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<h4>Activity (20 minutes)</h4>
<p>In the previous activity, students analyzed and
interpreted points on a graph relative to an equation and a situation.
In this activity, they write a linear equation to model a situation, use
graphing technology to graph the equation, and then use the graph to
solve problems. Each given situation involves an initial value and a
constant rate of change.</p>
<p>Before students begin the activity, introduce them to
the graphing technology available in the classroom. Offer a quick
tutorial on how to graph equations, adjust the graphing window, and plot
points. This tutorial could happen independently of the activity as
long as it precedes the activity.</p>
<h4>Launch</h4>
<p>Give all students access to graphing technology. Tell
students that in this course they will frequently use technology to
create a graph that represents an equation and use the graph to solve
problems.</p>
<p>Demonstrate how to use the technology available in your
classroom to create and view graphs of equations. Explain how to enter
equations, adjust the graphing window, and plot a point. If using
Desmos, please see the digital version of this activity for suggested
instructions.</p>
<p>Arrange students in groups of 2–4. Assign one
situation to each group. Ask students to answer the first few questions,
including writing an equation, and then graph the equation and answer
the last question.</p>
<br>
<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 7 Compare and Connect: Representing, Conversing</p>
</div>
<div class="os-raise-extrasupport-body">
<p>As
students share their responses with the class, call attention to the
different ways the quantities are represented graphically and within the
context of each situation. Take a close look at both graphs to
distinguish what the points represent in each situation. Wherever
possible, amplify student words and actions that describe the
connections between a specific feature of one mathematical
representation and a specific feature of another representation.</p>
<p class="os-raise-text-italicize">Design Principle(s): Maximize
meta-awareness; Support sense-making</p>
<p class="os-raise-extrasupport-title">Learn more about this routine</p>
<p> <a href="https://www.youtube.com/watch?v=PF8fRA107OA" target="_blank">View the instructional video</a> and <a href="https://k12.openstax.org/contents/raise/resources/94a1159e7b81493c647515711f325771076d99b8" target="_blank">follow along with the materials</a> to assist you with learning this routine. </p>
<p class="os-raise-extrasupport-title">Provide support for students</p>
<p> <a href="https://k12.openstax.org/contents/raise/resources/0b8a1a4ac3425e84a1d5452b3a5dffa38deb6b13" target="_blank">Distribute graphic organizers</a> to the students to assist them with participating in this routine. </p>
</div>
</div>
<br>
<br>
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title">Support for Students with Disabilities</p>
<p class="os-raise-extrasupport-name">Action and Expression: Internalize Executive Functions</p>
</div>
<div class="os-raise-extrasupport-body">
<p> Representation: Access for Perception.</em> Provide students with a physical copy of written directions for using
graphing technology and read them aloud. Include step-by-step directions
for how to enter equations, adjust the graphing window, and plot a
point. </p>
<p class="os-raise-text-italicize">Supports accessibility for: Language; Memory</p>
</div>
</div>
<br>
<h4>Student Activity</h4>
<p>Use the following scenario for questions 1-5:
A student has a savings account with $475 in it. She deposits $125 of her paycheck into the account every week. Her
goal is to save $7000 for college. </p>
<ol class="os-raise-noindent">
<li>How much will be in the account after 3 weeks?</li>
</ol>
<p><strong>Answer:</strong> $850</p>
<ol class="os-raise-noindent" start="2">
<li>How many weeks will it take until she has $1350? </li>
</ol>
<p><strong>Answer:</strong> 7 weeks</p>
<ol class="os-raise-noindent" start="3">
<li>Write an equation that represents the relationship between the dollar amount in her account and the number of
weeks of saving.</li>
</ol>
<p><strong>Answer:</strong> \(a=475+125w\), where \(a\) is the dollar amount in the account and \(w\) is the number of
weeks of savings.</p>
<ol class="os-raise-noindent" start="4">
<li>Graph your equation using graphing technology. Mark the points on the graph that represent the amount after 3
weeks and the week she has $1350. Write down the coordinates. </li>
</ol>
<p><strong>Answer:</strong></p>
<img alt="Graph of a linear function. amount in account (dollars), number of weeks." height="194"
src="https://k12.openstax.org/contents/raise/resources/fb79cc0520f9d5754497ccaff23da6041fa25ab4" width="251"> <br>
<br>
<ol class="os-raise-noindent" start="5">
<li>What is the \(x\)-intercept? </li>
</ol>
<p><strong>Answer:</strong> The correct answer is \((475, 0)\).</p>
<ol class="os-raise-noindent" start="6">
<li>What other information does the \(x\)-intercept identify in the function?</li>
</ol>
<p><strong>Answer:</strong> The \(x\)-intercept can identify the zero of the function since it represents where the function equals zero \((y = 0)\).</p>
<ol class="os-raise-noindent" start="7">
<li>What is the the \(y\)-intercept?</li>
</ol>
<p><strong>Answer:</strong> The correct answer is \((0, -3.8)\).</p>
<ol class="os-raise-noindent" start="8">
<li>What is the slope?</li>
</ol>
<p><strong>Answer:</strong> The correct answer is \(-\frac{1}{125}\).</p>
<br>
<h4>Student Facing Extension</h4>
<h4>Are you ready for more?</h4>
<br>
<blockquote>
<p>Use the following information to answer questions 1-5. A 450-gallon tank full of water is draining at a rate of 20 gallons per minute.</p>
</blockquote>
<ol class="os-raise-noindent">
<li>Write an equation that represents the relationship between the gallons of water in the tank and hours the tank has
been draining.</li>
</ol>
<p><strong>Answer:</strong> \(a=450-1,200h\) where \(h\) is time in hours.</p>
<ol class="os-raise-noindent" start="2">
<li>Write an equation that represents the relationship between the gallons of water in the tank and seconds the tank
has been draining.</li>
</ol>
<p><strong>Answer:</strong> \(a=450-\frac{s}{3}\) where \(s\) is time in seconds.</p>
<ol class="os-raise-noindent" start="3">
<li>Graph each of your new equations.</li>
</ol>
<p><strong>Answer:</strong> </p>
<img alt="Linear function." height="195"
src="https://k12.openstax.org/contents/raise/resources/54873f473fb892340f99aa047fc492c3531f2c16" width="245"> <br>
<br>
<ol class="os-raise-noindent" start="4">
<li>In what way are all of the graphs the same? In what way are they all different? Write down your answer. </li>
</ol>
<p><strong>Answer:</strong> Sample responses:</p>
<ul>
<li>They all have the same \(y\)-intercept. </li>
<li>They all decrease. </li>
<li>The slopes are different. </li>
</ul>
<ol class="os-raise-noindent" start="5">
<li>How would these graphs change if we used quarts of water instead of gallons? What would stay the same?</li>
</ol>
<p><strong>Answer:</strong> Sample responses:<br>
The graphs would still be lines with negative slopes that are different. They would have the same horizontal
intercepts. The vertical intercepts would be different, but they would represent the same amount of water.</p>
<h4>Activity Synthesis</h4>
<p>Select a group that analyzed the first situation and a
group that analyzed the second situation to share their responses.
Display their graphs for all to see.</p>
<p>Focus the discussion on two things: what the points on the graph mean and how the graph could be used to answer
questions about the quantities in each situation. Discuss questions such
as:</p>
<ul>
<li>“How did
you find the answers to the first two questions?” (By calculation, for
example, computing \( 475 + 125(3) \), or finding \( 1,\!350 - 475 \),
and then dividing by 125.)</li>
<li>“How did
you find the answer to the last question?” (By calculation, for
instance, finding \( 7,\!000 - 475 \), and then dividing by 125. Or,
alternatively, by using the graph.)</li>
</ul>
<p>Highlight how
the graph of the equations could be
used to answer the questions. If not already mentioned by students,
discuss how the graph of \( y=475-125x \) can be used to find the
answers to all the questions about the student’s savings account, and
the graph of \( y=450-20x \) can help us with the questions about
draining the water tank.</p>
<p>Keep the graphs of the two equations displayed for the lesson summary.</p>
<h3>1.5.4: 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>Which of the following points is a solution to the graph below?</p>
<img alt="Graph showing x=2, y=3(2)+1=7."
class="img-fluid atto_image_button_text-bottom" height="267"
src="https://k12.openstax.org/contents/raise/resources/9f553b521fcf4e3642d34b964b3189782107cd93" width="300">
</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>\( (2, 7) \)</td>
<td>That’s correct! Check yourself: When \( x = 2 \), \( y = 3(2) + 1 = 7 \).</td>
</tr>
<tr>
<td>\( (0, 3) \)</td>
<td>Incorrect. When \(x =0\) and \(y= 3\), the point is not on the line, so it is not a solution. Let’s try
again a different way: </span>Find a point that is on the line. The answer is \((2, 7)\).</td>
</tr>
<tr>
<td>\( (7, 2) \)</td>
<td>Incorrect. Make sure to always write the \(x\)-coordinate first in the ordered pair. Let’s try again a different way:</span> Find a point that is on the line. The answer is \((2, 7)\).</td>
</tr>
<tr>
<td>\( (-2, -6) \)</td>
<td>Incorrect. When \(x = -2\) and \(y =-6\), the point is not on the line, so it is not a solution. Let’s
try again a different way: Find a point that is on the line. </span>The answer is \((2, 7)\). </td>
</tr>
</tbody>
</table>
<br>
<h3>1.5.4: 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>Writing Equations Using Graphs in Situations </h4>
<p>An equation that contains two unknown quantities or two quantities that vary is called an equation in two variables. A solution to such an equation is a pair of numbers that makes the equation true. </p>
<p>Suppose Tyler spends $40 on T-shirts and socks. A T-shirt costs $10 and a pair of socks costs $2.50. If \(t\) represents the number of T-shirts and \(p\) represents the number of pairs of socks that Tyler buys, what is an equation that represents the equation?</p>
<p><strong>Example 1</strong></p>
<p><strong>Step 1</strong> - Create a two-variable equation. <br>
The cost is $10 per t-shirt \((10t)\) plus $2.50 per pair of socks \((2.50p)\) which equals $40.<br>
\(10t +2.50p = 40\)</p>
<p>Now, we have to graph the equation. We will let \(t=x\) and \(p=y\).</p>
<p><strong>Step 2</strong> - Find the \(x\)-intercept.<br>
To find the \(x\)-intercept, let \(p=0\).</p>
<p>\(10t +2.50p = 40\)<br>
\(10t +2.50(0) = 40\)<br>
\(10t +0 = 40\)<br>
\(t = 4\)</p>
<p>\((4, 0)\) is the \(x\)-intercept.<br>
The \(x\)-intercept is also called a solution or zero. </p>
<p>In this scenario, 4 represents the number of T-shirts Tyler can buy if he doesn't purchase any socks with $40.</p>
<p><strong>Step 3</strong> - Find the \(y\)-intercept.<br>
To find the \(y\)-intercept, let \(t=0\).
<p>\(10t +2.50p = 40\)<br>
\(10(0) +2.50p = 40\)<br>
\((0) +2.50p = 40\)<br>
\(2.50p = 16\)</p>
<p>\((0, 16)\) is the \(y\)-intercept.</p>
<p>In this scenario, 16 represents the number of socks Tyler can buy if he doesn't purchase any T-shirts with $40.</p>
<p><strong>Step 4</strong> - Graph the line by connecting the intercepts. </p>
<p>Let’s look at the graph of this equation: </p>
<img height="355" src="https://k12.openstax.org/contents/raise/resources/727f67f57bbc9d66c10ba9b0c622b45e9c0e4f66"
width="499"> <br>
<br>
<p>Let’s reflect about the graph and what it means. </p>
<p><strong>Example 2</strong></p>
<p>What is the slope of the graph? </p>
<p>Solution </p>
<p>\(m = -4\)</p>
<p><strong>Example 3</strong>
<p>What does the point \((4, 6)\) mean on this graph?</p>
<p><strong>Answer:</strong> If Tyler bought 4 T-shirts and 6 pairs of socks, it would cost more than $40.</p>
<br>
<h4>Try It: Writing Equations Using Graphs in Situations</h4>
<p>Use Desmos or a graphing calculator to create a graph for \( 40x +20y = 180 \).</p>
<ol class="os-raise-noindent">
<li>If \(x\) represents the number of pairs of shoes and \(y\) represents the number of pairs of jeans, what is one
combination that is a solution?</li>
</ol>
<p><strong>Answer:</strong> Your answer may vary, but here are some examples: \((2, 5)\), \((1, 7)\), \((3, 3)\), \((4,
1)\)</p>
<ol class="os-raise-noindent" start="2">
<li>What does the combination you identified mean on the graph? </li>
</ol>
<p><strong>Answer:</strong> Your answer may vary, but here is an example: For \((2, 5)\), that means buying 2 pairs of
shoes and 5 pairs of jeans will cost $180. </p>
<p>This is how to find a solution and determine its meaning:</p>
<ol class="os-raise-noindent">
<li>Find a point on the graph of the line. </li>
<li>The point \((2, 5)\) is on the graph. Since 2 is the coordinate of the \(x\)-coordinate, it represents 2 pairs of
shoes. </li>
<li>Since 5 is the \(y\)-coordinate, it represents 5 pairs of jeans. </li>
<li>The line represents all combinations that equal $180. </li>
</ol>