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<h4>Activity (15 minutes)</h4>
<p>In this activity, students continue to explore how quadratic functions can model the movement of a falling object.
They evaluate the function seen earlier \((d=16t^2)\) at a non-integer input, and then build a new function to
represent the distance from the ground of a falling object \(t\) seconds after it is dropped. To find a new expression
that describes the height of the object, students reason repeatedly about the height of the object at different times
and look for regularity in their reasoning.</p>
<p>The number 576 is chosen as the height (in feet) from which the object is dropped to make it more apparent for
students that the values in the two tables (distance fallen and distance from ground) record distances measured from
opposite ends. (Any value of at a whole-number \(t\) <span>could work. In this case, \(t=6\) is selected.)</span>
</p>
<h4>Launch</h4>
<p>Arrange students in groups of two, and suggest that they check in with each other after trying each question. To
facilitate peer discussion, consider displaying sentence stems or questions that students could use, such as:</p>
<ul>
<li> “Why do you think the object will have fallen that distance in 0.5 seconds?” </li>
<li> “How do you think the values in the first table are changing? What about in the second table?” </li>
<li> “How are the two tables alike? How are they different?” </li>
</ul>
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title">Support for English Language Learners</p>
<p class="os-raise-extrasupport-name">MLR 2 Collect and Display: Conversing</p>
</div>
<div class="os-raise-extrasupport-body">
<p>As students discuss their expressions with a partner, listen for and
collect the language students use to identify and describe what is the same and what is different between Elena and
Diego’s tables. Write the students’ words and phrases on a visual display and update it with connections to the
graphs introduced during the synthesis. Remind students to borrow language from the display as needed. This will
help students read and use mathematical language during their paired and whole-group discussions. </p>
<p class="os-raise-text-italicize">Design
Principle(s): Optimize output (for explanation); Maximize meta-awareness
</p>
<p class="os-raise-extrasupport-title">Provide support for students</p>
<p>
<a href="https://k12.openstax.org/contents/raise/resources/3765a3cff09304aea8d8db39295b1c00ffd0c12f" target="_blank">Distribute graphic organizers</a>
to the students to assist them with participating in this routine.
</p>
</div>
</div>
<br>
<h4>Student Activity</h4>
<p>Galileo Galilei, an Italian scientist, and other medieval scholars studied the motion of free-falling objects. The
law they discovered can be expressed by the equation \(d=16t^2\), which gives the distance fallen in feet as a
function of time, \(t\), in seconds.</p>
<p>An object is dropped from a height of 576 feet.</p>
<ol class="os-raise-noindent">
<li>How many feet does it fall in 0.5 seconds?<br>
<br>
<strong>Answer: </strong>4
</li>
</ol>
<ol class="os-raise-noindent" start="2">
<li>To find out where the object is the first few seconds after it was dropped, Elena and Diego created different
tables.</li>
</ol>
<p>Elena’s table:</p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">
Time (Seconds)
</th>
<th scope="col">
Distance Fallen (Feet)
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
0
</td>
<td>
0
</td>
</tr>
<tr>
<td>
1
</td>
<td>
16
</td>
</tr>
<tr>
<td>
2
</td>
<td>
64
</td>
</tr>
<tr>
<td>
3
</td>
<td></td>
</tr>
<tr>
<td>
4
</td>
<td></td>
</tr>
<tr>
<td>
\(t\)
</td>
<td></td>
</tr>
</tbody>
</table>
<br>
<p>Diego’s table:</p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">
Time (Seconds)
</th>
<th scope="col">
Distance from the Ground (Feet)
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
0
</td>
<td>
576
</td>
</tr>
<tr>
<td>
1
</td>
<td>
560
</td>
</tr>
<tr>
<td>
2
</td>
<td>
512
</td>
</tr>
<tr>
<td>
3
</td>
<td></td>
</tr>
<tr>
<td>
4
</td>
<td></td>
</tr>
<tr>
<td>
t
</td>
<td></td>
</tr>
</tbody>
</table>
<br>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> How are the two tables alike? <br>
<br>
<strong>Answer:</strong> They both have time in seconds and distance in feet as the two quantities being measured.
They both show 0–4 seconds and \(t\) seconds for time. They describe the same object falling.
</li>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> How are the two tables different? <br>
<br>
<strong>Answer:</strong> Elena’s table measures distance fallen, and Diego’s table measures distance
from the ground. They record measurements from opposite ends. Elena’s table starts with 0 feet for 0
seconds, and Diego’s starts with 576 feet for 0 seconds.
</li>
</ol>
</ol>
<ol class="os-raise-noindent" start="3">
<li>Complete Elena’s table. Be prepared to explain your reasoning.</li>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> What is the distance the object has fallen 3 seconds after it was dropped? <br>
<br>
<strong>Answer:</strong> 144
</li>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> What is the distance the object has fallen 4 seconds after it was dropped? <br>
<br>
<strong>Answer: </strong>256
</li>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="3" type="a">
<li> What is the distance the object has fallen \(t\) seconds after it was dropped? </li>
</ol>
</ol>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p><strong>Answer:</strong> \(16t^2\)</p>
<ol class="os-raise-noindent" start="4">
<li>Complete Diego’s table. Be prepared to explain your reasoning.</li>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> What is the object’s distance from the ground 3 seconds after it was dropped? <br>
<br>
<strong>Answer: </strong>432
</li>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> What is the object’s distance from the ground 4 seconds after it was dropped? <br>
<br>
<strong>Answer: </strong>320
</li>
</ol>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="3" type="a">
<li> What is the object’s distance from the ground \(t\) seconds after it was dropped? </li>
</ol>
</ol>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<p><strong>Answer: </strong>\(576-16t^2\)</p>
<br>
<h4>Video: Distance as a Quadratic Function of Elapsed Time</h4>
<p>Watch the following video to learn more about quadratic functions.</p>
<div class="os-raise-d-flex-nowrap os-raise-justify-content-center">
<div class="os-raise-video-container"><video controls="true" crossorigin="anonymous">
<source src="https://k12.openstax.org/contents/raise/resources/005a0fd918a3812ba158602b14d462d900fd0661">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/99b550bc7dd378abebde6439ebf81a0f0c4fedd3" srclang="en_us">
https://k12.openstax.org/contents/raise/resources/005a0fd918a3812ba158602b14d462d900fd0661
</video></div>
</div>
<br>
<br>
<h4>Activity Synthesis</h4>
<p>To help students make sense of the two functions, compare and contrast their representations (tables, equations, and
graphs) and discuss the connections between them. Ask questions such as:</p>
<ul>
<li> “How did you complete the missing values in the first table?” (Substituting 3 and 4 for \(t\) in
\(16t^2\) gives the distances fallen after 3 and 4 seconds.) </li>
<li> “What about those in the second table?” (The distance from the ground is 576 minus the distance
fallen, so we can use the values for \(t=3\) and \(t=4\) from the first table to calculate the values in the second
table.) </li>
<li> “Why do the values in the first table increase and those in the second table decrease?” (The distance
from the top of the building increases as the object falls farther and farther from it. The distance from the ground
decreases as the object falls closer and closer to it.) </li>
<li> “The expression representing the distance fallen shows \(16t^2\), and the other shows \(576-16t^2\). Why is
that?” (In the first function, the distance fallen, measured from where the object is dropped, will always be
positive. In the second function, what’s measured is the height from the ground, so the distance fallen needs
to be subtracted from the height of the building.) </li>
<li> “If we graph the two equations that represent distance fallen and distance from the ground over time, what
would the graphs look like? Try sketching the graphs.” </li>
</ul>
<p>Display graphs that represent the two functions and make sure students can interpret them. For example, they should
see that the \(y\)-intercept of each graph corresponds to the starting value of each function before the object is
dropped.</p>
<p>They should also notice that the difference in distance between successive seconds gets larger in both cases, hence
the curving graphs. (If the differences were constant, the graphs would have been straight lines.)</p>
<p><img alt="Graph with \(x\)-axis labeled time in seconds and \(y\)-axis labeled distance fallen in feet." height="305" src="https://k12.openstax.org/contents/raise/resources/2d0ccb37bd12cef6c2260ff9ae6a94a190371d0f" width="315"></p>
<p><img alt="Graph with \(x\)-axis labeled with time in seconds and \(y\)-axis with distance above the ground in feet." height="305" src="https://k12.openstax.org/contents/raise/resources/2cd4885abefe2984476e859b4087f4cf2b2cfa37" width="313"></p>
<p>Display the embedded applet for all to see. Ask students how the graph of the height of the object is related to the
path that the object takes as it falls.</p>
<h3>7.5.3: Self Check </h3>
<p class="os-raise-text-bold"><em>After the activity, students will answer the following question to check their understanding of the
concepts explored in the activity.</em></p>
<p class="os-raise-text-bold">QUESTION:</p>
<p>A small ball is dropped from a 350-foot-tall building. Which equation could represent the ball’s height,
\(h(t)\), in feet, relative to the ground, as a function of time, \(t\), in seconds?</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>
\(h(t)=350-16t\)
</td>
<td>
Incorrect. Let’s try again a different way: The \(t\) is squared in the formula. The answer is
\(h(t)=350-16t^2\).
</td>
</tr>
<tr>
<td>
\(h(t)=350-16t^2\)
</td>
<td>
That’s correct! Check yourself: The ball will fall from 350 feet, so subtract 16 times the time squared to
find the height at a given time.
</td>
</tr>
<tr>
<td>
\(h(t)=350+16t^2\)
</td>
<td>
Incorrect. Let’s try again a different way: Be sure to subtract, not add. The answer is
\(h(t)=350-16t^2\).
</td>
</tr>
<tr>
<td>
\(h(t)=\frac{350}{16t^2}\)
</td>
<td>
Incorrect. Let’s try again a different way: Subtract the \(16t^2\) to find the height. The answer is
\(h(t)=350-16t^2\).
</td>
</tr>
</tbody>
</table>
<br>
<h3>7.5.3: Additional Resources</h3>
<p class="os-raise-text-bold"><em>The following content is available to students who would like more support based on their experience with
the self check. Students will not automatically have access to this content, so you may wish to share it with
those who could benefit from it. </em></p>
<h4>Height as a Quadratic Function</h4>
<p>The general formula for a free falling object is:</p>
<p>\(h(t)=-16t^2+v_0t+h_0\).</p>
<p>Let’s look at the meaning of the coefficients in the formula:</p>
<p>In the formula, \(-16\) is the constant that has been determined by a combination of Newton’s Laws and
Earth’s weight and is used with the units feet/second\(^2\).</p>
<p>\(v_0\) is the vertical speed of the object. If the object is simply dropped, the vertical speed is 0. If the object
were thrown upward, this value would be positive, and if the object were thrown down, this value would be negative.
</p>
<p>\(h_0\) is the initial height of the object.</p>
<p>A man drops his keys from a restaurant at the top of a 196-foot-tall building. If the keys are free falling, will
they reach the ground within 3 seconds?</p>
<p><strong>Step 1</strong> - Write an equation for the height at a given time, \(t\).<br>
\(h(t)=196-16t^2\)</p>
<p><strong>Step 2</strong> - Substitute \(t=3\) into the function.<br>
\(h(3)=196-16(3)^2\)</p>
<p><strong>Step 3</strong> - Evaluate.<br>
\(h(3)=196-144=52\)</p>
<p>At \(t=3\) seconds, the keys will still have 52 feet to travel to hit the ground.</p>
<h4>Try It: Height as a Quadratic Function</h4>
<p>Will the keys in the situation above reach the ground after 5 seconds?</p>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<h5> Solution</h5>
<p>Here is how to determine if the keys reach the ground in 5 seconds:</p>
<p><strong>Step 1</strong> - Write an equation for the height at a given time, \(t\).<br>
\(h(t)=196-16t^2\)</p>
<p><strong>Step 2</strong> - Substitute \(t=5\) into the function.<br>
\(h(5)=196-16(5)^2\)</p>
<p><strong>Step 3</strong> - Evaluate.<br>
\(h(5)=196-400=-204\)</p>
<p>The keys reach the ground before \(t=5\) seconds since substituting in 5 creates a negative height, which implies
they have fallen below ground level and does not make sense for this scenario.</p>