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<h4>Activity (15 minutes)</h4>
<div>
<p>In this
activity, students write equations in one variable to represent the
constraints in a situation. They then reason about the solutions and
interpret the solutions in context. </p>
<p>To solve the equation, some students may try different values of \( h \)</span> until they find one that
gives a true equation. Others may perform the same operations to each side of the equation to isolate \( h
\). Identify students who use different strategies and ask them to share later.</p>
<h4>Launch</h4>
<p>Arrange students
in groups of 2 and encourage students to use Desmos to model the
scenario in question 1 as an equation. Let students in different groups
compare their equations, or discuss as a whole class before moving on. </p>
<p>For
questions 2–4, give students a few minutes of quiet work time and then
time to discuss their responses. Ask them to share with their partner
their explanations for why 4 and 7 are or are not solutions.</p>
<p>If
students are unsure how to interpret “take-home earnings,” clarify that
it means the amount Jada takes home after paying job-related expenses
(in this case, the bus fare).</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 5 Co-Craft Questions: Speaking, Reading</p>
</div>
<div class="os-raise-extrasupport-body">
<p>Use
this routine to help students interpret the language of writing equations and to increase awareness of language used to talk about
representing situations with equations. Display only the task statement
that describes the context, without revealing the questions that follow.
Invite students to discuss possible mathematical questions that could
be asked about the situation. Listen for and amplify any questions
involving equations that connect the quantities in this situation.</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=P_NQJdG92iA;&rel=0" target="_blank">View the instructional video</a>
and
<a href="https://k12.openstax.org/contents/raise/resources/4e340aa86ff7eda8a1076cbe2ff84123e50e8012" 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/77a07fd176bcc1a05392967ab523ab95586bfc98" 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">Action and Expression: Internalize Executive Functions</p>
</div>
<div class="os-raise-extrasupport-body">
<p>
Chunk this task into more manageable parts for students who benefit
from support with organizational skills in problem solving. Check in
with students after the first 2–3 minutes of work time. Invite 1–2
students to share how they determined an equation that represents Jada’s
take-home earnings. Record their thinking on a display and keep the
work visible as students continue to work. </p>
<p class="os-raise-text-italicize">Supports accessibility for: Organization; Attention</p>
</div>
</div>
<br>
<h4>Student Activity</h4>
<p>Jada has time on the weekends to earn some money. A local bookstore is looking for someone to help sort books and
will pay $12.20 an hour. To get to and from the bookstore on a work day, however, Jada would have to
spend $7.15 on bus fare. </p>
<ol class="os-raise-noindent">
<li>Write an equation that represents Jada’s take-home earnings in dollars, \( E \), if she
works at the bookstore for \( h \) hours in one day.</li>
</ol>
<p><strong>Answer:</strong> \( E= 12.20h - 7.15 \)</p>
<ol class="os-raise-noindent" start="2">
<li>One day, Jada takes home $90.45 after working \( h \) hours and after paying the bus
fare. Write an equation to represent this situation.
</li>
</ol>
<p><strong>Answer:</strong> \( 90.45 = 12.20h - 7.15 \) (or equivalent)</p>
<ol class="os-raise-noindent" start="3">
<li>Is 4 a solution to the last equation you wrote?
</li>
</ol>
<ul>
<li>If so, be prepared to explain how you know it is a solution.
</li>
<li>If not, be prepared to explain why it is not a solution. Then, find the solution.
</li>
</ul>
<p><strong>Answer:</strong> Your answer may vary, but here is a sample. </p>
<p>Substituting 4 into the equation gives \(90.45=12.20(4)-7.15\) or \(90.45=41.65\), which is not a true statement.
The solution is 8. Substituting 8 into the equation gives \(90.45=90.45\), which is true.</p>
<ol class="os-raise-noindent" start="4">
<li>Is 7 a solution to the last equation you wrote?
</li>
</ol>
<ul>
<li>If so, be prepared to explain how you know it is a solution.
</li>
<li>If not, be prepared to explain why it is not a solution. Then, find the solution.
</li>
</ul>
<p><strong>Answer:</strong> Your answer may vary, but here is a sample. </p>
<p>Substituting 7 into the equation gives \(90.45=78.25\), which is also false. The solution is 8. Substituting 8
into the equation gives \(90.45=90.45\), which is true.
</p>
<ol class="os-raise-noindent" start="5">
<li>In this situation, what does the solution to the equation tell us?
</li>
</ol>
<p><strong>Answer:</strong> It tells us the number of hours that Jada worked that allowed her to take home $90.45
after paying for her bus fare.</p>
<h4>Student Facing Extension</h4>
<h4>Are you ready for more?</h4>
<p>Jada has a second option to earn money: she could help some neighbors with errands and computer work for
$11 an hour. After reconsidering her schedule, Jada realizes that she has about 9 hours available to
work one day of the weekend.</p>
<p>Which option should she choose—sorting books at the bookstore or helping her neighbors? Be prepared to show
your reasoning.</p>
<h4>Extension Student Response</h4>
<p>Students could argue either way, depending on their assumptions. </p>
Compare your answer:
<ul>
<li>Jada should work at the bookstore because she would earn more there. Her pay would be $109.80,
and after subtracting $7.15 for the bus fare, she would still earn $102.65. She would
earn $99 from the other option.</li>
<li>Jada should
help her neighbors. Working 9 hours at the bookstore would mean a few
more dollars than working 9 hours helping her neighbors, but it would
also mean losing some personal time because of the travel involved.</li>
</ul>
<h4>Anticipated Misconceptions</h4>
<p>If students struggle to write equations in the first question, ask them how they might find out Jada’s
earnings if she works 1 hour, 2 hours, 5 hours,
and so on. Then, ask them to generalize the computation process for \( h \) hours.</p>
<h4>Activity Synthesis</h4>
<p>Ask a student to share the equation that represents Jada earning $90.45. Make sure students
understand why \( 90.45 = 12.20h - 7.15 \) describes that constraint. </p>
<p>Next, invite
students to share how they knew if 4 and 7 are or are not solutions to
the equation. Highlight that substituting those values into the equation
and evaluating them lead to false equations. </p>
<p>Then, select
students using different strategies to share how they found the
solution. Some students might notice that the solution must be greater
than 7 (because when \( h=7 \), the expression \( 12.20h-7.15 \) has a value less than
90.45) and start by checking if \( h=8 \) is a solution. If no students mention this, ask them about
it. </p>
<p>Make sure
students understand what the solution means in context. Emphasize that 8
is the number of hours that meets all the constraints in the situation.
Jada gets paid $12.20 an hour, pays $7.15 in bus fare, and takes home
$90.45. For all of these to be true, she must have worked 8 hours.</p>
<h3>1.4.2: 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>Jada found a third job option and was offered a position working at a camp for $10.50 per hour. The total bus
fare to and from the camp is $8.40. Write an equation that represents how much she will make in a day, \( d \),
after working h hours.</p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>\( d = 10.50h - 8.40 \)</td>
<td>That’s correct! Check yourself: Jada will make $10.50 per hour (\(10.50h \)) but must subtract the
bus fare for the day of 8.40. </td>
</tr>
<tr>
<td>\( d = 10.50h + 8.40 \)</td>
<td>Incorrect Let’s try again a different way. The bus fare should be subtracted from what Jada is
making per hour. The answer is \( d = 10.50h - 8.40 \).</td>
</tr>
<tr>
<td>\( d= \frac{10.50h}{8.40} \)</td>
<td>Incorrect. Let’s try again a different way: The bus fare should be subtracted from what Jada is
making per hour. The answer is \( d = 10.50h - 8.40 \).</td>
</tr>
<tr>
<td>\( d = 8.40h + 10.50 \)</td>
<td>Incorrect. Let’s try again a different way: The amount Jada makes per hour is 10.50, so that should
be multiplied by the number of hours, \(h\). The answer is \( d = 10.50h - 8.40 \).</td>
</tr>
</tbody>
</table>
<br>
<h3>1.4.2 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>Verify a Solution of an Equation</h4>
<p>Solving an <span data-schema-version="1.0" data-store="glossary-tooltip">equation</span> is like discovering the
answer to a puzzle. The purpose in solving an equation is to find the value or values of the <span
data-schema-version="1.0" data-store="glossary-tooltip">variable</span>
that make each side of the equation the same so that we end up with a
true statement. Any value of the variable that makes the equation true
is called a <span data-schema-version="1.0" data-store="glossary-tooltip">solution of an equation</span>. It is
the answer to the puzzle!</p>
<p>How to determine whether a number is a solution to an equation:</p>
<p><strong>STEP 1:</strong> Substitute the number in for the variable in the equation.</p>
<p><strong>Step 2 </strong> -Simplify the expressions on both sides of the equation</p>
<p><strong>Step 3 </strong> -Determine whether the resulting equation is true (the left side is equal to the right
side).<br>
If it is true, the number is a solution.<br>
If it is not true, the number is not a solution.</p>
<p>For example, the equation \( 2l+2w=72 \) represents the relationship between the length, \( l \), and the width,
\( w \), of a rectangle that has a perimeter of 72 units. If we know that the length is 15 units, what is the
value of the width?
</p>
<p><strong>STEP 1:</strong> Substitute the number in for the variable in the equation.</p>
<p>\( 2(15)+2w=72 \)</p>
<p>This is an equation in one variable because \(w\) is the only quantity that we don't know. To solve this equation
means to find a value of \(w\) that makes the equation true.</p>
<p><strong>Step 2 </strong> -Simplify the expressions on both sides of the equation<br>
In this case, 21 is the solution because substituting 21 for \( w \) in the equation results in a true statement.
</p>
<p>\( 2(15)+2w=72 \)</p>
<p>\( 2(15)+2(21)=72 \)</p>
<p>\( 30+42=72 \)</p>
<p><strong>Step 3 </strong> -Determine whether the resulting equation is true (the left side is equal to the right
side).<br>
If it is true, the number is a solution.<br>
If it is not true, the number is not a solution.</p>
<p>\( 72=72 \)</p>
<h4>Try It: Verify a Solution of an Equation</h4>
<p>The equation \(2l + 2w = 48\) represents the perimeter of a rectangle.
</p>
<ol class="os-raise-noindent">
<li>If the width of the rectangle is 10, what is the new equation?
</li>
</ol>
<p><strong>Answer:</strong> \(2l + 2(10) = 48\) or \(2l + 20 = 48\)</p>
<ol class="os-raise-noindent" start="2">
<li>What is the length of the rectangle that has a width of 10?
</li>
</ol>
<p><strong>Answer:</strong> 14 is the length of the rectangle that makes the statement \(2l + 20 = 48\) true.</p>
<ol class="os-raise-noindent" start="3">
<li>Verify the solution.
</li>
</ol>
<p><strong>Answer:</strong> To verify the solution, substitute \(w = 10\) and \(l = 14\) into \(2l + 2w = 48\).<br>
\(2(14) + 2(10) = 48\)<br>
\(28 + 20 = 48\)<br>
\(48 = 48\) is a true statement, so the solution works!
</p>