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<h4>Activity (10 minutes)</h4>
<p>In this activity, students write equations to represent quantities and relationships in two situations. In each
situation, students express the same relationship multiple times: initially using numbers and variables and later
using only variables. The progression helps students see that quantities can be known or unknown, and they can stay
the same or vary, but both kinds of quantities can be expressed with numbers or letters.</p>
<h4>Launch</h4>
<p>Let students know that they will now begin exploring a variety of different scenarios beyond Geometry. Each scenario
will have the theme of representing quantities and relationships with expressions and equations.<br></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 1 Stronger and Clearer Each Time: Writing, Listening, Conversing</p>
</div>
<div class="os-raise-extrasupport-body">
<p>Use this routine to help students
improve their written responses to the final question. Give students time to meet with one to two partners to share
and get feedback on their responses. Display feedback prompts that will help students strengthen their ideas and
clarify their language. For example, “Can you describe the quantities?,” “What operation was used?,” and “Can you
try to explain this using a different example?” Invite students to go back and revise their written explanation
based on the feedback from peers. This will help students understand situations in which quantities are related
through communicating their reasoning with a partner.
</p>
<p class="os-raise-text-italicize"> Design Principle(s): Optimize output (for explanation); Cultivate conversation</p>
<p class="os-raise-extrasupport-title">Provide support for students</p>
<p>
<a href="https://k12.openstax.org/contents/raise/resources/a5ae5bd09b27a5f53239a539c6009c19c92f7db7" target="_blank">Distribute graphic organizers</a>
to the students to assist them with participating in this routine.
</p>
</div>
</div>
<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">Engagement: Provide Access by Recruiting Interest</p>
</div>
<div class="os-raise-extrasupport-body">
<p>Leverage choice around perceived challenge. Invite
students to write equations for three to four of the situations they select. Chunking this task into more manageable
parts may also benefit students who benefit from additional processing time.
</p>
<p class="os-raise-text-italicize">Supports accessibility for: Organization; Attention; Social-emotional skills</p>
</div>
</div>
<br>
<h4>Student Activity</h4>
<p>Write an equation to represent each situation.</p>
<ol class="os-raise-noindent">
<li>Blueberries are $4.99 a pound. Diego buys \( b \) pounds of blueberries and pays $14.95.
</li>
<p><strong>Answer:</strong> \( 4.99 \cdot b = 14.95 \)</p>
<li>Blueberries are $4.99 a pound. Jada buys \( p \) pounds of blueberries and pays \( c \) dollars.
</li>
<p><strong>Answer:</strong> \( 4.99 \cdot p = c \)</p>
<li>Blueberries are \( d \) dollars a pound. Lin buys \( q \) pounds of blueberries and pays \( t \) dollars.
</li>
<p><strong>Answer:</strong> \( d \cdot q = t \)</p>
<li>Noah earned \( n \) dollars over the summer. Mai earned $275, which is $45 more than Noah did.
</li>
<p><strong>Answer:</strong> \( n + 45 = 275 \) or \( 275 - 45 = n \) (or equivalent)</p>
<li>Noah earned \( v \) dollars over the summer. Mai earned \( m \) dollars, which is 45 dollars more than Noah did.
</li>
<p><strong>Answer:</strong> \( v + 45 = m \) or \( m - 45 = v \) (or equivalent)</p>
<li>Noah earned \( w \) dollars over the summer. Mai earned \( x \) dollars, which is \( y \) dollars more than Noah
did.
</li>
<p><strong>Answer:</strong> \( w + y = x \) or \( x - y = w \) (or equivalent)</p>
<li>How are the equations you wrote for the blueberry purchases like the equations you wrote for Mai’s and
Noah’s summer earnings? How are they different?
</li>
<p><strong>Answer:</strong> </p>
<p>The two sets of equations are alike in that in each set:</p>
<ul>
<li>Each equation involves three quantities.</li>
<li>The first equation has two known quantities, the second has one known quantity, and the last one has no known
quantities.</li>
</ul><br>
<p>They are different in that: </p>
<ul>
<li>The three quantities in each set are different. In the first set, they are unit price, pounds of blueberries,
and total cost. In the second set, they are Noah’s earnings, Mai’s earnings, and the difference
between the two.</li>
<li>In the first set, the relationship involves multiplication. In the second, it involves addition (or
subtraction).</li>
</ul>
</ol>
<h4>Anticipated Misconceptions</h4>
<p>Students may translate “Mai earned \( m \) dollars, which is 45 more dollars than Noah did” as \( m + 45
= v \), not paying attention to where the plus sign should go. As with other problems throughout this unit, encourage
students to try using numbers in their equation to see if the equation really says what they want it to say.</p>
<h4>Activity Synthesis</h4>
<p>Focus the discussion on students’ observations about how the two sets of equations are alike. Then, ask how the
equations within each set are different. If students mention that some quantities are known or are fixed and others
are not, ask them to specify which ones are which.</p>
<p>Highlight the idea that sometimes we know how quantities are related, but the value of each quantity may be unknown
or may change. We often use letters to represent those unknown or changing quantities. </p>
<p>There might be times, however, when we use letters to represent quantities that are known or are constant. Doing so
may help us focus on the relationship rather than the numbers. Tell students we will look at examples of such
situations in upcoming activities.</p>
<h3>1.2.3: Self Check</h3>
<p class="os-raise-text-bold"><em>Following 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>Max and Jules are renting a car for their 3-day trip. The rental company charges $50 per day and $0.40 per mile
driven. Which is a possible equation for their cost of the rental car?</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">
Answers
</th>
<th scope="col">
<strong>Feedback
</strong>
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<p>\( C=50(3)+40m \)</p><br>
</td>
<td>
<p>Incorrect. Let’s try again a different way: The cost per mile should represent 40 cents, which is 0.40.
The correct answer is \( C=50(3)+0.40m \).</p>
</td>
</tr>
<tr>
<td>
<p>\( C=50m+40(3) \)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: This equation shows they traveled 3 miles. The number of
miles is unknown, \( m \), and the cost per mile should be 0.40. The cost per day is $50, and the number of
days is known, 3. The correct answer is \( C=50(3)+0.40m \).</p>
</td>
</tr>
<tr>
<td>
<p>\( C=50(3)+40(3)+0.40m \)</p>
</td>
<td>
<p>Incorrect. Let’s try again a different way: Look for the known and unknown values. The cost per day is
known, $50. The number of days of the trip is known, 3. The number of miles driven is unknown, \( m \). The
cost per mile is known, $0.40. The correct answer is \( C=50(3)+0.40m \).</p>
</td>
</tr>
<tr>
<td>
<p>\( C=50(3)+0.40m \)</p>
</td>
<td>
<p>That’s correct! Check yourself: They will pay $50 times 3 days plus 0.40 multiplied by the number of
miles, \( C=50(3)+0.40m \).</p><br>
</td>
</tr>
</tbody>
</table>
<br>
<h3>1.2.3: 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>Video: Writing an Equation to Represent a Real-World Problem</h4>
<p>Watch the following video to see how this is done. You’ll need to answer the question(s) that appear during the
video to proceed to the next portion of the lesson.<br></p>
<br>
<p>Here are the steps to writing an equation to represent a real-world scenario:</p>
<div class="os-raise-graybox">
<p><strong>Step 1</strong> - Read the problem. <br>
<strong>Step 2</strong> - Identify the variables and known values. If needed, sketch a picture of the scenario.<br>
<strong>Step 3</strong> - Write a sentence using the relationship among the values. <br>
<strong>Step 4</strong> - Translate the sentence into an equation.
</p>
</div>
<br>
<h4>Try It: Writing an Equation to Represent a Real-World Problem</h4>
<p>Translate the following scenario into an equation:</p>
<p>A married couple together earns $110,000 a year. The wife earns $16,000 less than twice what her husband earns. What
does the husband earn?</p>
<p>Here is how to turn this scenario into an equation using the four-step process:</p>
<p><strong>Step 1</strong>. After reading the problem, what are some of the known variables identified in the description?</p>
<p><strong>Answer:</strong> <br>
<li>Together the husband and wife earn $110,000 a year.</li>
<li>The wife earns $16,000 less than twice what her husband earns.</li>
</p>
<p><strong>Step 2</strong>. Identify a variable and describe what it represents for this scenario. </p>
<p><strong>Answer:</strong> <br>
<p>
<li>\(2h - 16,000\) = the amount the wife earns.</li>
<li>Let \(h\) = the amount the husband earns.</li>
</p>
<p><strong>Step 3</strong>. What sentence can be used to describe the relationship among the values?</p>
<p><strong>Answer:</strong> <br>
<p>Together the husband and wife earn $110,000.</p>
<p><strong>Step 4</strong>. What equation can be used to represent the scenario?</p>
<p><strong>Answer:</strong> <br>
<p>\(h + 2h - 16,000 = $110,000\)</p>
<h4>Video: Writing an Equation to Represent a Real-World Problem</h4>
<p>Watch the following video to learn more about how to write an equation to represent a real-world problem.</p>
<br>
<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/de016c385bf8f86f2d1466cad8f4028db53393a4">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/fc73823f0356a44b5dec37ffa33bfa4b81a5199f" srclang>
https://k12.openstax.org/contents/raise/resources/de016c385bf8f86f2d1466cad8f4028db53393a4
</video></div>
</div>
</p>
</p>
</p>