e39a5f15-9508-4ff9-9d00-5200ceb11b50.html

<h3>Activity (10 minutes)</h3>
<p>To introduce the task, make sure students can easily visualize what shape is made by water coming out of a jet or fountain. Ask students what shape water makes when it comes out of a drinking fountain, and either sketch the shape for all to see or invite a student to sketch it. Ask if they have seen fountains with jets that shoot water. Consider showing examples of a few, including videos if possible. Here is an example image:</p>
<p><img height="310" src="https://k12.openstax.org/contents/raise/resources/8dce6d05c183e18c4fa9e66e657820b3fc6fc88a" width="465"></p>
<p>An important point to highlight is that the water makes a curved path through the air that can be modeled by a quadratic function.</p>
<p>Then take students through a simplified version of the task, with one statue and one water jet. Here is an image to use if needed. The trapezoid represents the statue and the red dot is the location of the water jet.</p>
<h4>Student Activity </h4>
<p>The trapezoid represents the statue and the red dot is the location of the water jet.</p>
<p><img height="77" src="https://k12.openstax.org/contents/raise/resources/8399bc87a4780f101a00eceb82f8dce8db6bd47c" width="202"></p>
<ol>
  <li> What would you need to know in order to write an equation for a path for the water that would make it go over the statue? </li>
  <li> What is a reasonable height of the statue and a reasonable distance from the jet to the statue? </li>
  <li> What strategy would you use to find the equation for the path of the water? </li>
</ol>
<h4>Student Response</h4>
<ol>
  <li> I would need to know the distance from the jet to the statue and the height of the statue. </li>
  <li> Possible answer: The height of the statue is 6 feet and the jet is 8 feet from the statue. </li>
  <li> Use the given distances to represent the statue and the location of the jet on a coordinate grid, then write a vertex form equation by choosing a reasonable vertex and adjusting parameters as needed. </li>
<ol>
<p>Have students share their ideas with a partner before inviting students to share with the class. If no student suggests estimating a reasonable vertex for the water&rsquo;s path and writing a vertex form equation, ask students to discuss this idea. Vertex form is a very convenient way to write the equation if we know the size and distance of the statue so we can represent it on a coordinate grid.</p>
<p>Sample answer for trapezoid example: If the fountain is 8 feet from the statue and the statue is 6 feet tall and 4 feet wide, then a possible vertex is \((-10, 7)\). Using vertex form to solve for \(a\): \(y=a(x+10)^2+7\). Using \((0,0)\) as a point to solve for a: \(0=a(0+10)^2+7\), \(-7=100a\), \(a=\frac{-7}{100}\). Equation is \(y=\frac{-7}{100}(x+10)^2+7\). Have students verify the height is over 6 feet as it goes over the statue. \((-8, 6.72)\) and \((-10, 7)\)</p>
<h4>Anticipated Misconceptions</h4>
<p>Students may have trouble getting started. It may help for students to use grid paper to sketch the trapezoid and write some measurements for the distance from the fountain to the statue and the height of the statue. They may also need to be reminded that the view of the fountain is from the side.</p>
<h4>Activity Synthesis</h4>
<p>The purpose of this activity is to prepare students to design their own fountain in the next activity. Students will be required to communicate how they used mathematics to choose the design of their fountains. A review of using the vertex form to write a quadratic equation will be helpful. Students may also use technology to test their equations.</p>