02.+Engaging+Questions+and+Big+ideas

toc =Engaging Questions and Big Ideas =

**Concepts**
• solutions of quadratic equations (with real roots) - by factoring - by completing the square - by using quadratic formula • understanding of the equivalence of the methods

**Skills**
• find (solutions to quadratic equations with real roots) - factor (quadratic equations) - complete (as in completing the square) - use (quadratic formula) • demonstrate (an understanding of the equivalence of the methods)

**Big Ideas**
• Algebraic methods for solving equations provide exact solutions. • Quadratic equations can be solved in different ways. • Many real world phenomena are modeled by quadratic equations. • Why use algebraic methods to solve equations? • What is a quadratic equation? • How can we solve quadratic equations? • Why do different ways of solving equations yield the same answers? • What kinds of situations can a quadratic equation model?
 * Essential Questions**


 Nationals vs. Red Sox. You are watching a baseball game at RFK stadium. It’s the Nationals versus the Red Sox. One of the players hits a baseball at a height of 3 ft and with an initial upward velocity of 88 ft/sec. As the ball flies through the air you wonder, “Is there any mathematical explanation for its path?”

Let x represent time in seconds after the ball is hit, and let y represent the height of the ball in feet. Write an equation for the height as a function of time using the model 2 y = −16x − v0x + h0, where 0 v is the initial velocity and 0 h is the initial height. 1. Write an equation to find the time(s) when the ball is 24ft. above the ground. 2. Use the quadratic formula to solve the equation from Step 1. What is the real-world meaning of each of your solutions? Why are there two solutions? 3. The vertex of this parabola has a y-coordinate of 124. Explain why the ball will reach a height of 124ft only once. 4. Write and solve an equation to find the time when the ball reaches a height of 124ft. 5. At what point in the solution process does it become obvious that there is only one solution to the equation in Step 4? Explain how the quadratic formula confirms that there is only one solution to the equation in step 4. 6. Write an equation to find the time when the ball reaches a height of 200ft. What happens when you try to solve this impossible situation with the quadratic formula?