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<h3>Activity (20 minutes)</h3>
<p>In the Numbers Screen activity, students will explore an arithmetic function using the terms “input” and “output.” As students build a function, the equation representation updates dynamically and includes a simplified representation. Students can observe the resulting outputs as in the previous activity, and as an explicit table of ordered pairs. <br><br>
In the Equations Screen activity, students build on their experiences by shifting from “input” and “output” to “x” and “y” and adding the representation of a graph for the function values. This permits students to wonder and explore how changing different parameters in the function will impact the outputs, the table of values, and the graph. </p>
<h3>Launch</h3>
<ol>
<li>Circulate as students work on part 3. If some groups seem behind, offer assistance if they have questions. </li>
<li>After 5 minutes or so, give students a time update that they should be on to #4. </li>
<li>When groups have finished, bring the class together for a whole-class discussion. Have four different students share their responses to #6.</li>
</ol>
<h3>Student Activity</h3>
<div class="os-raise-familysupport">
<p><a href="https://phet.colorado.edu/sims/html/function-builder/latest/function-builder_en.html" target="_blank">Access the interactive simulator</a> to begin this project.</p>
</div>
<p>For questions 1-3, use the following instructions. </p>
<blockquote>Use the simulation screen labeled “Numbers”. Note you can switch between simulation screens using the toolbar located below the function builder.</blockquote>
<ol class="os-raise-noindent">
<p><li>Build a function. Fill in the function builder and table. Access the input and output information by clicking the gray tabs above and below the function builder.</li></p>
<p class="os-raise-text-center"><img height=400 src="https://k12.openstax.org/contents/raise/resources/2effc7c16d2e7d366079e2c971e7ca01afb6d372"></p>
<p><strong>Answer:</strong><br>Answers will vary but here is a sample: \(Output = \frac{Input}{2} + 2\)</p>
<p><li> What is the output when the input is 10? Switch papers
with a teammate and check that you found the correct
output. 💬</li></p>
<p><strong>Answer:</strong><br>Response using sample listed above: \(7 = \frac{10}{2} + 2\)</p>
<p><li>
Describe how to find the output for your function if given
any input. <strong> Challenge yourself to describe in multiple ways. </strong>
</li>
</p>
<p><strong>Answer:</strong><br>
Response using sample equation above: Divide the input by two and then add two. Two more than half the input. Etc.
</p>
<p>For questions 4-8, use the following instructions.</p>
<blockquote>Use the simulation screen labeled “Equations”. Note you can switch between simulation screens using the toolbar located below the function builder.</blockquote>
<p>
<li>
Build a custom function. Fill in the function builder and representations. Access the table and graph information by clicking the gray tabs above and below the function builder.
<p class="os-raise-text-center"><br><img height=400 src="https://k12.openstax.org/contents/raise/resources/b252c80348b84c3a925f4ed0aeb179aec3bc68d7"></p>
</li>
</p>
<p><strong>Answer:</strong><br>Answers may vary but here is a sample: \(y = -1(x - 3)+ 1\) or \(y = -x + 4\)</p>
<p><li>What is y when x is 100?</li></p>
<p><strong>Answer:</strong><br>Response using sample equation in question 4: \(y = -100 + 4 = -96\)</p>
<p><li>
Manipulate your function in different ways. Describe the effects on the table, graph, and equation that each of your actions has.
</li></p>
<table class="os-raise-wideequaltable">
<thead>
<tr>
<th scope="col">Action</th>
<th scope="col">Effect on table</th>
<th scope="col">Effect on graph</th>
<th scope="col">Effect on equation</th>
</tr>
</thead>
<tbody>
<tr>
<td>Click the up arrow on the addition operation <img height="50" src="https://k12.openstax.org/contents/raise/resources/e0a877d1de82ef133f366ee3025d8c563e4e7f66"></td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td><br><br></td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td><br><br></td>
<td> </td>
<td> </td>
<td> </td>
</tr>
<tr>
<td><br><br></td>
<td> </td>
<td> </td>
<td> </td>
</tr>
</tbody>
</table>
<br>
<p><strong>Answer:</strong></p>
<table class="os-raise-wideequaltable">
<thead>
<tr>
<th scope="col">Action</th>
<th scope="col">Effect on table</th>
<th scope="col">Effect on graph</th>
<th scope="col">Effect on equation</th>
</tr>
</thead>
<tbody>
<tr>
<td>Click the up arrow on the addition operation <img height="50" src="https://k12.openstax.org/contents/raise/resources/e0a877d1de82ef133f366ee3025d8c563e4e7f66"></td>
<td>When I clicked up, the \(y\) values increased by 1 for each click. </td>
<td>The graph moved up the number of spaces that I had clicked. </td>
<td>The number at the end of the equation changed (constant term).</td>
</tr>
<tr>
<td>Click the down arrow on the addition operation <img height="50" src="https://k12.openstax.org/contents/raise/resources/e0a877d1de82ef133f366ee3025d8c563e4e7f66"></td>
<td>If I clicked down, it decreased 1 unit for each click. </td>
<td>The graph moved down the number of spaces that I had clicked.</td>
<td>The number at the end of the equation changed (constant term).</td>
</tr>
<tr>
<td>Click on the up or down arrows on the multiplication operation <img height="50" src="https://k12.openstax.org/contents/raise/resources/040d740d29548a03b2ea17a0572458038ef67646"></td>
<td>The numbers in the table got farther apart when I went up or down. At one point they were 2 units apart then after I clicked it, they were 3 units apart. Something weird happened when the multiplier was 0. At zero, everything just equaled 1 because the zero got rid of everything else.</td>
<td>This changed the slope or tilt of the line. When I clicked up, the direction changed to where the line tilted up as I moved to the right. When I clicked down, the line got steeper when it slanted down as I moved to the right. </td>
<td>Both numbers in the equation changed - the one at the end (constant term) and the one in front of the x (coefficient).</td>
</tr>
<tr>
<td>Click on the up or
down arrows on the
subtraction operation <img height="50" src="https://k12.openstax.org/contents/raise/resources/b2dc2fb014d98e80b6eab494c60bbe22921e552c"></td>
<td>When I click down, all the \(y\)-values move down 1 unit, too. </td>
<td>The graph slid down the \(y\)-axis by 1 unit each click. The slope or tilt of the line did not change. </td>
<td>In the equation, the value inside the parentheses changed with each click. In the simplified equation, this changed the last number (constant term). </td>
</tr>
</tbody>
</table>
<p><li>What does your graph look like? What other graphs can you make? </li></p>
<p><strong>Answer:</strong><br>I can make a line. I can only make lines with this function.</p>
<p><li>Briefly describe how different operations impact the graph of your function.</li></p>
<ol class="os-raise-noindent" type="a">
<li>Addition
<p><strong>Answer:</strong><br>The addition operation impacted the graph by moving it up or down. Adding more makes the graph move up.</p>
</li>
<li>Subtraction
<p><strong>Answer:</strong><br>The subtraction operation impacted the graph by moving it up or down. Subtracting more makes the graph move down.</p>
</li>
<li>Multiplication
<p><strong>Answer:</strong><br>The multiplication operation impacted the graph by changing how steep the line was and whether it increased or decreased (moving from left to right on the graph). Multiplying more makes the graph steeper up and down.</p>
</li>
<li>Division
<p><strong>Answer:</strong><br>The division operation impacted the graph by changing how steep the line was and whether it increased or decreased (moving to the right on the graph). Dividing more makes the graph flatter.</p>
</li>
</ol>
</ol>
<h4>Extension </h4>
<p>Play with the game! Challenge yourself to figure out the mystery functions. They get tricky when you have two or three operations. Does your rule match the simulation's rule? Can there be more than one answer? </p>
<div class="os-raise-familysupport">
<p><a href="https://phet.colorado.edu/sims/html/function-builder/latest/function-builder_en.html?screens=4" target="_blank">Access the mystery functions game </a>for an extra challenge</p>
</div>
<h3>Project Synthesis</h3>
<p>
Ask students to answer the following question on an exit ticket. Review after class to determine student misunderstandings and address in a subsequent day's warm up. <br><br>
1. What is a function?<br>
<strong>Answer:</strong> A function describes the relationship between an input and an output. Every input has a single output, no matter how many times you put it through the function. Different inputs could have the same output though. A function is a rule that takes an input, does something to it, and makes an output.<br>
<br>
2. What are the different ways a function can be represented? <br>
<strong>Answer:</strong> A function can be represented with words, table of values, list of points, a graph, or an equation.</p>