of the parabola.) 5. Construct a point at the vertex of the parabola and any other point. 6. Select these two points and then go to **Measure - Coordinates**.

Calculating the equation for the curve using the vertex and a point
Using the two points on your curve and the vertex form of a parabola, calculate the equation for the parabola in your picture. Use a screencasting software to record you explaining your calculation and how you arrived at it

Using GSP to check your equation
1. Go to **Graph - Plot New Function** and type in your equation.

Possible Extensions
1. Have students expand their equations into standard form and then pair up. Each partner gives their standard form equation to the other and then attempts to calculate the vertex of their partner's parabola. 2. Extend this lesson to connect to transformations of graphs, relating the shifts to the changes in the parent graph and its equation. 3. Extend this lesson to have students plot a number of points of the graph and then use the plot feature of the TI to derive a best fit equation, examining how the R^2 value relates to its fit.

=Project Submission= 1. A screencast explaining the derivation of your equation. 2. A GSP file with your picture, vertex and point coordinates defined, and equation displayed and plotted. Additionally identify the vertex, whether it is a minimum or maximum, the intercepts, and the axis of symmetry.

=Reflection= Answer the following questions in a blog post tagged with **morrisonalg2, quadequations, realworld** : 1. How well did your quadratic equation fit the parabola in your image? Can you make any conjectures about why it is or is not a good fit? 2. How do you think we might go about finding the equation for a parabola if we are not given the vertex, but rather only points on the parabola? Is there a minimum number of points we would need to find the equation if we are not given the vertex? 3. When is the vertex form of a quadratic equation useful and when may it not be as useful? What other forms might be helpful and when?

=Project Rubric= //edited from Leslie Williams//
 * (to do: continue to modify rubric to assess reflection)**
 * ** Content & Skills ** || ** Advanced ** || ** On-target ** || ** Novice ** ||
 * **Calculation** || * Derivation of formula is clear and correct.
 * Derivation of formula makes sense and is correct.
 * Derivation of formula is easy to follow and is correct.
 * Arithmetic has no flaws. || * Derivation of formula is clear.
 * Derivation of formula makes sense.
 * Derivation of formula is easy to follow.
 * Arithmetic has few flaws. || * Derivation of formula is unclear.
 * Derivation of formula makes little sense.
 * Derivation of formula is difficult to follow.
 * Arithmetic has many flaws. ||
 * ** Understanding of Concepts ** || * Understands quadratic equations and can expand on the understanding.
 * Understands all the parts of a parabola and how to identify them.
 * Understands how to apply the parts of the vertex form of the equation of a quadratic equation. || * Understands quadratic eqautions.
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 12px;">Understands all the parts of a parabola and how to identify them.
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 12px;">Understands how to use the vertex form of the equation of a quadratic equation. || * <span style="font-family: 'Comic Sans MS',cursive; font-size: 12px;">Does not completely understand quadratic equations.
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 12px;">Does not completely understand all the parts of a parabola and how to identify them.
 * <span style="font-family: 'Comic Sans MS',cursive; font-size: 12px;">Does not completely understand how to apply the parts the vertex form of a quadratic equation. ||